An Introduction to Motivic Feynman Integrals

An Introduction to Motivic Feynman Integrals††This paper is a contribution to
the Special Issue on Representation Theory and Integrable Systems in honor of
Vitaly Tarasov on the 60th birthday and Alexander Varchenko on the 70th
birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Tarasov-Varchenko.html

Claudia RELLA

C. Rella

Section de Mathématiques, Université de Genève, Genève, CH-1211 Switzerland
<EMAIL_ADDRESS>
Received August 30, 2020, in final form March 03, 2021; Published online March
26, 2021

This article gives a short step-by-step introduction to the representation of
parametric Feynman integrals in scalar perturbative quantum field theory as
periods of motives. The application of motivic Galois theory to the algebro-
geometric and categorical structures underlying Feynman graphs is reviewed up
to the current state of research. The example of primitive log-divergent
Feynman graphs in scalar massless $\phi^{4}$ quantum field theory is analysed
in detail.

scattering amplitudes; Feynman diagrams; multiple zeta values; Hodge
structures; periods of motives; Galois theory; Tannakian categories

81-02; 14-02; 81Q30; 81T18; 81T15; 14C15; 14C30; 14F40; 11R32

###### Contents

  1. 1 Introduction
  2. 2 Scalar Feynman graphs
    1. 2.1 Perturbative quantum field theory
    2. 2.2 Feynman parametrisation
    3. 2.3 Graph polynomials
    4. 2.4 Primitive log-divergent graphs
    5. 2.5 Multiple zeta values
  3. 3 Cohomology theory of algebraic varieties
    1. 3.1 Singular homology
      1. 3.1.1 Some properties of homology
      2. 3.1.2 Relative singular homology
    2. 3.2 De Rham cohomology
      1. 3.2.1 Algebraic de Rham cohomology
      2. 3.2.2 Relative de Rham cohomology
    3. 3.3 Comparison isomorphism
    4. 3.4 Pure Hodge structures
    5. 3.5 Mixed Hodge structures
  4. 4 Periods of motives
    1. 4.1 Periods
    2. 4.2 Algebra of motivic periods
    3. 4.3 Period map
    4. 4.4 Examples
      1. 4.4.1 Motivic $2\pi{\rm i}$
      2. 4.4.2 Motivic $\log(z)$
      3. 4.4.3 Elementary relations
  5. 5 Feynman motives
    1. 5.1 Singularities and the blow up
    2. 5.2 Motivic multiple zeta values
    3. 5.3 Motivic Feynman integrals
    4. 5.4 Tannakian formalism
    5. 5.5 Motivic Galois theory
    6. 5.6 Coaction conjecture
    7. 5.7 Weights and the small graph principle
  6. 6 Conclusions
    1. Acknowledgements

##  1 Introduction

In Section 2, we describe the graph-theoretic framework for the investigation
of the algebraic information contained in the topology of scalar Feynman
diagrams. Perturbative quantum field theories possess an inherent algebraic
structure, which underlies the combinatorics of recursion governing
renormalisation theory, and are thus deeply connected to the theory of graphs.

In Section 3, we broadly review preliminary notions in algebraic geometry and
algebraic topology. An algebraic variety over $\mathbb{Q}$ gives rise to two
distinct rational structures via algebraic de Rham cohomology and Betti
cohomology, which are compatible with each other only after complexification.
The coexistence of these two cohomologies and their peculiar compatibility are
linked to a specific class of complex numbers, known as periods. The
cohomology of an algebraic variety is equipped with two filtrations, and the
mixed Hodge structure arising from their interaction constitutes the bridge
between the theory of periods and the theory of motives.

In Section 4, we introduce the set of periods, lying between
$\bar{\mathbb{Q}}$ and $\mathbb{C}$, among which are the numbers that come
from evaluating parametric Feynman integrals, and we briefly review their
remarkable properties. Suitable cohomological structures are exploited to
derive non-trivial information about these numbers.

In Section 5, we describe how Feynman integrals are promoted to periods of
motives. Technical issues arising from the presence of singularities are
tackled by blow up. We adopt the category-theoretic Tannakian formalism where
motivic periods, and motivic Feynman integrals in particular, reveal their
most intriguing properties. We present an overview of the current progress of
research towards the general understanding of the structure of scattering
amplitudes via the theory of motivic periods, giving particular attention to
recent results in massless scalar $\phi^{4}$ quantum field theory.

##  2 Scalar Feynman graphs

###  2.1 Perturbative quantum field theory

A quantum field theory encodes in its Lagrangian every admissible interaction
among particles, but it does it in a way that makes decoding this information
a difficult task. The probability amplitude associated to the interaction
process between given initial and final states, called its Feynman amplitude,
is determined by the set of kinematic and interaction terms in the Lagrangian.
However, individual Lagrangian terms correspond to propagators and interaction
vertices which can be linked together in infinitely many distinct ways to
connect the same pair of initial and final states. Each of these admissible
realisations of the same interaction process has to be accounted for in an
infinite sum of contributions to the probability amplitude. We associate to
each of these possibilities a graphical representation, called its Feynman
diagram, whose individual contribution to the probability amplitude is
explicitly written in the form of a Feynman integral by applying the formal
correspondence between Lagrangian terms and graphical components, which is
established by convention through the set of Feynman rules of the theory. It
is only the sum of the contributing Feynman integrals to a given process that
has a physical meaning and not the individual integrals, which are themselves
interrelated by the gauge symmetry of the Lagrangian.

In perturbative quantum field theory, the sum of individual Feynman integrals
is a perturbative expansion in some small parameter of the theory, typically a
suitable coupling constant. Thus, the Feynman amplitude can be expanded in a
formal power series, which has been shown to be divergent111Serone et al [61]
characterised the conditions under which some class of asymptotic perturbative
series are Borel resummable, leading to exact results without introducing non-
perturbative effects in the form of trans-series. by Dyson [32]. The
divergence does not, however, undermine the accuracy of predictions that can
be made with the theory. Indeed, although a Feynman amplitude receives
contributions to any order in perturbation theory, practical calculations are
made by truncating the sum at a certain order and directly evaluating only the
remaining finitely many terms. Moreover, the explicit calculation of a Feynman
amplitude only includes those diagrams which are one-particle irreducible, or
1PI, that is, diagrams which cannot be divided in two by cutting through a
single propagator. See Fig. 1. The contribution from a non-1PI diagram at some
given order can be expressed as a combination of lower-order 1PI
contributions, which have already been accounted for in the formal series.

(a) One-particle irreducible

(b) One-particle reducible

Figure 1: Examples of 1PI and non-1PI diagrams.

The leading order terms in the perturbative expansion of a Feynman amplitude
are called tree-level contributions. Higher order diagrams are obtained from
tree-level diagrams by adding internal loops. Each independent loop in a
diagram is associated to an unconstrained momentum and integrals over
unconstrained loop momenta are the origin of singularities in Feynman
integrals. We distinguish two classes of singularities. The ultraviolet (UV)
divergences arise in the limit of infinite loop momentum, a regime that is far
beyond the energy scale that we have currently experimental access to and
where we expect new physical phenomena to become relevant and corresponding
new terms to enter the Lagrangian. Sensitivity to the high loop momentum
region is treated by means of renormalisation theory. For a renormalizable
theory, a suitable adjustment of the Lagrangian parameters allows to
systematically re-express the predictions of the theory in terms of
renormalized physical couplings, so that they decouple from UV physics. Thus,
the theory gives a finite and well-defined relation between physical
observables. The infrared (IR) divergences only arise in theories with
massless particles as they originate in the limit of infinitesimal loop
momentum. They cannot be removed by renormalisation and introduce numerous
subtleties in the evaluation of Feynman integrals which we are not dealing
with in the present text. For a detailed and comprehensive presentation of
perturbative quantum field theory we refer to Zee [73] and Srednicki [65].

Evaluating Feynman integrals over loop momenta has been of fundamental concern
since the early days of perturbative quantum field theory. Since the first
insights into the problem of UV divergences in a quantum field theory
presented by Dyson [31, 32], Salam [55, 56] and Weinberg [72] in the 1950s and
60s, our understanding has vastly improved. In 2004, Smirnov [64] summarised
more than fifty years of advancements in the field, providing an overview of
the most powerful, successful and well-established methods available at the
time for evaluating Feynman integrals in a systematic way, showing how the
problem of evaluation had become more and more critical. What could be easily
evaluated had already been evaluated years ago. Nowadays, new approaches,
based on the symmetry properties of the loop integrands222We refer to Elvang
and Huang [33] for a review of the subject, including unitarity methods, BCFW
recursion relations, and the methods of leading singularities and maximal
cuts. and the complementary perspective of differential equations,333Henn [39]
gives an overview of the method of differential equations, using tools such as
Chen iterated integrals, multiple polylogarithms, and the Drinfeld associator.
are available and vastly studied. Despite progress, the mathematical
understanding and the computation of Feynman integrals are still far from
being complete. Overlapping divergences can be treated iteratively, thus
revealing in the first place the recursive nature of renormalisation theory.
However, this combinatorics of subdivergences is only the first hint to a more
fundamental algebraic structure inherent in all renormalizable quantum field
theories and deeply connected to the theory of graphs.444A first discussion
about the appearance of transcendental numbers in Feynman integrals and its
relation to the topology of Feynman graphs is presented by Kreimer [48] in the
framework of knot theory and link diagrams. A recent review on the theory of
numbers and single-valued functions on the complex plane which arise in
quantum field theory is presented by Schnetz [60] in the modern context of the
theory of motivic periods.

###  2.2 Feynman parametrisation

We consider a scalar quantum field theory in an even number $D$ of space-time
dimensions with Euclidean metric555It is common practice to compute amplitudes
in Euclidean space. Moving to Minkowski space involves performing an extension
by analytic continuation known as Wick rotation. See for example [64] and
[65]. and allow different propagators to have different masses. A Feynman
diagram is a connected directed graph where each edge represents a propagator
and is assigned a momentum and a mass and each vertex stands for a tree-level
interaction. External half-edges, also known as external legs, represent
incoming or outgoing particles, while internal edges are the internal
propagators of the diagram. We define the loop number to be the number of
independent cycles of the graph. We adopt the convention for which all
external legs have arrows pointing inwards, and consequently distinguish
incoming and outgoing particles depending on the momentum being positive or
negative, respectively. Momentum is positive when it points in the same
direction of the arrow of the corresponding directed edge, and it is negative
otherwise. We fix momenta on external lines and for each internal loop we
choose an arbitrary orientation of the edges which is consistent with momentum
conservation at each vertex of the graph and globally. Momentum conservation
leaves one unconstrained free momentum variable for each loop. Thus, the loop
number is equal to the number of independent loop momentum vectors. We only
consider graphs that are one-particle irreducible.

Let $G$ be such a Feynman graph with $m$ external legs, $n$ internal edges,
and $l$ independent loops. Its Feynman integral, up to numerical prefactors,
is

$\displaystyle
I_{G}=\big{(}\mu^{2}\big{)}^{n-lD/2}\int\prod_{r=1}^{l}\frac{{\rm
d}^{D}k_{r}}{{\rm
i}\pi^{D/2}}\;\prod_{j=1}^{n}\frac{1}{-q_{j}^{2}+m_{j}^{2}-{\rm
i}\varepsilon},$ (2.1)

where $\varepsilon$ is a small positive parameter,666The $-{\rm i}\varepsilon$
term is required by the choice of Feynman pole prescription for the
computation of the propagators and it allows to perform a Wick rotation to
Minkowski space. In what follows, however, the $-{\rm i}\varepsilon$ term does
not play a role. We set $\varepsilon=0$ for simplicity of notation. $\mu$ is a
scale introduced to make the expression dimensionless,777In what follows, the
scale $\mu$ remains factored out. We set $\mu^{2}=1$ for simplicity of
notation. $k_{1},\dots,k_{l}$ are the independent loop momenta,
$m_{1},\dots,m_{n}$ are the masses of the internal lines, and
$q_{1},\dots,q_{n}$ are the momenta flowing through them, which can be
expressed as

$\displaystyle
q_{j}=\sum_{i=1}^{l}\lambda_{ji}k_{i}+\sum_{i=1}^{m}\sigma_{ji}p_{i},$ (2.2)

where $p_{1},\dots,p_{m}$ are the external momenta and $\lambda_{ji}$,
$\sigma_{ji}\in\\{-1,0,1\\}$ are constants depending on the particular graph
structure.

Feynman [34] introduced the well-known manipulation consisting of defining a
set of parameters $x_{1},\dots,x_{n}$, called Feynman parameters, one for each
internal edge of the graph, and applying the formula

$\displaystyle\prod_{j=1}^{n}\frac{1}{P_{j}}=\Gamma(n)\int_{\\{x_{j}\geq
0\\}}{\rm
d}^{n}x\,\delta\bigg{(}1-\sum_{j=1}^{n}x_{j}\bigg{)}\frac{1}{\left(\sum_{j=1}^{n}x_{j}P_{j}\right)^{n}}$
(2.3)

with the choice $P_{j}=-q_{j}^{2}+m_{j}^{2}$ for $j=1,\dots,n$. Here, $\Gamma$
is the Euler gamma function and $\delta$ is the Dirac delta distribution.
Indeed, we can write

$\displaystyle\sum_{j=1}^{n}x_{j}\big{(}{-}q_{j}^{2}+m_{j}^{2}\big{)}=-\sum_{r=1}^{l}\sum_{s=1}^{l}k_{r}\cdot(M_{rs}k_{s})+\sum_{r=1}^{l}2k_{r}\cdot
Q_{r}+J,$ (2.4)

where $M$ is an $l\times l$-matrix with scalar entries, $Q$ is an $l$-vector
with momentum vectors as entries and $J$ is a scalar. $M$, $Q$ and $J$ can be
suitably expressed in terms of the graph parameters
$\\{x_{j},q_{j},m_{j}\\}_{j=1}^{n}$. Applying Feynman parametrisation to
(2.1), the $l$-dimensional integral over the loop momenta becomes an
$(n-1)$-dimensional integral over the Feynman parameters

$\displaystyle I_{G}=\Gamma\bigg{(}n-\frac{lD}{2}\bigg{)}\int_{\\{x_{j}\geq
0\\}}{\rm
d}^{n}x\,\delta\bigg{(}1-\sum_{j=1}^{n}x_{j}\bigg{)}\frac{\mathcal{U}^{n-(l+1)D/2}}{\mathcal{F}^{n-lD/2}},$
(2.5)

which is characterised by the polynomials $\mathcal{U}=\det(M)$ and
$\mathcal{F}=\det(M)\big{(}J+QM^{-1}Q\big{)}$, called first and second
Symanzik polynomials of the Feynman graph, respectively. Notice that the
dimension $D$ of space-time, entering the exponents in the integrand of (2.5),
acts as regularisation. We use dimensional regularisation888The dimensional
regularisation procedure has been first introduced by ’t Hooft and Veltman
[67]. with $D=4-2\epsilon$ and $\epsilon$ a small parameter. A detailed
description of Feynman parametrisation can be found in Srednicki [65].

######  Example 2.1.

Consider a one-loop diagram with $m=n$ external legs, as the one shown in Fig.
2.

$m_{n}$$m_{6}$$m_{5}$$m_{4}$$m_{3}$$m_{2}$$m_{1}$$q_{n}$$q_{6}$$q_{5}$$q_{4}$$q_{3}$$q_{2}$$q_{1}$$p_{n}$$p_{n-1}$$p_{5}$$p_{4}$$p_{3}$$p_{2}$$p_{1}$

Figure 2: Example of a one-loop Feynman diagram with $m=n$ external legs.

Its Symanzik polynomials are

$\displaystyle\mathcal{U}_{\text{1-loop}}=\sum_{j=1}^{n}x_{j},$ (2.6)
$\displaystyle\mathcal{F}_{\text{1-loop}}=\mathcal{U}_{\text{1-loop}}\sum_{j=1}^{n}m_{j}^{2}x_{j}+\sum_{\begin{subarray}{c}i,j=1\\\
i<j\end{subarray}}^{n}(q_{i}-q_{j})^{2}x_{i}x_{j},$ (2.7)

where the internal momenta are given by $q_{1}=k$,
$q_{i}=k+p_{1}+\dots+p_{i-1}$ for $1<i\leq n$. Here, $k$ is the unique loop
momentum of the graph, and $p_{1}+\dots+p_{m}=0$ by global momentum
conservation.

###  2.3 Graph polynomials

Re-expression of Feynman integrals in parametric form shows that the
correspondence between scalar Feynman diagrams and Feynman integrals can be
reformulated in different terms. The information contained in a Feynman graph
is shared out among its multiple components, which can be identified as the
underlying graph structure, the directionality of edges and the various edge
labels. If we destructure a Feynman graph in these layers and momentarily
neglect the extra information apart from the graph structure, we observe that
its integral is insensitive to changes of the graph which leave its topology
unaltered. Focusing on the underlying graph topology, the Symanzik polynomials
can be suitably re-interpreted and they are commonly called graph polynomials
in this context.

Let $G$ be a finite graph without isolated vertices. $G$ is specified by the
pair $(V_{G},E_{G})$, where $V_{G}$ is the collection of vertices and $E_{G}$
is the collection of edges. We choose an arbitrary orientation of its edges
and define the map

$\displaystyle\mathbb{Z}^{E_{G}}$
$\displaystyle\longrightarrow\mathbb{Z}^{V_{G}},$ (2.8) $\displaystyle e$
$\displaystyle\longmapsto t(e)-s(e),$ (2.9)

where $e\in E_{G}$ is an edge and $s(e),t(e)\in V_{G}$ are its source and
target endpoints with respect to the edge orientation. Let us extend this map
to the exact sequence

$\displaystyle 0\rightarrow
H_{1}(G,\mathbb{Z})\rightarrow\mathbb{Z}^{E_{G}}\rightarrow\mathbb{Z}^{V_{G}}\rightarrow
H_{0}(G,\mathbb{Z})\rightarrow 0,$ (2.10)

where $H_{0}(G,\mathbb{Z})$ and $H_{1}(G,\mathbb{Z})$ are the zeroth and first
homology groups of the graph. As a consequence, the graph loop number $l_{G}$
is related to the number of edges $n_{G}$, the number of vertices $v_{G}$, and
the number of connected components $c_{G}$ by999The loop number is
equivalently defined as the rank of the first homology group of the graph,
while the number of connected components corresponds to the rank of the zeroth
homology group of the graph.

$\displaystyle
l_{G}=\text{rank}(H_{1}(G,\mathbb{Z}))=|E_{G}|-|V_{G}|+\text{rank}(H_{0}(G,\mathbb{Z}))=n_{G}-v_{G}+c_{G}.$
(2.11)

Assume $G$ is a graph of Feynman type, that is, finite, connected and one-
particle irreducible. Let the valence of a vertex be the number of edges
attached to it. Since they do not contribute to the braid pattern of Feynman
graphs, both vertices of valence one, corresponding to the source endpoints of
external legs, and vertices of valence two, corresponding to mass insertions,
do not play a role here. To such a graph $G$ we wish to assign an integral
$I_{G}$ which corresponds to the one previously defined in (2.5) when the
neglected extra information is re-inserted. We start by associating a variable
$x_{e}$ to every internal edge $e\in E_{G}$ of the graph. These variables are
known as Schwinger parameters and they are the graph-theoretic analogues of
Feynman parameters. Let $\mathcal{T}_{1}$ be the set of spanning trees101010A
graph of zero loop number with $k$ connected components is called a
$k$-forest. When $k=1$, the forest is called a tree. Given an arbitrary
connected graph $G$, a spanning $k$-forest of $G$ is a subgraph $T\subseteq G$
such that $V_{T}=V_{G}$ and $T$ is a $k$-forest. A spanning $k$-forest of $G$
is usually denoted by the collection of its trees. A connected graph has
always at least one spanning tree. of $G$. The first graph polynomial of $G$
is defined as

$\displaystyle\Psi_{G}=\sum_{\begin{subarray}{c}T\in\mathcal{T}_{1}\end{subarray}}\prod_{e\notin
E_{T}}x_{e}.$ (2.12)

It is a homogeneous polynomial of degree $l_{G}$ in the Schwinger parameters.
Note that each monomial of $\Psi_{G}$ has coefficient one, and $\Psi_{G}$ is
linear in each Schwinger parameter.

######  Example 2.2.

The first graph polynomial of the Feynman graph shown in Fig. 3 is
$\Psi_{G}=x_{1}\cdots
x_{n}\big{(}\frac{1}{x_{1}}+\dots+\frac{1}{x_{n}}\big{)}$.

$x_{1}$$x_{2}$$x_{3}$$x_{n}$

Figure 3: Example of a scalar Feynman graph with $n$ internal propagators.

By construction, the first Symanzik polynomial $\mathcal{U}$ of a Feynman
graph $G$ does not depend on momenta and masses involved in the diagram, but
is only dependent on the graph topology. Indeed, it explicitly identifies with
the first graph polynomial $\Psi_{G}$ of the corresponding pure graph
structure. The same is not true for the second Symanzik polynomial
$\mathcal{F}$, which is a function of external momenta and internal masses.
However, we can re-express $\mathcal{F}$ in a way that clearly separates the
contribution to $\mathcal{F}$ coming from the graph topology from its other
dependences. To this end, momenta and masses edge labels must re-enter our
discussion. Let $\mathcal{T}_{2}$ be the set of spanning $2$-forests of $G$
and $P_{T_{i}}$ be the set of external momenta of $G$ attached to its tree
$T_{i}$. The second graph polynomial of $G$ is defined as

$\displaystyle\Xi_{G}(\\{p_{j},m_{e}\\})=\bigg{(}\sum_{e\in
E_{G}}m_{e}^{2}x_{e}\bigg{)}\Psi_{G}\;-\sum_{(T_{1},T_{2})\in\mathcal{T}_{2}}\bigg{(}\prod_{e\notin
E_{T_{1}}\cup E_{T_{2}}}x_{e}\bigg{)}\Bigg{(}\sum_{\begin{subarray}{c}p_{j}\in
P_{T_{1}}\\\ p_{k}\in P_{T_{2}}\end{subarray}}p_{j}\cdot p_{k}\Bigg{)}.$
(2.13)

It is a homogeneous polynomial of degree $l_{G}+1$ in the Schwinger
parameters. Note that, if all internal masses are zero, then $\Xi_{G}$ is
linear in each Schwinger parameter. It follows from their definitions that the
second Symanzik polynomial and the second graph polynomial of a Feynman graph
are, indeed, the same. Moreover, having fixed the momenta of external
particles and the masses of internal propagators, we are left with the
explicit dependence of $\mathcal{F}$ on the graph structure given in terms of
spanning $2$-forests.

$q_{3}$$q_{2}$$q_{4}$$q_{1}$$m_{3}$$m_{2}$$m_{4}$$m_{1}$$p_{3}$$p_{2}$$p_{4}$$p_{1}$

(a) Full Feynman diagram

$x_{3}$$x_{2}$$x_{4}$$x_{1}$

(b) Underlying graph structure

Figure 4: Box diagram with four legs.

######  Example 2.3.

To explicitly see how the individual terms in the graph polynomials arise from
the knot structure of the diagram, we look closer at the one-loop Feynman
graph with $m=4$ external legs, also called box diagram,111111This gives a
next-to-leading order contribution to the two-to-two particle scattering
process. Srednicki [65] gives a detailed discussion of two particles elastic
scattering at one-loop using standard methods in perturbative quantum field
theory. which is shown in Fig. 4. Its Symanzik polynomials are

$\displaystyle\mathcal{U}_{\text{box}}=x_{1}+x_{2}+x_{3}+x_{4},$ (2.14)
$\displaystyle\mathcal{F}_{\text{box}}=\big{[}(x_{1}+x_{2}+x_{3}+x_{4})\big{(}m_{1}^{2}x_{1}+m_{2}^{2}x_{2}+m_{3}^{2}x_{3}+m_{4}^{2}x_{4}\big{)}+x_{1}x_{2}p_{1}^{2}+x_{2}x_{3}p_{2}^{2}+x_{3}x_{4}p_{3}^{2}$
(2.15)
$\displaystyle\hphantom{\mathcal{F}_{\text{box}}=}{}+x_{4}x_{1}p_{4}^{2}+x_{1}x_{3}(p_{1}+p_{2})^{2}+x_{2}x_{4}(p_{2}+p_{3})^{2}\big{]}.$
(2.16)

Neglecting mass terms, the remaining monomials correspond to the spanning
forests shown in Figs. 5 and 6.

$x_{3}$$x_{2}$$x_{4}$

(a) $+x_{1}$

$x_{3}$$x_{4}$$x_{1}$

(b) $+x_{2}$

$x_{2}$$x_{4}$$x_{1}$

(c) $+x_{3}$

$x_{3}$$x_{2}$$x_{1}$

(d) $+x_{4}$

Figure 5: Spanning trees in the box diagram with four legs and corresponding
terms in $\mathcal{U}_{\text{box}}$.

$x_{4}$$x_{3}$

(a) $+x_{1}x_{2}p_{1}^{2}$

$x_{4}$$x_{1}$

(b) $+x_{2}x_{3}p_{2}^{2}$

$x_{2}$$x_{1}$

(c) $+x_{3}x_{4}p_{3}^{2}$

$x_{3}$$x_{2}$

(d) $+x_{4}x_{1}p_{4}^{2}$

$x_{4}$$x_{2}$

(e) $+x_{1}x_{3}(p_{1}+p_{2})^{2}$

$x_{3}$$x_{1}$

(f) $+x_{2}x_{4}(p_{2}+p_{3})^{2}$

Figure 6: Spanning $2$-forests in the box diagram with four legs and
corresponding terms in $\mathcal{F}_{\text{box}}$.

Thus, the Symanzik or graph polynomials capture the algebraic information
contained in the topology of a Feynman diagram and they prove to be the first
tool to be used in the tentative investigation of renormalisation theory via
the algebraic manipulation of concatenated one-loop integrals. For a more
detailed overview of the properties of Feynman graph polynomials we refer to
Bogner and Weinzierl [8].

###  2.4 Primitive log-divergent graphs

The parametric Feynman integral in (2.5) can be written in a slightly
different notation, which turns out to be particularly useful henceforth.
Neglecting prefactors and assuming $D=4$, it is equivalent to the projective
integral

$\displaystyle
I_{G}(\\{p_{j},m_{e}\\})=\int_{\sigma}\frac{\Omega}{\Psi_{G}^{2}}\bigg{(}\frac{\Psi_{G}}{\Xi_{G}(\\{p_{j},m_{e}\\})}\bigg{)}^{n_{G}-2l_{G}},$
(2.17)

where $\sigma$ is the real projective simplex given by

$\displaystyle\sigma=\big{\\{}[x_{1}:\dots:x_{n_{G}}]\in\mathbb{P}^{n_{G}-1}(\mathbb{R})\,|\,x_{e}\geq
0,\,e=1,\dots,n_{G}\big{\\}}$ (2.18)

and $\Omega$ is the top-degree differential form on $\mathbb{P}^{n_{G}-1}$
expressed in local coordinates as

$\displaystyle\Omega=\sum_{e=1}^{n_{G}}(-1)^{e}x_{e}\,{\rm
d}x_{1}\wedge\dots\wedge\widehat{{\rm d}x_{e}}\wedge\dots\wedge{\rm
d}x_{n_{G}}.$ (2.19)

One can check that the integrand is homogeneous of degree zero, so that the
integral in projective space is well-defined and equivalent, under the affine
constraint $x_{n_{G}}=1$, to the previous parametric integral in affine space.
Integral (2.17) is in general divergent, as singularities may arise if the
zero sets of the graph polynomials $\Psi_{G}$ and $\Xi_{G}$ intersect the
domain of integration.

Graphs satisfying the condition $n_{G}=2l_{G}$ are called logarithmically
divergent and constitute a particularly interesting class of graphs. In fact,
their Feynman integral simplifies to

$\displaystyle I_{G}=\int_{\sigma}\frac{\Omega}{\Psi_{G}^{2}},$ (2.20)

where the dependence on the second Symanzik polynomial, and consequently on
momenta and masses, has vanished. Being uniquely sensitive to the graph
topology, such a Feynman graph describes a so-called single-scale
process.121212Among other contexts, the feature of no-scaling also occurs in
the evaluation of Feynman diagrams concerning the anomalous magnetic moment of
the electron, as presented by Laporta and Remiddi [50]. For a logarithmically
divergent graph $G$, we define the graph hypersurface as the zero set of its
first Symanzik polynomial

$\displaystyle
X_{G}=\big{\\{}[x_{1}:\dots:x_{n_{G}}]\in\mathbb{P}^{n_{G}-1}\,|\,\Psi_{G}(x_{1},\dots,x_{n_{G}})=0\big{\\}}$
(2.21)

which describes the singularities of its Feynman integral $I_{G}$. The
following theorem on the convergence of logarithmically divergent graphs is
proven by Bloch, Esnault and Kreimer [5].

######  Theorem 2.4.

Let $G$ be logarithmically divergent. The integral $I_{G}$ converges if and
only if every proper subgraph $\varnothing\neq\gamma\subset G$ satisfies the
condition $n_{\gamma}>2l_{\gamma}$.

A logarithmically divergent graph $G$ such that $I_{G}$ is convergent is
called primitive log-divergent, or simply primitive. We give particular
attention to the class of primitive log-divergent graphs in scalar massless
$\phi^{4}$ quantum field theory. They are called $\phi^{4}$-graphs, and have
vertices with valence at most four. Feynman amplitudes in $\phi^{4}$ theory
have been computed to much higher loop orders than most other quantum field
theories thanks to the work of Broadhurst and Kreimer [11, 12], and Schnetz
[58]. Some of the simplest $\phi^{4}$-graphs are shown in Fig. 7 along with
the values of the associated Feynman integrals. Here, $\zeta$ is the Riemann
zeta function, and
$P_{3,5}=-\frac{216}{5}\zeta(3,5)-81\zeta(5)\zeta(3)+\frac{522}{5}\zeta(8)$.

(a)

(b)

(c)

(d)

Figure 7: Examples of $\phi^{4}$-graphs with 3, 4, 5 and 6 loops.

###  2.5 Multiple zeta values

The Riemann zeta function is defined on the half-plane of complex numbers
$s\in\mathbb{C}$ with $\mathop{\rm Re}(s)>1$ by the absolutely convergent
series

$\displaystyle\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$ (2.22)

and extended to a meromorphic function on the whole complex plane with a
single pole at $s=1$. The first tentative attempts to find polynomial
relations among zeta values by multiplying terms of the form (2.22) have led
to a generalisation of the notion of Riemann zeta value. Multiple zeta values,
or MZVs, are the real numbers

$\displaystyle\zeta(s_{1},\dots,s_{l})=\sum_{n_{1}>n_{2}>\dots>n_{l}\geq
1}\frac{1}{n_{1}^{s_{1}}\cdots n_{l}^{s_{l}}}$ (2.23)

associated to tuples of integers $\mathbf{s}=(s_{1},\dots,s_{l})$, called
multi-indices. To guarantee the convergence of the infinite series, only
multi-indices such that $s_{i}\geq 1$ for $i=1,\dots,l$ and $s_{1}\geq 2$ are
considered. They are called admissible multi-indices. The integers
$\mathop{\rm wt}(\mathbf{s})=s_{1}+\dots+s_{l}$ and $l$ are called weight and
length of the multi-index $\mathbf{s}$, respectively.

Following the early observations that products of two zeta values are
$\mathbb{Q}$-linear combinations of zeta and double zeta values, and that
products of more than two zeta values are analogously expressed in terms of
multiple zeta values of higher length, linear relations among MZVs have been
the object of a more and more extensive investigation by many mathematicians,
including Brown, Cartier, Deligne, Drinfeld, Écalle, Goncharov, Hain, Hoffman,
Kontsevich, Terasoma, Zagier, Broadhurst and Kreimer. Indeed, the
$\mathbb{Q}$-linear relations among multiple zeta values directly provide
insights on the widely sought-after algebraic relations among Riemann zeta
values.

The $\mathbb{Q}$-vector space spanned by multiple zeta values forms an algebra
under the so-called stuffle product. Analytic methods, like partial fraction
expansions, provide only a few of the known relations among MZVs. Many more
are obtained, although conjecturally, by performing extensive numerical
experiments, as described by Blümlein et al [6]. However, enormous progress
followed the analytic discovery of a crucial feature of multiple zeta values,
that is, beside their representation as infinite series, they admit an
alternative representation as iterated integrals over simplices of weight-
dimension. Let $\Delta^{p}=\\{(t_{1},\dots,t_{p})\in\mathbb{R}^{p}\,|\,1\geq
t_{1}\geq t_{2}\geq\dots\geq t_{p}\geq 0\\}$ and define the measures on the
open interval $(0,1)$

$\displaystyle\omega_{0}(t)=\frac{{\rm d}t}{t},\qquad\omega_{1}(t)=\frac{{\rm
d}t}{1-t}.$ (2.24)

If $\mathbf{s}$ is an admissible multi-index, write $r_{i}=s_{1}+\dots+s_{i}$
for each $i=1,\dots,l$ and set $r_{0}=0$. Define the measure
$\omega_{\mathbf{s}}$ on the interior of the simplex $\Delta^{\mathop{\rm
wt}(\mathbf{s})}$ by

$\displaystyle\omega_{\mathbf{s}}=\prod_{i=1}^{l}\underbrace{\omega_{0}(t_{r_{i-1}+1})\cdot\cdot\cdot\omega_{0}(t_{r_{i}-1})}_{s_{i}-1\text{
times }}\omega_{1}(t_{r_{i}}).$ (2.25)

The following insight is due to Kontsevich.

######  Proposition 2.5.

Let $\mathbf{s}=(s_{1},\dots,s_{l})$ be an admissible multi-index. The
multiple zeta value $\zeta(\mathbf{s})$ can be obtained by the convergent
improper integral

$\displaystyle\zeta(\mathbf{s})=\zeta(s_{1},\dots,s_{l})=\int_{\Delta^{\mathop{\rm
wt}(\mathbf{s})}}\omega_{\mathbf{s}}.$ (2.26)

This different way of writing multiple zeta values yields a new algebra
structure associated with the so-called shuffle product. Many other linear
relations among MZVs are obtained systematically in this alternative
framework. However, relations are also and most interestingly derived by the
comparison of the two representations given by (2.23) and (2.26). The
coexistence of the stuffle and shuffle algebra structures on the
$\mathbb{Q}$-vector space of MZVs proved to be the most productive source of
information about these numbers. For a more detailed discussion of the
classical theory of multiple zeta values we refer to Fresán and Burgos Gil
[35]. Making concrete use of the many $\mathbb{Q}$-linear relations that MZVs
are known to satisfy is not an easy task, particularly at high weights. For
example, there is no algorithm that leads to the reduction of any given MZV
into a chosen $\mathbb{Q}$-basis. A boost in our understanding originated from
the exact-numerical decomposition algorithm by Brown [13]. Developed from the
non-classical perspective of the theory of motives, it conjecturally provides
a general strategy to handle MZVs, and more generally polylogarithmic numbers,
by converting them into the so-called f-alphabet.131313The elements of the
shuffle algebra on the $\mathbb{Q}$-vector space of MZVs are interpreted as
words in letters of certain weights. Precisely, there is one letter for each
odd weight greater than 1. Words in these letters span finite-dimensional
subspaces of definite weight. Notice, however, that the conversion into the
$f$-alphabet depends on the choice of algebra basis.

We observe the remarkable fact that $\mathbb{Q}$-linear combinations of
multiple zeta values are ubiquitous in the evaluation of Feynman amplitudes in
perturbative quantum field theories. It was conjectured by Broadhurst and
Kreimer [11] and then proved by Brown and Schnetz [20] that Feynman integrals
of the infinite family of zig-zag graphs in $\phi^{4}$ theory (see Fig. 8) are
certain known rational multiples of the odd values of the Riemann zeta
function.

(a)

(b)

Figure 8: Examples of zig-zag graphs with 5 and 6 loops.

######  Theorem 2.6.

Let $Z_{l}$ be the zig-zag graph with $l$ loops. Its Feynman integral is

$\displaystyle
I_{Z_{l}}=4\frac{(2l-2)!}{l!(l-1)!}\left(1-\frac{1-(-1)^{l}}{2^{2l-3}}\right)\zeta(2l-3).$
(2.27)

Another example is given by the anomalous magnetic moment of the electron in
quantum electrodynamics. The tree level Feynman diagram representing a slow-
moving electron emitting a photon is depicted in Fig. 9 along with its one-
loop correction. The two-loop correction comes from the contributions of seven
distinct two-loop diagrams. The total two-loop Feynman amplitude has been
evaluated by Petermann [53], giving
$\frac{197}{144}+\frac{1}{2}\zeta(2)-3\zeta(2)\log(2)+\frac{3}{4}\zeta(3)$,
which involves the logarithm of 2 and again values of the Riemann zeta
function.

(a) Tree-level contribution

(b) One-loop contribution

Figure 9: Up to one-loop Feynman diagrams contributing to the anomalous
magnetic moment of the electron.

Many more examples are given by Broadhurst [10]. Due to a vast amount of
evidence, it was believed for a long time that all primitive amplitudes of the
form (2.20) in massless $\phi^{4}$ theory should be $\mathbb{Q}$-linear
combinations of MZVs. Only recently this conjectural statement was proved
false in the motivic setup141414Outside of the motivic framework, the
statement relies on transcendentality conjectures. by Brown and Schnetz [19].
Explicit examples of $\phi^{4}$-amplitudes at high loop orders not expressible
in terms of multiple zeta values have been found by Panzer and Schnetz [52].
In the same work, explicit computation of all $\phi^{4}$-amplitudes with loop
order up to $7$ suggests that not all MZVs appear among them. For example, no
$\phi^{4}$-graph is known to evaluate to $\zeta(2)$ or $\zeta(2)^{2}$.
Remarkably, the integral representation of MZVs partially clarify the presence
of these numbers in perturbative calculations in quantum field theory. Indeed,
both expressions (2.20) and (2.26) are suitably interpreted as periods of
algebraic varieties.

##  3 Cohomology theory of algebraic varieties

###  3.1 Singular homology

We follow the expositions by Weibel [71] and Hartshorne [38]. Let $M$ be a
topological space. For each integer $k\geq 0$, the standard $k$-simplex is

$\displaystyle\Delta^{k}_{st}=\bigg{\\{}(t_{0},\dots,t_{k})\in\mathbb{R}^{k+1}\,\bigg{|}\,\sum_{i=0}^{k}t_{i}=1,\;t_{i}\geq
0,\;i=0,\dots,k\bigg{\\}}.$ (3.1)

For each $i=0,\dots,k$, the face map
$\delta_{i}^{k}\colon\Delta^{k-1}_{st}\rightarrow\Delta^{k}_{st}$ is defined
by

$\displaystyle\delta^{k}_{i}(t_{0},\dots,t_{k-1})=(t_{0},\dots,t_{i-1},0,t_{i},\dots,t_{k-1}).$
(3.2)

A singular k-chain in $M$ is a continuous151515If $M$ is a differentiable
manifold, we can assume the singular chains to be piecewise smooth, or smooth,
without altering the homology groups. map
$\sigma\colon\Delta^{k}_{st}\rightarrow M$. For each $k\geq 0$, let

$\displaystyle C_{k}(M)=\bigoplus_{\sigma}\mathbb{Z}\sigma$ (3.3)

be the free abelian group generated by singular $k$-chains. Elements of
$C_{k}(M)$ are finite $\mathbb{Z}$-linear combinations of the continuous maps
$\sigma\colon\Delta^{k}_{st}\rightarrow M$. For each $k\geq 1$, the boundary
map $\partial_{k}\colon C_{k}(M)\rightarrow C_{k-1}(M)$ is defined by

$\displaystyle\partial_{k}(\sigma)=\sum_{i=0}^{k}(-1)^{i}\big{(}\sigma\circ\delta_{i}^{k}\big{)},$
(3.4)

where the alternating signs in the sum guarantee that boundary maps satisfy
the condition $\partial_{k-1}\circ\partial_{k}=0$. The pair
$(C_{\bullet}(M),\partial_{\bullet})$ is called a homological chain complex
and is graphically represented as

$\displaystyle\dots\xlongrightarrow{\partial_{k+1}}C_{k}(M)\xlongrightarrow{\partial_{k}}C_{k-1}(M)\xlongrightarrow{\partial_{k-1}}\dots\xlongrightarrow{\partial_{2}}C_{1}(M)\xlongrightarrow{\partial_{1}}C_{0}(M)$

######  Definition 3.1.

The singular homology of the topological space $M$ is the homology of the
complex $(C_{\bullet}(M),\partial_{\bullet})$, that is

$\displaystyle
H_{k}^{\text{s}}(M,\mathbb{Z})=\begin{cases}C_{0}(M)/\Im(\partial_{1}),&k=0,\\\
\mathop{\rm Ker}(\partial_{k})/\Im(\partial_{k+1}),&k\geq 1.\end{cases}$ (3.5)

In degree $k$, chains in the kernel of the boundary map $\partial_{k}$ are
called $($closed$)$ cycles and chains in the image of the boundary map
$\partial_{k+1}$ are called $($exact$)$ boundaries.

######  Example 3.2.

Let $M=\mathbb{C}^{*}$ be the punctured complex plane. The singular chains

$\displaystyle\gamma_{0}\colon\quad\Delta^{0}_{st}\rightarrow\mathbb{C}^{*},\qquad
1\mapsto 1,$ (3.6)
$\displaystyle\gamma_{2}\colon\quad\Delta^{1}_{st}\rightarrow\mathbb{C}^{*},\qquad(t,1-t)\mapsto{\rm
e}^{2\pi{\rm i}t}$ (3.7)

generate the singular homology groups
$H_{0}^{\text{s}}(\mathbb{C}^{*},\mathbb{Z})$ and
$H_{1}^{\text{s}}(\mathbb{C}^{*},\mathbb{Z})$, respectively. These are both
free groups of rank one. All the other homology groups vanish.

For each $k\geq 0$, the free abelian group of singular n-cochains is defined
by

$\displaystyle C^{k}(M)=\mathop{\rm Hom}(C_{k}(M),\mathbb{Z}).$ (3.8)

Analogously, applying vector duality, we introduce the coboundary maps
$d^{k}\\!\colon C^{k}(M)\\!\rightarrow\\!C^{k+1}(M)$, which satisfy the
condition $d^{k+1}\circ d^{k}=0$. The corresponding cohomological chain
complex $(C^{\bullet}(M),d^{\bullet})$ is graphically represented as

$\displaystyle\cdots\xlongleftarrow{d^{k+1}}C^{k+1}(M)\xlongleftarrow{d^{k}}C^{k}(M)\xlongleftarrow{d^{k-1}}\cdots\xlongleftarrow{d^{1}}C^{1}(M)\xlongleftarrow{d^{0}}C^{0}(M)$

######  Definition 3.3.

The singular cohomology of the topological space $M$ is the cohomology of the
complex $(C^{\bullet}(M),d^{\bullet})$, that is

$\displaystyle H^{k}_{\text{s}}(M,\mathbb{Z})=\begin{cases}\mathop{\rm
Ker}\big{(}d^{0}\big{)},&k=0,\\\ \mathop{\rm
Ker}\big{(}d^{k}\big{)}/\Im(d^{k-1}),&k\geq 1.\end{cases}$ (3.9)

Definitions 3.1 and 3.3 of singular homology and cohomology of topological
spaces, given here with respect to $\mathbb{Z}$, extend naturally to other
coefficient rings. For our purposes, we assume the ring of coefficients to be
$\mathbb{Q}$. This allows us to identify singular cohomology groups with the
vector duals of the corresponding singular homology groups161616This
isomorphism is true for real or complex coefficients as well, but it does not
hold for integer coefficients.

$\displaystyle H^{k}_{\text{s}}(M,\mathbb{Q})\simeq\mathop{\rm
Hom}(H_{k}^{\text{s}}(M,\mathbb{Q}),\mathbb{Q}),$ (3.10)

that is, classes of a cohomology group can be interpreted as classes of linear
functionals on the corresponding homology group. The singular cohomology of
the topological space underlying a complex algebraic variety is of particular
interest.

######  Definition 3.4.

Let $X$ be an algebraic variety over a subfield $\mathbb{K}$ of $\mathbb{C}$.
Its set of complex points $X(\mathbb{C})$ canonically carries the complex
analytic topology, and the corresponding topological space171717Equipped with
the canonical structure sheaf, $X^{\rm an}$ is a complex analytic space,
called the analytification of $X$. The relationship between algebraic spaces
over the complex numbers and complex analytic spaces is described by a series
of results, known as GAGA-type theorems. These developments followed the work
by Serre [62] on the existence and faithfulness of the analytification of a
complex algebraic variety. is written as $X^{\rm an}$. The Betti cohomology of
$X$ is the singular cohomology of the underlying topological space $X^{\rm
an}$, that is

$\displaystyle H^{k}_{{\rm B}}(X,\mathbb{Q})=H^{k}_{\text{s}}\big{(}X^{\rm
an},\mathbb{Q}\big{)}$ (3.11)

for $k\geq 0$.

######  Example 3.5.

Let $\mathbb{G}_{m}=\mathop{\rm Spec}\mathbb{Q}[x,1/x]$ be the multiplicative
group. $\mathbb{G}_{m}$ is an algebraic variety over $\mathbb{Q}$ and its
underlying topological space of complex points is $\mathbb{G}_{m}^{\rm
an}=\mathbb{C}^{*}$. For each $k\geq 0$, the $k$-th Betti cohomology group of
$\mathbb{G}_{m}$ is $H_{\rm
B}^{k}(\mathbb{G}_{m},\mathbb{Q})=H_{\text{s}}^{k}(\mathbb{C}^{*},\mathbb{Q})$.

####  3.1.1 Some properties of homology

We briefly recall some properties of singular homology and cohomology.

  * ($a$)

Homotopy invariance. If $M_{1}$ and $M_{2}$ are homotopically equivalent
topological spaces, then $H_{k}^{\text{s}}(M_{1},\mathbb{Q})\simeq
H_{k}^{\text{s}}(M_{2},\mathbb{Q})$ for each $k\geq 0$. An analogous statement
holds for singular cohomology.

  * ($b$)

Mayer–Vietoris sequences. For any two open subspaces $U,V\subseteq M$ of a
given topological space $M$, such that $M=U\cup V$, there is a long exact
sequence of the following form:

${\cdots}$${H^{s}_{k}(U\cap V,\mathbb{Q})}$${H^{s}_{k}(U,\mathbb{Q})\oplus
H^{s}_{k}(V,\mathbb{Q})}$${H^{s}_{k}(M,\mathbb{Q})}$${H^{s}_{k-1}(U\cap
V,\mathbb{Q})}$${\cdots.}$ (3.14)

An analogous statement holds for singular cohomology.

  * ($c$)

Künneth formula. For any two topological spaces $M_{1}$, $M_{2}$, for each
$k\geq 0$, there is a natural isomorphism

$\displaystyle H_{k}^{\text{s}}(M_{1}\times
M_{2},\mathbb{Q})\simeq\bigoplus_{i+j=k}H_{i}^{\text{s}}(M_{1},\mathbb{Q})\otimes
H_{j}^{\text{s}}(M_{2},\mathbb{Q}).$ (3.15)

An analogous statement holds for singular cohomology.

  * ($d$)

Push-forward. Let $f\colon M_{1}\rightarrow M_{2}$ be a continuous map between
two topological spaces $M_{1}$, $M_{2}$. Then, $f$ induces a morphism of chain
complexes

$\displaystyle f_{*}\colon\ C_{\bullet}(M_{1})\rightarrow C_{\bullet}(M_{2})$
(3.16)

called push-forward, sending $\sigma_{1}\in C_{k}(M_{1})$ to
$\sigma_{2}=f\circ\sigma_{1}\in C_{k}(M_{2})$. Equivalently, the following
diagram:

${\Delta_{st}^{k}}$${M_{1}}$${M_{2}}$$\scriptstyle{\sigma_{1}}$$\scriptstyle{\sigma_{2}}$$\scriptstyle{f}$
(3.19)

commutes. Hence, $f$ induces also a group homomorphism between the
corresponding singular homology groups

$\displaystyle f_{*}\colon\ H_{k}^{\text{s}}(M_{1},\mathbb{Q})\rightarrow
H_{k}^{\text{s}}(M_{2},\mathbb{Q})$ (3.20)

for each $k\geq 0$.

  * ($e$)

Pull-back. Let $f\colon M_{1}\rightarrow M_{2}$ be a continuous map between
two topological spaces $M_{1}$, $M_{2}$. Then, $f$ induces a morphism of
cochain complexes

$\displaystyle f^{*}\colon\ C^{\bullet}(M_{2})\rightarrow C^{\bullet}(M_{1})$
(3.21)

called pull-back, sending $\omega_{2}\in C^{k}(M_{2})$ to
$\omega_{1}=\omega_{2}\circ f_{*}\in C^{k}(M_{1})$. Equivalently, the
following diagram:

${C_{k}(M_{1})}$${\mathbb{Q}}$${C_{k}(M_{2})}$$\scriptstyle{\omega_{1}}$$\scriptstyle{f_{*}}$$\scriptstyle{\omega_{2}}$
(3.24)

commutes. Hence, $f$ induces also a group homomorphism between the
corresponding singular cohomology groups

$\displaystyle f^{*}\colon\ H^{k}_{\text{s}}(M_{2},\mathbb{Q})\rightarrow
H^{k}_{\text{s}}(M_{1},\mathbb{Q})$ (3.25)

for each $k\geq 0$.

####  3.1.2 Relative singular homology

Let $M$ be a topological space and $\iota\colon N\hookrightarrow M$ the
canonical inclusion of a topological subspace $N\subseteq M$. Denote by
$(C_{\bullet}(N),\partial^{N}_{\bullet})$ and
$(C_{\bullet}(M),\partial^{M}_{\bullet})$ their homological chain complexes,
and by $\iota_{*}\colon C_{\bullet}(N)\rightarrow C_{\bullet}(M)$ the
corresponding injective morphism obtained via push-forward. For each $k\geq
1$, we define the total chain complex $C_{\bullet}(M,N)$ to be the mapping
cone181818Note that, for any morphism of chain complexes $f_{*}\colon
C_{\bullet}(M_{1})\rightarrow C_{\bullet}(M_{2})$, the mapping cone
$C_{k}(M_{2},M_{1})=C_{k-1}(M_{1})\oplus C_{k}(M_{2})$ can be defined.
However, injectivity of the morphism $\iota_{*}$ implies that the cone
$C_{\bullet}(M,N)$ is quasi-isomorphic to the quotient
$C_{\bullet}(M)/C_{\bullet}(N)$. of the morphism $\iota_{*}$, that is

$\displaystyle C_{k}(M,N)=C_{k-1}(N)\oplus C_{k}(M),$ (3.26)

and the differential $\partial_{k}\colon C_{k}(M,N)\rightarrow C_{k-1}(M,N)$
to act as

$\displaystyle\partial_{k}(\sigma_{N},\sigma_{M})=\big{(}{-}\partial_{k-1}^{N}(\sigma_{N}),-\iota_{*}(\sigma_{N})+\partial_{k}^{M}(\sigma_{M})\big{)},$
(3.27)

where $(\sigma_{N},\sigma_{M})\in C_{k}(M,N)$.

######  Definition 3.6.

The relative homology of the pair of topological spaces $(M,N)$ is the
homology of the total chain complex $(C_{\bullet}(M,N),\partial_{\bullet})$.
For $k\geq 1$, we denote the relative singular homology groups as
$H_{k}^{\text{s}}(M,N,\mathbb{Q})$.

Relative homology fits into the following long exact sequence:

${\cdots}$${H_{k}^{\text{s}}(M,\mathbb{Q})}$${H_{k}^{\text{s}}(M,N,\mathbb{Q})}$${H_{k-1}^{\text{s}}(N,\mathbb{Q})}$${H_{k-1}^{\text{s}}(M,\mathbb{Q})}$${H_{k-1}^{\text{s}}(M,N,\mathbb{Q})}$${\cdots,}$
(3.30)

where the connecting morphisms are the push-forward maps $\iota_{*}\colon
H_{k}^{\text{s}}(N,\mathbb{Q})\rightarrow H_{k}^{\text{s}}(M,\mathbb{Q})$
induced by the inclusion $\iota\colon N\hookrightarrow M$. Consider an element
of the relative homology group $H_{k}^{\text{s}}(M,N,\mathbb{Q})$. This is
represented by a pair $(\sigma_{N},\sigma_{M})$ of singular chains
$\sigma_{N}\in C_{k-1}(N)$ and $\sigma_{M}\in C_{k}(M)$ satisfying

$\displaystyle\partial_{k-1}^{N}\sigma_{N}=0,\qquad\partial_{k}^{M}\sigma_{M}=\iota_{*}\sigma_{N}.$
(3.31)

Note that, since $\iota_{*}$ is injective, the latter condition implies the
former. Thus, relative homology classes are represented by chains in $M$ whose
boundary is contained in $N$. Relative cohomology groups
$H^{k}_{\text{s}}(M,N,\mathbb{Q})$ are defined similarly.

######  Example 3.7.

Let $M=\mathbb{C}^{*}$ be the punctured complex plane and $N=\\{p,q\\}\subset
M$ be the subspace consisting of two points $p,q\in\mathbb{C}^{*}$ with $p\neq
q$. Let $\gamma_{1}\colon\Delta^{1}_{st}\rightarrow M$ be any continuous
map191919When it does not pass through the origin, the oriented segment
starting at $p$ and ending at $q$ is an example of such a map. such that
$\gamma_{1}(0,1)=p$, $\gamma_{1}(1,0)=q$, and it does not encircle the origin.
Then

$\displaystyle\partial_{1}^{M}\gamma_{1}=p-q\in C_{0}(N).$ (3.32)

Consequently, $\gamma_{1}$ defines a relative chain. It follows from the long
exact sequence (3.30) that the only non-trivial relative homology group is
$H_{1}^{\text{s}}(M,N,\mathbb{Q})$. A basis of this group is given by the
chain $\gamma_{1}$ and the chain $\gamma_{2}$, introduced in Example 3.2,
consisting of a counterclockwise circle containing the origin. Such a basis is
graphically represented in Fig. 10.

$\gamma_{2}$$O$$p$$\gamma_{1}$$q$

Figure 10: Basis of $H_{1}^{\text{s}}(\mathbb{C}^{*},\\{p,q\\},\mathbb{Q})$.

###  3.2 De Rham cohomology

We start by reviewing a classical construction in differential geometry. Let
$M$ be a differentiable manifold of dimension $n$. A differential $p$-form on
$M$ is written in local coordinates as

$\displaystyle\sum_{1\leq i_{1}<\dots<i_{p}\leq n}f_{i_{1},\dots,i_{p}}\;{\rm
d}x_{i_{1}}\wedge\dots\wedge{\rm d}x_{i_{p}},$ (3.33)

where $f_{i_{1},\dots,i_{p}}$ are $\mathcal{C}^{\infty}$-functions. Let
$\Omega^{p}(M)$ denote the $\mathbb{R}$-vector space of differential $p$-forms
on $M$ and define the space of differential forms on $M$ as

$\displaystyle\Omega(M)=\bigoplus_{p=0}^{n}\Omega^{p}(M).$ (3.34)

The exterior derivative ${\rm d}\colon\Omega(M)\rightarrow\Omega(M)$ is the
unique $\mathbb{R}$-linear map which sends $p$-forms into $(p+1)$-forms and
satisfies the following axioms:

  * (1)

Let $f$ be a smooth function. Then, ${\rm d}f=\sum_{i=1}^{n}\frac{\partial
f}{\partial x_{i}}{\rm d}x_{i}$ is the ordinary differential of $f$.

  * (2)

${\rm d}\circ{\rm d}=0$.

  * (3)

Let $\alpha$ be a $p$-form on $M$ and $\beta$ any differential form in
$\Omega(M)$. Denote by $\alpha\wedge\beta$ their exterior product. Then, ${\rm
d}(\alpha\wedge\beta)={\rm d}\alpha\wedge\beta+(-1)^{p}\alpha\wedge{\rm
d}\beta$.

The associated cochain complex is

$\displaystyle 0\rightarrow\Omega^{0}(M)\xrightarrow{\rm
d}\Omega^{1}(M)\xrightarrow{\rm d}\cdots\xrightarrow{\rm
d}\Omega^{n}(M)\rightarrow 0$ (3.35)

and its cohomology, denoted $H^{\bullet}_{\rm dR}(M,\mathbb{R})$, is called
the smooth de Rham cohomology of $M$. A differential $p$-form $\omega$ is
closed if ${\rm d}\omega=0$ and it is exact if there exists a differential
$(p-1)$-form $\eta$ such that $\omega={\rm d}\eta$. A classical
theorem202020De Rham’s theorem was first presented in his PhD thesis,
published in 1931, when cohomology groups had not been introduced yet. He did
not state the theorem in the way it is described today, but gave an equivalent
version involving Betti numbers and integration of closed differential forms
over cycles. by de Rham [24] asserts that the singular cohomology
$H^{\bullet}_{\text{s}}(M,\mathbb{R})$ can be computed using differential
forms.212121We refer to Bott and Tu [9] for a comprehensive investigation of
differential forms in algebraic topology.

######  Theorem 3.8.

Let $M$ be a differentiable manifold of dimension $n$. For $0\leq k\leq n$,
the map

$\displaystyle H^{k}_{\mathrm{dR}}(M,\mathbb{R})$
$\displaystyle\longrightarrow
H^{k}_{\mathrm{s}}(M,\mathbb{R})\simeq\mathop{\rm
Hom}\big{(}H_{k}^{\text{s}}(M,\mathbb{R}),\mathbb{R}\big{)},$ (3.36)
$\displaystyle{}[\omega]$ $\displaystyle\longmapsto\int\omega,$ (3.37)

which sends the class of a differential form $\omega$ to the integration
functional

$\displaystyle\int\omega\colon\ H_{k}^{\text{s}}(M,\mathbb{R})$
$\displaystyle\longrightarrow\mathbb{R},$ (3.38) $\displaystyle{}[\gamma]$
$\displaystyle\longmapsto\displaystyle\int_{\gamma}\omega,$ (3.39)

is an isomorphism.

####  3.2.1 Algebraic de Rham cohomology

A notion of de Rham cohomology for general algebraic varieties over fields of
characteristic zero has been introduced by Grothendieck [37]. Let $\mathbb{K}$
be a subfield of $\mathbb{C}$ and let $X$ be an algebraic variety over
$\mathbb{K}$.

######  Definition 3.9.

Consider $U\subseteq X$ an open affine subset in the Zariski topology. The
ring of regular functions on $U$, denoted by $\mathcal{O}(U)$, is a finitely-
generated $\mathbb{K}$-algebra, and precisely a quotient of a polynomial ring
over $\mathbb{K}$. We say that $X$ is smooth or nonsingular of dimension $n$
if, for every closed point $x\in X$, the limit $\lim_{\begin{subarray}{c}\\\
\overrightarrow{U\ni x}\end{subarray}}\mathcal{O}(U)$, indexed over all
Zariski open affine subsets $U\subseteq X$ containing $x$, and with ordering
defined by reverse inclusion, is a regular local ring of dimension $n$.

Let $X$ be smooth of dimension $n$ and affine. We can write $X=\mathop{\rm
Spec}R$, where $R=\mathcal{O}(X)$ is the ring of regular functions on $X$. A
$\mathbb{K}$-linear algebraic $p$-form on $X$ is a differential $p$-forms on
$X$ with coefficients in $R$. In a local coordinate chart, it is given by an
expression of the form

$\displaystyle\sum_{1\leq i_{1}<\dots<i_{p}\leq n}f_{i_{1},\dots,i_{p}}\;{\rm
d}x_{i_{1}}\wedge\dots\wedge{\rm d}x_{i_{p}},$ (3.40)

where $f_{i_{1},\dots,i_{p}}$ are $\mathbb{K}$-polynomial functions on $X$. We
denote by $\Omega^{p}(X)$ the $\mathbb{K}$-vector space of algebraic $p$-forms
on $X$ and we define the space of algebraic forms on $X$ as

$\displaystyle\Omega(X)=\bigoplus_{p=0}^{n}\Omega^{p}(X).$ (3.41)

A derivation ${\rm d}\colon\Omega(X)\rightarrow\Omega(X)$, satisfying
properties that are analogous to the ones described in Section 3.2 for the
exterior derivative, can be defined. It canonically yields a cochain complex

$\displaystyle 0\rightarrow R\simeq\Omega^{0}(X)\xrightarrow{\rm
d}\Omega^{1}(X)\xrightarrow{\rm d}\cdots\xrightarrow{\rm
d}\Omega^{n}(X)\rightarrow 0$ (3.42)

called the algebraic de Rham complex of $X$. The associated cohomology,
denoted $H^{\bullet}_{\rm dR}(X,\mathbb{K})$, is called the algebraic de Rham
cohomology of $X$.

######  Remark 3.10.

If $X$ is smooth of dimension $n$, but not necessarily affine, at each closed
point $x\in X$, we can choose some Zariski open affine neighbourhood $U$ of
$x$ and some regular functions $x_{1},\dots,x_{n}\in\mathcal{O}(U)$ in such a
way to define a system of local parameters222222If we do not assume $X$ to be
smooth, then we can find local coordinates in an affine open neighbourhood $U$
of a closed point $x\in X$ if and only if the rank of the Jacobian matrix at
$x$ is equal to the dimension of $U$. at $x$. Viewed as a subvariety of the
affine $\mathbb{K}$-space $\mathbb{A}^{n}$, $U$ inherits its local coordinate
structure. Intuitively, by choosing a covering of $X$ composed of Zariski open
affine subsets, the algebraic variety is charted with affine spaces. Observe
that the morphism $U\rightarrow\mathbb{A}^{n}$ defined by the local
coordinates $x_{1},\dots,x_{n}$ is always an étale map,232323Étale maps can be
interpreted as the algebraic analogue of local isomorphisms in the complex
analytic topology. However, open sets in the Zariski topology are not small
enough for étale maps to be local isomorphisms. but not generally an
embedding.242424For complex neighbourhoods, local coordinates define local
isomorphisms. Indeed, smooth algebraic varieties over $\mathbb{C}$ can be
locally embedded as submanifolds of the complex affine space. Conceptually,
the $\mathbb{K}$-linear algebraic forms of degree $p$ on $X$ are obtained by
suitably gluing252525The assignment of algebraic forms to smooth affine
varieties via local coordinates is well-behaved under gluing, and hence it
globalises. the algebraic $p$-forms defined locally, as in (3.40), on each
subset $U$ of an affine open covering of $X$. The notion of algebraic de Rham
cohomology thus Čeck-style generalises to arbitrary smooth algebraic
$\mathbb{K}$-varieties. Such an intuition does not, however, capture the full
picture. The algebraic substitute for the smooth differential form is
rigorously defined through the notions of Kähler differential and exterior
power, while the rigorous construction of the algebraic de Rham cohomology of
any smooth algebraic $\mathbb{K}$-variety requires the use of sheaf cohomology
and hypercohomology. We do not present these concepts here, since an intuitive
understanding is sufficient to our purpose, but we refer to Kashiwara and
Schapira [44], and Hartshorne [38]. Moreover, we mention that several
constructions are available to adapt the definition of algebraic de Rham
cohomology to the case of singular varieties giving well-behaved theories.
Details are reported by Huber and Müller-Stach [43].

######  Example 3.11.

Consider $X=\mathbb{G}_{m}=\mathop{\rm Spec}\mathbb{Q}[x,1/x]$. The only non-
vanishing spaces of $\mathbb{Q}$-linear algebraic forms are

$\displaystyle\Omega^{0}(\mathbb{G}_{m})=\mathbb{Q}[x,1/x],$ (3.43)
$\displaystyle\Omega^{1}(\mathbb{G}_{m})=\mathbb{Q}[x,1/x]\cdot{\rm d}x.$
(3.44)

Consequently, the two groups

$\displaystyle H^{0}_{\rm dR}(\mathbb{G}_{m},\mathbb{Q})=\mathbb{Q},$ (3.45)
$\displaystyle H^{1}_{\rm
dR}(\mathbb{G}_{m},\mathbb{Q})=\frac{\mathbb{Q}[x,1/x]\cdot{\rm d}x}{{\rm
d}\mathbb{Q}[x,1/x]}=\mathbb{Q}\bigg{[}\frac{{\rm d}x}{x}\bigg{]}$ (3.46)

are the only non-trivial algebraic de Rham cohomology groups of $X$.

####  3.2.2 Relative de Rham cohomology

The definition of algebraic de Rham cohomology extends to the relative
setting. Let $\mathbb{K}$ be a subfield of $\mathbb{C}$ and let $X$ be a
smooth algebraic variety over $\mathbb{K}$ of dimension $n$. Recall the
following definition.

######  Definition 3.12.

A codimension-1 closed subvariety $D\subset X$ is called a divisor with normal
crossings if, for every point $x\in D$, there is an open affine neighbourhood
$U\subseteq X$ of $x$ and some local coordinates $x_{1},\dots,x_{n}$ on $U$
such that:

  * (1)

The morphism $U\rightarrow\mathbb{A}^{n}$ defined by $x_{1},\dots,x_{n}$ is
étale.

  * (2)

The restriction $D_{|U}$ is locally described by an equation of the form
$x_{1}\cdot x_{2}\cdots x_{r}$ for some $1\leq r\leq n$.

Moreover, $D$ is called a divisor with simple normal crossings262626$D$ looks
locally like a collection of coordinate hyperplanes. if, in addition, its
irreducible components are smooth.

For simplicity,272727We illustrate here the construction of relative algebraic
de Rham cohomology in a particularly simple framework. The construction can,
however, be adapted for the general case of a closed subvariety of a smooth
algebraic $\mathbb{K}$-variety. For a general discussion, we refer to Huber
and Müller-Stach [43]. let $X$ be affine and $D\subset X$ a divisor with
simple normal crossings. Denote by $D_{i}$, for $i=1,\dots,r$, the smooth
irreducible components of $D$. For $I\subseteq\\{0,\dots,r\\}$, we set

$\displaystyle D_{I}=\bigcap_{i\in I}D_{i},\qquad
D^{p}=\begin{cases}X,&p=0,\\\ \coprod_{|I|=p}D_{I},&p\geq 1.\end{cases}$
(3.47)

The associated double cochain complex of $\mathbb{K}$-vector spaces
$K^{p,q}=\Omega^{q}(D^{p})$ is graphically represented as

${\cdots}$${\cdots}$${\cdots}$${\Omega^{2}(X)}$${\bigoplus_{i}\Omega^{2}(D_{i})}$${\bigoplus_{i<j}\Omega^{2}(D_{i}\cap
D_{j})}$${\cdots}$${\Omega^{1}(X)}$${\bigoplus_{i}\Omega^{1}(D_{i})}$${\bigoplus_{i<j}\Omega^{1}(D_{i}\cap
D_{j})}$${\cdots}$${\Omega^{0}(X)}$${\underbrace{\bigoplus_{i}\Omega^{0}(D_{i})}_{|I|=1}}$${\underbrace{\bigoplus_{i<j}\Omega^{0}(D_{i}\cap
D_{j})}_{|I|=2}}$${\cdots,}$$\scriptstyle{\rm d}$$\scriptstyle{-{\rm
d}}$$\scriptstyle{\rm d}$$\scriptstyle{\rm d}$$\scriptstyle{-{\rm
d}}$$\scriptstyle{\rm d}$$\scriptstyle{\rm d}$$\scriptstyle{-{\rm
d}}$$\scriptstyle{\rm d}$ (3.52)

where the vertical differential ${\rm d}_{\rm ver}\colon K^{p,q}\rightarrow
K^{p,q+1}$ is given by

$\displaystyle{\rm d}_{\rm ver}=(-1)^{p}{\rm d}$ (3.53)

and the horizontal differential ${\rm d}_{\rm hor}\colon K^{p,q}\rightarrow
K^{p+1,q}$ is given by

$\displaystyle{\rm d}_{\rm hor}=\bigoplus_{\begin{subarray}{c}|I|=p\\\
|J|=p+1\\\ I\subset J\end{subarray}}(-1)^{l}{\rm d}_{IJ},$ (3.54)

where $J=\\{j_{0},\dots,j_{p}\\}$ with $j_{0}<\dots<j_{p}$,
$I=\big{\\{}j_{0},\dots,\widehat{j_{l}},\dots,j_{p}\big{\\}}$, and ${\rm
d}_{IJ}\colon\Omega^{q}(D_{I})\rightarrow\Omega^{q}(D_{J})$ is the restriction
map.282828We observe that the two-row sequence in (3.52) is exact. For the
vertical lines $K^{p,q}\rightarrow K^{p,q+1}$, it follows from the property
${\rm d}\circ{\rm d}=0$ of the differential ${\rm d}$. For the horizontal
lines $K^{p,q}\rightarrow K^{p+1,q}$, it follows from the fact that the
differential ${\rm d}_{\rm hor}\colon K^{p,q}\rightarrow K^{p+1,q}$ is
surjective for even values of $p$ and trivial for odd values of $p$, as a
consequence of the surjectivity of the restriction maps ${\rm d}_{IJ}$. Note
that the sign factor $(-1)^{p}$ in the definition of ${\rm d}_{\rm ver}$
implies that the vertical and horizontal differentials anticommute. Moreover,
since $D_{I}$ has dimension equal to $n-|I|$, the double complex is trivial
for $p+q>n$. We denote by $(\Omega^{\bullet}(X,D),\delta)$ the total cochain
complex associated to $K^{p,q}$, that is

$\displaystyle\Omega^{\bullet}(X,D)=\bigoplus_{p+q=\bullet}K^{p,q},\qquad\delta={\rm
d}_{\rm ver}+{\rm d}_{\rm hor}.$ (3.55)

For each $k\geq 0$, the space $\Omega^{k}(X,D)$ corresponds to the direct sum
of the spaces on the $k$-th diagonal of the double cochain complex $K^{p,q}$
represented in (3.52). The total complex is indeed explicitly written as

$\displaystyle\Omega^{0}(X,D)\simeq\Omega^{0}(X)\xlongrightarrow{\delta^{0}}\Omega^{1}(X,D)\simeq\Omega^{1}(X)\oplus\bigoplus_{i}\Omega^{0}(D_{i})\xlongrightarrow{\delta^{1}}\cdots.$
(3.56)

The relative algebraic de Rham cohomology $H_{\rm
dR}^{\bullet}(X,D,\mathbb{K})$ is the cohomology of the total cochain complex
$\Omega^{\bullet}(X,D)$, that is

$\displaystyle H^{k}_{\rm dR}(X,D,\mathbb{K})=\begin{cases}\mathop{\rm
Ker}\big{(}\delta^{0}\big{)},&k=0,\\\ \mathop{\rm
Ker}\big{(}\delta^{k}\big{)}/\Im\big{(}\delta^{k-1}\big{)},&k\geq
1.\end{cases}$ (3.57)

The following proposition is a consequence of the surjectivity of the
restriction maps ${\rm d}_{IJ}$.

######  Proposition 3.13.

Let $X$ be a smooth affine variety over $\mathbb{K}$ of dimension $n$ and
$D\subset X$ a divisor with simple normal crossings. Each class in the top-
degree cohomology group $H_{\rm dR}^{n}(X,D,\mathbb{K})$ has a representative
in $\Omega^{n}(X)$.

######  Example 3.14.

Let $X=\mathbb{G}_{m}=\mathop{\rm Spec}\mathbb{Q}[x,1/x]$ and $D=\\{1,z\\}$
with $z\in\mathbb{Q}$, $z\neq 1$. The corresponding double algebraic de Rham
complex is

${0}$${\displaystyle\mathbb{Q}\bigg{[}x,\frac{1}{x}\bigg{]}{\rm
d}x}$${0}$${\displaystyle\mathbb{Q}\bigg{[}x,\frac{1}{x}\bigg{]}}$${\mathbb{Q}\oplus\mathbb{Q}}$${0,}$$\scriptstyle{\rm
d}$$\scriptstyle{\rm d}$$\scriptstyle{-{\rm d}}$ (3.61)

where the only non-trivial horizontal differential is the evaluation map

$\displaystyle\mathbb{Q}\bigg{[}x,\frac{1}{x}\bigg{]}$
$\displaystyle\longrightarrow\mathbb{Q}\oplus\mathbb{Q},$ (3.62)
$\displaystyle f$ $\displaystyle\longmapsto(f(1),f(z)).$ (3.63)

The corresponding total complex is

$\displaystyle\mathbb{Q}\bigg{[}x,\frac{1}{x}\bigg{]}$
$\displaystyle\xlongrightarrow{\delta^{0}}\mathbb{Q}\bigg{[}x,\frac{1}{x}\bigg{]}{\rm
d}x\oplus\mathbb{Q}\oplus\mathbb{Q},$ (3.64) $\displaystyle f(x)$
$\displaystyle\longmapsto(f^{\prime}(x){\rm d}x,f(1),f(z)),$ (3.65)

where the only non-trivial differential is explicitly written. The non-trivial
relative algebraic de Rham cohomology groups are

$\displaystyle H^{0}_{\rm dR}(X,D,\mathbb{Q})=\mathop{\rm
Ker}\big{(}\delta^{0}\big{)}=0,$ (3.66) $\displaystyle H^{1}_{\rm
dR}(X,D,\mathbb{Q})=\mathop{\rm
coKer}\big{(}\delta^{0}\big{)}=\frac{\mathbb{Q}\left[x,\frac{1}{x}\right]{\rm
d}x\oplus\mathbb{Q}\oplus\mathbb{Q}}{\Im(\delta^{0})}$ (3.67)

and a basis of $H^{1}_{\rm dR}(X,D)$ is given by the classes
$\big{[}\big{(}\frac{{\rm d}x}{x},0,0\big{)}\big{]}=\big{[}\frac{{\rm
d}x}{x}\big{]}$ and $\big{[}\big{(}\frac{{\rm
d}x}{z-1},0,0\big{)}\big{]}=\big{[}\frac{{\rm d}x}{z-1}\big{]}$.

###  3.3 Comparison isomorphism

The following fundamental theorem is due to Grothendieck [37].

######  Theorem 3.15.

Let $\mathbb{K}$ be a subfield of $\mathbb{C}$ and let $X$ be a smooth
algebraic variety over $\mathbb{K}$. There is a canonical isomorphism

$\displaystyle\mathop{\rm comp}\colon\ H^{\bullet}_{\rm
dR}(X,\mathbb{K})\otimes_{\mathbb{K}}\mathbb{C}\xlongrightarrow{\sim}H^{\bullet}_{{\rm
B}}(X,\mathbb{Q})\otimes_{\mathbb{Q}}\mathbb{C}$ (3.68)

known as comparison isomorphism. Moreover, if $Y\subset X$ is closed
subvariety, we have

$\displaystyle\mathop{\rm comp}\colon\ H^{\bullet}_{\rm
dR}(X,Y,\mathbb{K})\otimes_{\mathbb{K}}\mathbb{C}\xlongrightarrow{\sim}H^{\bullet}_{{\rm
B}}(X,Y,\mathbb{Q})\otimes_{\mathbb{Q}}\mathbb{C}.$ (3.69)

As mentioned in Definition 3.4, any algebraic $\mathbb{K}$-variety $X$
canonically yields an analytic complex space $X^{\rm an}$ associated with the
space of complex points $X(\mathbb{C})$. If $X$ is smooth, then $X^{\rm an}$
is an analytic complex manifold. The classical theory of de Rham cohomology,
discussed in Section 3.2 for differentiable manifolds, extends to complex
geometry as well.

######  Definition 3.16.

Let $M$ be a complex manifold of dimension $n$. For $p,q\geq 0$, a
differential form of holomorphic degree $p$ and antiholomorphic degree $q$ on
$M$, also called a differential $(p,q)$-form, is written in local analytic
coordinates as

$\displaystyle\sum_{I,J}f_{IJ}\,{\rm d}z_{i_{1}}\wedge\dots\wedge{\rm
d}z_{i_{p}}\wedge{\rm d}\bar{z}_{j_{1}}\wedge\dots\wedge{\rm
d}\bar{z}_{j_{q}},$ (3.70)

where the sum runs over the index subsets
$I=\\{i_{1},\dots,i_{p}\\},J=\\{j_{1},\dots,j_{q}\\}\subseteq\\{1,\dots,n\\}$
and $f_{IJ}$ are $\mathcal{C}^{\infty}$-functions. Let $\Omega^{p,q}(M)$
denote the $\mathbb{C}$-vector space of differential $(p,q)$-forms on $M$. The
standard de Rham differential splits as ${\rm d}=\partial+\bar{\partial}$,
where $\partial\colon\Omega^{p,q}\rightarrow\Omega^{p+1,q}$ and
$\bar{\partial}\colon\Omega^{p,q}\rightarrow\Omega^{p,q+1}$. The Dolbeault
cohomology of $M$, denoted by $H^{\bullet,\bullet}_{{\rm D}}(M,\mathbb{C})$,
is the cohomology of the double cochain complex
$(\Omega^{\bullet,\bullet}(M),\bar{\partial})$, called the Dolbeault complex
of $M$.

######  Definition 3.17.

A holomorphic $t$-form on $M$ is a finite sum of differential $(p,q)$-forms on
$M$ with $p+q=t$ that are locally expressed as in (3.70) with the coefficient
functions $f_{IJ}$ being holomorphic. The holomorphic de Rham
cohomology292929A singular version of the holomorphic de Rham complex, called
logarithmic de Rham complex, is obtained by considering meromorphic forms
which are holomorphic in the bulk, but admit logarithmic poles towards the
compactification boundaries of $X$. of $M$, denoted by $H^{\bullet}_{\rm
dR}(M,\mathbb{C})$, is the cohomology of the cochain complex associated with
the graded $\mathbb{C}$-vector space of holomorphic forms on $M$ and the
holomorphic component $\partial$ of the de Rham differential.303030For a
formally rigorous definition of holomorphic de Rham cohomology, using the
tools of sheaf cohomology and hypercohomology, see Voisin [69].

The following proposition is the complex analogue of Theorem 3.8 by de Rham.

######  Proposition 3.18.

Let $M$ be a complex manifold of dimension $n$. For $0\leq k\leq n$, there is
an isomorphism

$\displaystyle H^{k}_{\rm
dR}(M,\mathbb{C})\xlongrightarrow{\sim}H^{k}_{\text{s}}(M,\mathbb{Q})\otimes_{\mathbb{Q}}\mathbb{C}.$
(3.71)

In particular, if $M=X^{\rm an}$, where $X$ is a smooth algebraic
$\mathbb{K}$-variety of dimension $n$, we have

$\displaystyle H^{k}_{\rm dR}(X^{\rm
an},\mathbb{C})\xlongrightarrow{\sim}H^{k}_{{\rm
B}}(X,\mathbb{Q})\otimes_{\mathbb{Q}}\mathbb{C}$ (3.72)

for $0\leq k\leq n$.

######  Remark 3.19.

The holomorphic de Rham complex of $X^{\rm an}$ is equivalently obtained by
analytification of the algebraic de Rham complex of $X$, and an equivalent
notion of holomorphic de Rham cohomology of $X^{\rm an}$ follows canonically.
Thus, the procedure of analytification of an algebraic variety and the
properties of holomorphic de Rham cohomology provide the conceptual link
between algebraic de Rham cohomology and Betti cohomology, which underlies
Grothendieck’s comparison isomorphism.

######  Remark 3.20.

An important observation follows from combining Grothendieck’s and de Rham’s
theorems. Given a smooth algebraic variety $X$ over a subfield $\mathbb{K}$ of
$\mathbb{C}$, the holomorphic de Rham cohomology of the underlying topological
space $X^{\rm an}$, equivalent to its singular cohomology, is isomorphic to
the algebraic de Rham cohomology of $X$ after complexification, that is

$\displaystyle H^{k}_{\rm dR}(X^{\rm an},\mathbb{C})\simeq H^{k}_{\rm
dR}(X,\mathbb{K})\otimes_{\mathbb{K}}\mathbb{C}$ (3.73)

for $k\geq 0$. The holomorphic de Rham cohomology can therefore be computed
considering algebraic forms only. In this way, a purely algebraic definition
of cohomology is obtained.

###  3.4 Pure Hodge structures

As a consequence of Theorem 3.15, the Betti cohomology of an algebraic variety
is endowed with a richer structure than the singular cohomology of a generic
topological space. Recall the following definition.

######  Definition 3.21.

Let $H$ be a finite-dimensional $\mathbb{Q}$-vector space and let
$H_{\mathbb{C}}=H\otimes_{\mathbb{Q}}\mathbb{C}$. Assume that $H_{\mathbb{C}}$
possesses a bigrading

$\displaystyle H_{\mathbb{C}}=\bigoplus_{p+q=k}H^{p,q}$ (3.74)

for some integer $k$, satisfying the property $\overline{H^{p,q}}=H^{q,p}$,
called Hodge symmetry. $H$ is called a pure Hodge structure of weight $k$ and
the given direct sum decomposition of its complexification $H_{\mathbb{C}}$ is
called a Hodge decomposition.

######  Remark 3.22.

An equivalent definition of pure Hodge structure of weight $k$ is obtained by
observing that the data encoded in the Hodge decomposition is equivalent to a
finite decreasing filtration $F^{\bullet}$ of $H_{\mathbb{C}}$, called Hodge
filtration, such that

$\displaystyle
F^{p}H_{\mathbb{C}}\oplus\overline{F^{k-p+1}H_{\mathbb{C}}}=H_{\mathbb{C}}$
(3.75)

for all integers $p$. The two equivalent descriptions are related by

$\displaystyle
H^{p,q}=F^{p}H_{\mathbb{C}}\cap\overline{F^{q}H_{\mathbb{C}}},\qquad
F^{p}H_{\mathbb{C}}=\bigoplus_{i\geq p}H^{i,k-i}$ (3.76)

for $p$, $q$ integers such that $p+q=k$.

Let $M$ be a compact Kähler313131A Kähler manifold is a manifold with a
complex structure, a Riemannian structure, and a symplectic structure which
are mutually compatible. manifold. For $p,q\geq 0$, its Dolbeault cohomology
classes in bidegree $(p,q)$ uniquely correspond323232This result, true for any
compact hermitian complex manifold, is known as Hodge isomorphism. to the
harmonic $(p,q)$-forms on $M$, and there are canonical maps

$\displaystyle H^{p,q}_{{\rm D}}(M,\mathbb{C})\rightarrow H^{p+q}_{\rm
dR}(M,\mathbb{C})\simeq H^{p+q}_{\text{s}}(M,\mathbb{C}).$ (3.77)

The following theorem by Hodge [42] marks the beginning of what is currently
known as Hodge theory.

######  Theorem 3.23.

Let $M$ be a compact Kähler manifold. The following direct sum decomposition

$\displaystyle H^{k}_{\rm dR}(M,\mathbb{C})=\bigoplus_{p+q=k}H^{p,q}_{{\rm
D}}(M,\mathbb{C})$ (3.78)

holds for $k\geq 0$.

######  Remark 3.24.

Note that the complex conjugate of $H^{p,q}(M)$ is $H^{q,p}(M)$. Following
equation (3.71), the ordinary cohomology group
$H^{k}_{\text{s}}(M,\mathbb{Q})$ is a pure Hodge structure of weight $k$, and
Hodge’s theorem gives a decomposition333333The Hodge decomposition of a
compact Kähler manifold is independent of the choice of Kähler metric,
although there is no analogous decomposition for arbitrary compact complex
manifolds. of its complexification as a direct sum of $\mathbb{C}$-vector
spaces.

Let $X$ be a smooth projective variety defined over a subfield $\mathbb{K}$ of
$\mathbb{C}$. $X^{\rm an}$ is a compact Kähler manifold, and thus the Hodge
decomposition and filtration are defined on $H^{k}_{\rm dR}(X^{\rm
an},\mathbb{C})$ for $k\geq 0$. Following equation (3.72), the Betti
cohomology groups of $X$ are then pure Hodge structures of weights equal to
their degrees. Moreover, it can be proven343434This result follows non-
trivially from the interpretation of the Hodge filtration in terms of
hypercohomology with coefficients in the complex of holomorphic forms, and the
GAGA theorem. We refer to Voisin [69]. that the Hodge filtration
$F^{\bullet}$, which makes the Betti cohomology group $H^{k}_{{\rm
B}}(X,\mathbb{Q})$ into a pure Hodge structure of weight $k$, directly acts on
the algebraic de Rham cohomology groups of $X$ over $\mathbb{K}$. Precisely,
there is an integer $n$ such that $F^{\bullet}$ is a finite decreasing
filtration on $H_{\mathbb{K}}=H^{k}_{\rm dR}(X,\mathbb{K})$ satisfying

$\displaystyle
F^{p}H_{\mathbb{K}}\oplus\overline{F^{n-p+1}H_{\mathbb{K}}}=H_{\mathbb{K}}$
(3.79)

for all integers $p$. To keep track of all these structures, we define the
formal assignment $X\mapsto H^{\bullet}(X)$, where $H^{k}(X)$ is the triple of
data given by

$\displaystyle H^{k}(X)=\big{(}\big{(}H^{k}_{\rm
dR}(X,\mathbb{K}),F^{\bullet}\big{)},H^{k}_{{\rm B}}(X,\mathbb{Q}),{\rm
comp}\big{)}$ (3.80)

for $k\geq 0$. We call $H^{k}(X)$ a (pure) de Rham and Betti system of
realisations, or shortly a (pure) $H$-system, of weight $n$ over $\mathbb{K}$.
Observe that the weight of $H^{k}(X)$ is defined by the action of the Hodge
filtration on the algebraic de Rham cohomology, and it does not generally
equal the degree $k$.

######  Definition 3.25.

Let $X,X^{\prime}$ be smooth projective $\mathbb{K}$-varieties. Write
$H=H^{k}(X)$ and $H^{\prime}=H^{k^{\prime}}(X^{\prime})$, where
$k,k^{\prime}\geq 0$. A morphism of pure $H$-systems $f\colon H\rightarrow
H^{\prime}$ is a pair $f=(f_{\rm dR},f_{{\rm B}})$ consisting of a
$\mathbb{K}$-linear map $f_{\rm dR}\colon H_{\rm dR}\rightarrow H_{\rm
dR}^{\prime}$ and a $\mathbb{Q}$-linear map $f_{{\rm B}}\colon H_{{\rm
B}}\rightarrow H_{{\rm B}}^{\prime}$ such that:

  * (1)

$f_{\rm dR}$ is filtered353535Let $\mathbb{K}$ be a subfield of $\mathbb{C}$
and $(V,F)$, $(V^{\prime},F)$ be filtered $\mathbb{K}$-vector spaces. A
morphism $f\colon V\rightarrow V^{\prime}$ is called filtered if
$f(F^{p}V)\subseteq F^{p}V^{\prime}$ for each $p\geq 0$. with respect to the
Hodge filtration, that is

$\displaystyle f_{\rm dR}(F^{\bullet}H_{\rm dR})\subseteq F^{\bullet}H_{\rm
dR}^{\prime}.$ (3.81)

  * (2)

The following diagram commutes:

${H_{\rm dR}\otimes_{\mathbb{K}}\mathbb{C}}$${H_{{\rm
B}}\otimes_{\mathbb{Q}}\mathbb{C}}$${H_{\rm
dR}^{\prime}\otimes_{\mathbb{K}}\mathbb{C}}$${H_{{\rm
B}}^{\prime}\otimes_{\mathbb{Q}}\mathbb{C}.}$$\scriptstyle{\mathop{\rm
comp}}$$\scriptstyle{f_{\rm
dR}\otimes_{\mathbb{K}}\text{Id}_{\mathbb{C}}}$$\scriptstyle{f_{{\rm
B}}\otimes_{\mathbb{Q}}\text{Id}_{\mathbb{C}}}$$\scriptstyle{\mathop{\rm
comp}^{\prime}}$ (3.84)

Observe that, if $H$ and $H^{\prime}$ have different weights, then every
morphism between them is zero. The following variant of Theorem 3.23 implies
that pure $H$-systems are functorial for morphisms of smooth projective
varieties.

######  Theorem 3.26.

Let $X$, $X^{\prime}$ be smooth projective varieties over $\mathbb{K}$. For
any morphism $f\colon X\rightarrow X^{\prime}$, the induced map on cohomology
$f^{*}\colon H^{\bullet}(X^{\prime})\rightarrow H^{\bullet}(X)$ is a morphism
of pure $H$-systems.

######  Example 3.27.

For each $n\in\mathbb{Z}$, we define

$\displaystyle\mathbb{Q}(n)=((\mathbb{K},F^{\bullet}),\mathbb{Q},\mathop{\rm
comp}),$ (3.85)

where the filtration yields $\mathbb{K}=F^{-n}\mathbb{K}\supseteq
F^{-n+1}\mathbb{K}=0$ and the isomorphism $\mathop{\rm
comp}\colon\mathbb{C}\rightarrow\mathbb{C}$ is given by multiplication by
$(2\pi{\rm i})^{-n}$. $\mathbb{Q}(n)$ is a one-dimensional pure $H$-system of
weight $-2n$ over $\mathbb{K}$ and is called a Tate–Hodge system. As an
example, $\mathbb{Q}(-1)$ is isomorphic to
$H^{1}(\mathbb{G}_{m})=\big{(}\big{(}H^{1}_{\rm
dR}(\mathbb{G}_{m}),F^{\bullet}\big{)},H^{1}_{{\rm B}}(\mathbb{G}_{m}),{\rm
comp}\big{)}$, where $F^{\bullet}$ is the trivial filtration concentrated in
degree $1$. Observe that $\mathbb{Q}(-1)$ is a pure $H$-system of weight $2$,
although $H^{1}(\mathbb{G}_{m})$ has degree $1$.

###  3.5 Mixed Hodge structures

The Betti cohomology in degree $k$ of a smooth projective $\mathbb{K}$-variety
$X$ carries canonically a pure Hodge structure of weight $k$. However, this is
no longer true when $X$ fails to be smooth or projective. The generalisation
of the notion of pure Hodge structure to the case of quasi-projective
varieties is due to Deligne [25, 26, 27], who proved that the Betti cohomology
of a quasi-projective variety over a subfield $\mathbb{K}$ of $\mathbb{C}$ is
an iterated extension of pure Hodge structures. Recall the following
definition.

######  Definition 3.28.

Let $H$ be a finite-dimensional $\mathbb{Q}$-vector space and let
$H_{\mathbb{C}}=H\otimes_{\mathbb{Q}}\mathbb{C}$. Assume that $H$ possesses a
finite increasing filtration $W_{\bullet}$, called weight filtration, and that
$H_{\mathbb{C}}$ possesses a finite decreasing filtration $F^{\bullet}$,
called Hodge filtration, such that, for all integers $m$, the $m$-th graded
quotient of $H$ with respect to $W_{\bullet}$

$\displaystyle{\rm Gr}^{W}_{m}H=W_{m}/W_{m-1}$ (3.86)

together with the filtration induced by $F^{\bullet}$ on its complexification

$\displaystyle F^{\bullet}{\rm Gr}^{W}_{m}H=(F^{\bullet}\cap
W_{m}\otimes\mathbb{C}+W_{m-1}\otimes\mathbb{C})/W_{m-1}\otimes\mathbb{C}$
(3.87)

is a pure Hodge structure of weight $m$. $H$ is called a mixed Hodge
structure.

######  Remark 3.29.

Let $H$ be a mixed Hodge structure. For all integers $m$, there is a short
exact sequence

$\displaystyle 0\rightarrow W_{m-1}\rightarrow W_{m}\rightarrow{\rm
Gr}^{W}_{m}H\rightarrow 0.$ (3.88)

Take $m=h$ to be the highest weight of $H$, defined by $W_{h}=H$. The short
exact sequence above gives $H$ as an extension of the pure Hodge structure
${\rm Gr}^{W}_{h}H$ by $W_{h-1}$. Analogously, taking $m=h-1$, $W_{h-1}$ is an
extension of ${\rm Gr}^{W}_{h-1}H$ by $W_{h-2}$, which in turn is an extension
of ${\rm Gr}^{W}_{h-2}H$ by $W_{h-3}$, and so on. In this way, mixed Hodge
structures are explicitly realised as iterated extensions of pure ones.

######  Theorem 3.30.

Let $X$ be a quasi-projective variety over a subfield $\mathbb{K}$ of
$\mathbb{C}$.

  * $(1)$

For $k\geq 0$, its Betti cohomology group $H^{k}_{{\rm B}}(X,\mathbb{Q})$ is a
mixed Hodge structure with respect to a weight filtration $W_{\bullet}$ and a
Hodge filtration $F^{\bullet}$ which satisfy

$\displaystyle W_{-1}=0\subseteq W_{0}\subseteq W_{1}\subseteq\dots\subseteq
W_{2k}=H^{k}_{{\rm B}}(X,\mathbb{Q}),$ (3.89) $\displaystyle F^{0}=H^{k}_{{\rm
B}}(X,\mathbb{Q})\otimes_{\mathbb{Q}}\mathbb{C}\supseteq
F^{1}\supseteq\dots\supseteq F^{k}\supseteq F^{k+1}=0.$ (3.90)

If $X$ is smooth, then ${\rm Gr}^{W}_{m}H^{k}_{{\rm B}}(X,\mathbb{Q})=0$ for
all $m<k$. If $X$ is projective, then ${\rm Gr}^{W}_{m}H^{k}_{{\rm
B}}(X,\mathbb{Q})=0$ for all $m>k$.

  * $(2)$

The Hodge filtration $F^{\bullet}$ acts on the algebraic de Rham cohomology
groups of $X$ over $\mathbb{K}$, and the weight filtration $W^{{\rm
B}}_{\bullet}=W_{\bullet}$ induces a corresponding weight filtration
$W_{\bullet}^{\rm dR}$ on the algebraic de Rham cohomology groups of $X$ over
$\mathbb{K}$.

  * $(3)$

The comparison isomorphism is filtered with respect to the weight filtration

$\displaystyle\mathop{\rm comp}(W^{\rm
dR}_{\bullet}\otimes_{\mathbb{K}}\mathbb{C})=W^{{\rm
B}}_{\bullet}\otimes_{\mathbb{Q}}\mathbb{C}.$ (3.91)

Again, to keep track of the several structures that we have introduced, we
define a formal assignment $X\mapsto H^{\bullet}(X)$, where $H^{k}(X)$ is the
triple of data given by

$\displaystyle H^{k}(X)=\big{(}\big{(}H^{k}_{\rm
dR}(X,\mathbb{K}),F^{\bullet},W_{\bullet}^{\rm dR}\big{)},\big{(}H^{k}_{{\rm
B}}(X,\mathbb{Q}),W^{{\rm B}}_{\bullet}\big{)},\mathop{\rm comp}\\!\big{)}$
(3.92)

for $k\geq 0$. We call $H^{k}(X)$ a (mixed) de Rham and Betti system of
realisations, or shortly a (mixed) $H$-system, over $\mathbb{K}$. Observe
that, for each integer $m$, the triple of data

$\displaystyle{\rm Gr}^{W}_{m}H=\big{(}\big{(}{\rm Gr}^{W}_{m}H_{\rm
dR},F^{\bullet}\big{)},{\rm Gr}^{W}_{m}H_{{\rm B}},\mathop{\rm
comp}\\!\big{)},$ (3.93)

where $H=H^{k}(X)$, is a pure $H$-system of weight $m$.

######  Definition 3.31.

Let $X$, $X^{\prime}$ be quasi-projective $\mathbb{K}$-varieties. Write
$H=H^{k}(X)$ and $H^{\prime}=H^{k^{\prime}}(X^{\prime})$, where
$k,k^{\prime}\geq 0$. A morphism of mixed $H$-systems $f\colon H\rightarrow
H^{\prime}$ is a pair $f=(f_{\rm dR},f_{{\rm B}})$ consisting of a
$\mathbb{K}$-linear map $f_{\rm dR}\colon H_{\rm dR}\rightarrow H_{\rm
dR}^{\prime}$ and a $\mathbb{Q}$-linear map $f_{{\rm B}}\colon H_{{\rm
B}}\rightarrow H_{{\rm B}}^{\prime}$ such that:

  * $(1)$

$f_{{\rm B}}$ is filtered with respect to the weight filtration, that is

$\displaystyle f_{{\rm B}}(W_{\bullet}^{{\rm B}}H_{{\rm B}})\subseteq
W_{\bullet}^{{\rm B}}H_{{\rm B}}^{\prime}.$ (3.94)

  * $(2)$

$f_{\rm dR}$ is filtered with respect to the weight and Hodge filtrations,
that is

$\displaystyle f_{\rm dR}\big{(}W_{\bullet}^{\rm dR}H_{\rm dR}\big{)}\subseteq
W_{\bullet}^{\rm dR}H_{\rm dR}^{\prime}.$ (3.95) $\displaystyle f_{\rm
dR}\big{(}F^{\bullet}H_{\rm dR}\big{)}\subseteq F^{\bullet}H_{\rm
dR}^{\prime},$ (3.96)

  * (3)

$f_{{\rm B}}$ and $f_{\rm dR}$ are compatible with the comparison isomorphism,
that is

$\displaystyle(f_{{\rm
B}}\otimes_{\mathbb{Q}}\text{Id}_{\mathbb{C}})\circ\mathop{\rm
comp}=\mathop{\rm comp}\nolimits^{\prime}\circ(f_{\rm
dR}\otimes_{\mathbb{K}}\text{Id}_{\mathbb{C}}).$ (3.97)

The following analogue of Theorem 3.26 holds.

######  Theorem 3.32.

Let $X$, $X^{\prime}$ be quasi-projective varieties over $\mathbb{K}$. For any
morphism $f\colon X\rightarrow X^{\prime}$, the induced map on cohomology
$f^{*}\colon H^{\bullet}(X^{\prime})\rightarrow H^{\bullet}(X)$ is a morphism
of mixed $H$-systems.

We denote by $\mathbf{MHSy}(\mathbb{Q})$ the category363636Further aspects of
de Rham and Betti systems of realisations are discussed by Brown [15]. of
mixed $H$-systems over $\mathbb{Q}$. Deligne [26] proved that
$\mathbf{MHSy}(\mathbb{Q})$ is an abelian category. Moreover, it is naturally
endowed with two forgetful functors

$\displaystyle\omega_{{\rm B}}\colon\
\mathbf{MHSy}(\mathbb{Q})\rightarrow\text{Vec}_{\mathbb{Q}},\qquad\omega_{\rm
dR}\colon\ \mathbf{MHSy}(\mathbb{Q})\rightarrow\text{Vec}_{\mathbb{Q}}$ (3.98)

called Betti and de Rham functors, sending the mixed system of realisations
$H\in\mathbf{MHSy}(\mathbb{Q})$ into the $\mathbb{Q}$-vector spaces $H_{{\rm
B}}$ and $H_{\rm dR}$, respectively.

##  4 Periods of motives

###  4.1 Periods

The following elementary definition was introduced by Kontsevich and Zagier
[47].

######  Definition 4.1.

A period is a complex number whose real and imaginary parts are values of
absolutely convergent integrals of the form

$\displaystyle\int_{\sigma}f(x_{1},\dots,x_{n})\,{\rm d}x_{1}\cdots{\rm
d}x_{n},$ (4.1)

where the integrand $f$ is a rational function with rational coefficients and
the domain of integration $\sigma\subseteq\mathbb{R}^{n}$ is defined by finite
unions and intersections of domains of the form $\\{g(x_{1},\dots,x_{n})$
$\geq 0\\}$ with $g$ a rational function with rational coefficients.

If rational functions and coefficients are replaced in Definition 4.1 by
algebraic functions and coefficients, the same set of numbers is obtained.
Indeed, algebraic functions in the integrand can be substituted by rational
functions by enlarging the set of variables. Note that, because the integral
of any real-valued function is equivalent to the volume subtended by its
graph, any period admits a representation as the volume of a domain defined by
polynomial inequalities with rational coefficients. Thus, the integrand can
always be assumed to be the constant function 1. However, this extremely
simplified framework does not prove to be particularly useful. Quite the
opposite, in what follows, we mostly work with an even more general
description of periods than the one given in Definition 4.1. We denote by
$\mathcal{P}$ the set of periods. Being
$\bar{\mathbb{Q}}\subset\mathcal{P}\subset\mathbb{C}$, periods are generically
transcendental numbers and nonetheless they contain only a finite amount of
information, which is captured by the integrand and domain of integration of
its integral representation as in (4.1). Indeed, just like $\bar{\mathbb{Q}}$,
$\mathcal{P}$ is countable. Many famous numbers belong to the class of
periods. Here are some examples:

  * $(a)$

Algebraic numbers are periods, e.g.,

$\displaystyle\sqrt{2}=\int_{2x^{2}\leq 1}{\rm d}x.$ (4.2)

  * $(b)$

Logarithms of algebraic numbers are periods, e.g.,

$\displaystyle\log 2=\int_{1}^{2}\frac{{\rm d}x}{x}.$ (4.3)

  * $(c)$

The transcendental number $\pi$ is a period, as given by

$\displaystyle\pi=\int_{-1}^{1}\frac{{\rm
d}x}{\sqrt{1-x^{2}}}=\int\displaylimits_{-\infty}^{+\infty}\frac{{\rm
d}x}{1+x^{2}}=\iint\displaylimits_{x^{2}+y^{2}\leq 1}{\rm d}x{\rm d}y$ (4.4)

and alternatively by

$\displaystyle 2\pi{\rm i}=\oint_{\gamma_{2}}\frac{{\rm d}z}{z},$ (4.5)

where $\gamma_{2}$ is a closed path encircling the origin in the complex
plane.

  * $(d)$

Values of the beta function at positive rational arguments are periods, as
given by

$\displaystyle B(u,v)=\int_{0}^{1}t^{u-1}(1-t)^{v-1}{\rm d}t,\qquad\mathop{\rm
Re}(u),\,\mathop{\rm Re}(v)>0,$ (4.6)

and values of the gamma function at positive rational arguments
satisfy373737The statement follows from the relation between the gamma and the
beta functions
$\Gamma(a_{1})\cdots\Gamma(a_{n})=\Gamma(a_{1}+\dots+a_{n})\prod_{i=1}^{n-1}B(a_{1}+\dots+a_{i-1},a_{i})$
with $\mathop{\rm Re}(a_{k})>0$, $k=1,\dots,n$.

$\displaystyle\Gamma\bigg{(}\frac{p}{q}\bigg{)}^{q}\in\mathcal{P},\qquad
p,q\in\mathbb{N}.$ (4.7)

  * $(e)$

The elliptic integral

$\displaystyle
2\int_{-b}^{b}\sqrt{1+\frac{a^{2}x^{2}}{b^{4}-b^{2}x^{2}}}\,{\rm d}x$ (4.8)

representing the perimeter of an ellipse with radii $a$ and $b$, is a period.
Note that it is not an algebraic function of $\pi$ for $a\neq b$,
$a,b\in\mathbb{Q}_{>0}$,

  * $(f)$

Values of the Riemann zeta function at integer arguments $s\geq 2$ are
periods, e.g.,

$\displaystyle\zeta(3)=1+\frac{1}{2^{3}}+\frac{1}{3^{3}}+\dots=\sum_{n=1}^{\infty}\frac{1}{n^{3}}=\iiint\displaylimits_{0<x<y<z<1}\frac{{\rm
d}x{\rm d}y{\rm d}z}{(1-x)yz},$ (4.9)

and more generally multiple zeta values are periods by means of their integral
representation (2.26).

  * $(g)$

Convergent Feynman integrals, as in (2.20), are periods. Moreover, removing
the convergence requirement, the statement suitably extends to a wider class
of Feynman integrals.383838Under some assumptions, Bogner and Weinzierl [7]
showed that the coefficients appearing in the Laurent series of any scalar
multi-loop integral are periods.

  * $(h)$

Special values at algebraic arguments of hypergeometric functions, values of
modular forms at suitable arguments, and values of various kinds of
L-functions are periods.

Because the integral representation of a period is not unique, it is possible
that a certain integral of a transcendental function admits a representation
as a period as well. For example, $\log(2)$ is a period, and yet it can be
written as the following integral of a transcendental function

$\displaystyle\int\displaylimits_{0}^{1}\frac{x}{\log\frac{1}{1-x}}\,{\rm
d}x.$ (4.10)

Indeed, there seems to be no general principle able to predict if a certain
infinite sum or integral of a transcendental function is a period according to
Definition 4.1, or able to determine whether two periods, given by explicit
integrals, are equal or different. A number in $\bar{\mathbb{Q}}$ also admits
apparently different expressions, but those same techniques that work for
checking the equality of algebraic numbers do not in general work for periods.
In fact, two different periods may be numerically very close and yet be
distinct.393939For example, the approximation
$\pi=\frac{6}{\sqrt{3502}}\log(2u)+7.37\times 10^{-82}$, where $u$ is the
product of four quartic units, has been found by Shanks [63]. However, the
following conjecture is presented by Kontsevich and Zagier [47].

######  Conjecture 4.2.

If a period has two different integral representations, then one expression
can be transformed into the other by application of the three integral
transformation rules of additivity, change of variables, and Stokes’ formula,
in which all integrands and domains of integration are algebraic with
algebraic coefficients.

We note that even a proof of Conjecture 4.2 would not solve the additional
problem of finding an algorithm to determine whether or not two given numbers
in $\mathcal{P}$ are equal, or whether or not a given real number belongs to
$\mathcal{P}$. Another fundamental open problem in the theory of periods is to
explicitly exhibit one number which does not belong to $\mathcal{P}$. Such
numbers must exist, because $\mathcal{P}$ is a countable subset of
$\mathbb{C}$, but the concrete identification of one of such numbers has only
been proposed conjecturally. Indeed, the basis of natural logarithms $e$ and
the Euler–Mascheroni constant $\gamma$ are conjecturally not periods. Several
further questions on the arithmetic nature and transcendence of periods are
open or only conjecturally answered.404040See Waldschmidt [70] for an overview
of the topic.

Before moving to a more sophisticated definition of periods written in the
language of algebraic geometry, which is essential to subsequent developments,
we mention the fruitful interplay between the theory of periods and the theory
of linear differential equations. When the integrands or the domains of
integration depend on some set of parameters, the integrals, as functions of
these parameters, usually satisfy linear differential equations with algebraic
coefficients. The solutions of these differential equations generate periods
when evaluated at algebraic arguments. The differential equations occurring in
this way are called Picard–Fuchs differential equations. The relation between
periods and Picard–Fuchs equations has proved to be particularly productive in
the case of elliptic curves, hypergeometric functions, modular forms and
L-functions.

###  4.2 Algebra of motivic periods

The theory of periods is alternatively developed within the formalism of
algebraic geometry. We refer to Huber and Müller-Stach [43].

######  Definition 4.3.

Let $X$ be a smooth quasi-projective variety defined over $\bar{\mathbb{Q}}$,
and $Y\subset X$ a closed subvariety. A period is a complex number which can
be expressed as an integral of the form $\int_{\gamma}\omega\in\mathbb{C}$,
where $\omega$ is a closed algebraic differential $k$-form on $X$ vanishing on
$Y$, and $\gamma$ is a singular $k$-chain on the complex manifold $X^{\rm an}$
with boundary contained in $Y^{\rm an}$ for some integer $k\geq 0$.

The equivalence of Definitions 4.3 and 4.1 follows from the observation that
the algebraic chain $\gamma$ can be deformed to a semi-algebraic chain and
then broken up into small pieces, which can be bijectively projected onto open
domains in $\mathbb{R}^{n}$ with algebraic boundary. Without loss of
generality, we work with coefficients in $\mathbb{Q}$ instead of
$\bar{\mathbb{Q}}$. We note that, like Definition 4.1, Definition 4.3 also
contains redundancy. The integral $\int_{\gamma}\omega$ can be formally
decomposed into the quadruple

$\displaystyle(X,Y,\omega,\gamma)$ (4.11)

and different quadruples can give the same resulting number. To get rid of
this redundancy, the various forms of topological invariance of the integral
must be suitably accounted for. Following Stokes’ theorem, the integral is
insensitive to the individual cycle and form, being instead determined by the
homology and cohomology classes of these. Let us associate to $\omega$ its
cohomology class in the $k$-th algebraic de Rham cohomology group of $X$
relative to $Y$, and to $\gamma$ its homology class in the $k$-th Betti
homology group of $X$ relative to $Y$. Then, the first step towards a unique
algebraic description of periods consists of the following substitutions

$\displaystyle\omega\longrightarrow[\omega]\in H^{k}_{\rm
dR}(X,Y,\mathbb{Q}),$ (4.12) $\displaystyle\gamma\longrightarrow[\gamma]\in
H_{k}^{{\rm B}}(X,Y,\mathbb{Q})$ (4.13)

into the quadruple $(X,Y,\omega,\gamma)$. The problem of the coexistence of
distinct, but similarly behaved, cohomologies associated to the same variety,
which seems to imply an arbitrary choice here and in many other situations,
has been tackled by Grothendieck414141 Grothendieck proposed the notion of a
motive in a letter to Serre in 1964. He himself did not author any publication
on motives, although he mentioned them frequently in his correspondence. The
first formal expositions of the theory of motives are due to Demazure [30] and
Kleiman [46], who based their work on Grothendieck’s lectures on the topic.
[23] with the introduction of the theory of motives. He suggested that there
should exist a universal cohomology theory taking values in a
$\mathbb{Q}$-category of motives. The notion of a motive is thus proposed to
capture the intrinsic cohomological essence of a variety. Without delving into
the category-theoretic details of the theory of motives424242For a thorough
introduction to the theory of motives, we refer to Voevodsky [68], André [4],
Deligne and Goncharov [28], and Murre et al [51]., we give here a conceptual
introduction to its specific application to the theory of periods. Further
discussion on the fundamental features of motives, as they appear in the study
of periods, is presented in Section 5 in a more rigorous formalism. Recall
from Theorem 3.15 that there is a comparison isomorphism

$\displaystyle\mathop{\rm comp}\colon\ H^{k}_{\rm
dR}(X,Y,\mathbb{Q})\otimes_{\mathbb{Q}}\mathbb{C}\xlongrightarrow{\sim}H^{k}_{{\rm
B}}(X,Y,\mathbb{Q})\otimes_{\mathbb{Q}}\mathbb{C}$ (4.14)

induced by the pairing

$\displaystyle H^{k}_{\rm dR}(X,Y,\mathbb{Q})\times
H_{k}^{\text{s}}\big{(}X^{\rm an},Y^{\rm an},\mathbb{Q}\big{)}$
$\displaystyle\longrightarrow\mathbb{C},$ (4.15)
$\displaystyle([\omega],[\gamma])$
$\displaystyle\longmapsto\int_{\gamma}\omega.$ (4.16)

Momentarily neglecting the presence of filtrations for simplicity, the de Rham
and Betti system of realisations of $X$ relative to $Y$ in degree $k$ is

$\displaystyle H^{k}(X,Y)=\big{(}H^{k}_{\rm dR}(X,Y,\mathbb{Q}),H^{k}_{{\rm
B}}(X,Y,\mathbb{Q}),{\rm comp}\big{)}.$ (4.17)

In the same way that the cohomology class of a differential form singles out
its cohomological behaviour, the $H$-system of an algebraic variety
intuitively selects the content shared by its coexisting algebraic de Rham and
Betti cohomologies, and it filters out everything else. It is, therefore, a
first approximation towards the realisation of Grothendieck’s idea of a
motive. We define the motivic version of the period $\int_{\gamma}\omega$ as
the triple

$\displaystyle\big{[}H^{k}(X,Y),[\omega],[\gamma]\big{]}^{\rm m},$ (4.18)

where m in the superscript stands for motivic. We call a period in this guise
a motivic period. This has proved to be the most profitable reformulation of
the original notion of a period. However, a second source of redundancy in the
description of periods via the integral formulation in Definition 4.3,
corresponding to the same transformation rules in Conjecture 4.2, has yet to
be factored out.

######  Definition 4.4.

The space $\mathcal{P}^{\rm m}$ of motivic periods is defined as the
$\mathbb{Q}$-vector space434343In what follows, we no longer display the field
$\mathbb{Q}$ among the arguments of the cohomology groups for simplicity of
notation. generated by the symbols $[H^{\bullet}(X,Y),[\omega],[\gamma]]^{\rm
m}$ after factorisation modulo the following three equivalence relations:

  * (1)

Bilinearity. $[H^{\bullet}(X,Y),[\omega],[\gamma]]^{\rm m}$ is bilinear in
$[\omega]$ and $[\gamma]$.

  * (2)

Change of variables. If $f\colon(X_{1},Y_{1})\rightarrow(X_{2},Y_{2})$ is a
$\mathbb{Q}$-morphism of pairs of algebraic varieties, $[\gamma_{1}]\in
H^{{\rm B}}_{\bullet}(X_{1},Y_{1})$ and $[\omega_{2}]\in H^{\bullet}_{\rm
dR}(X_{2},Y_{2})$, then

$\displaystyle[H^{\bullet}(X_{1},Y_{1}),f^{*}[\omega_{2}],[\gamma_{1}]]^{\rm
m}=[H^{\bullet}(X_{2},Y_{2}),[\omega_{2}],f_{*}[\gamma_{1}]]^{\rm m},$ (4.19)

where $f^{*}$ and $f_{*}$ are the pull-back and the push-forward of $f$,
respectively.

  * (3)

Stokes’ formula. Assume for simplicity that $X$ is a smooth affine algebraic
variety over $\mathbb{Q}$ of dimension $d$ and $D\subset X$ is a simple normal
crossing divisor. The normalisation444444$\tilde{D}$ is the disjoint union of
the irreducible components of $D$. $\tilde{D}$ of $D$ contains a simple normal
crossing divisor $\tilde{D}_{1}$ coming from double points in $D$. If
$[\omega]\in H^{d-1}_{\rm dR}\big{(}\tilde{D},\tilde{D}_{1}\big{)}$ and
$[\gamma]\in H^{{\rm B}}_{d}(X,D)$, then

$\displaystyle[H^{d}(X,D),\delta[\omega],[\gamma]]^{\rm
m}=\big{[}H^{d-1}(\tilde{D},\tilde{D}_{1}),[\omega],\partial[\gamma]\big{]}^{\rm
m},$ (4.20)

where $\delta\colon H^{d-1}_{\rm
dR}\big{(}\tilde{D},\tilde{D}_{1}\big{)}\rightarrow H^{d}_{\rm dR}(X,D)$ is
the coboundary operator acting on the algebraic de Rham cohomology and
$\partial\colon H_{d}^{{\rm B}}(X,D)\rightarrow H_{d-1}^{{\rm
B}}\big{(}\tilde{D},\tilde{D}_{1}\big{)}$ is the boundary operator acting on
the Betti homology.

We observe that the space of motivic periods $\mathcal{P}^{\rm m}$ is
naturally endowed with an algebra structure. Indeed, new periods are obtained
by taking sums and products of known ones.

###  4.3 Period map

We call period map the evaluation homomorphism

$\displaystyle\mathop{\rm per}\nolimits\colon\ \mathcal{P}^{\rm m}$
$\displaystyle\longrightarrow\mathcal{P},$ (4.21)
$\displaystyle{}\big{[}H^{k}(X,Y),[\omega],[\gamma]\big{]}^{\rm m}$
$\displaystyle\longmapsto[\gamma]\circ\mathop{\rm
comp}\circ[\omega]=\int_{\gamma}\omega.$ (4.22)

Following the construction in Section 4.2, the period map is explicitly
surjective. Its injectivity is, on the other hand, not proven. Indeed, a
period has a unique motivic realisation only conjecturally. Conjecture 4.2 is
equivalent to the period conjecture below.

######  Conjecture 4.5.

The period map $\mathop{\rm per}\nolimits\colon\mathcal{P}^{\rm
m}\rightarrow\mathcal{P}$ is an isomorphism.

Let us briefly discuss the key idea underlying the period conjecture. A
$\mathbb{Q}$-morphism $f\colon$ $(X_{1},Y_{1})\rightarrow(X_{2},Y_{2})$
between two pairs of algebraic varieties induces a change of coordinates
between the corresponding algebraic de Rham cohomologies by pull-back, that is

${(X_{1},Y_{1})}$${H^{\bullet}_{\rm
dR}(X_{1},Y_{1})}$${(X_{2},Y_{2})}$${H^{\bullet}_{\rm
dR}(X_{2},Y_{2}).}$$\scriptstyle{f}$$\scriptstyle{f^{*}}$ (4.25)

The same morphism $f$ acts on the topological spaces of complex points
underlying the given algebraic varieties, and it induces a change of
coordinates between the corresponding singular homologies by push-forward,
that is

${\big{(}X_{1}^{\rm an},Y_{1}^{\rm
an}\big{)}}$${H_{\bullet}^{\text{s}}\big{(}X_{1}^{\rm an},Y_{1}^{\rm
an}\big{)}}$${\big{(}X_{2}^{\rm an},Y_{2}^{\rm
an}\big{)}}$${H_{\bullet}^{\text{s}}\big{(}X_{2}^{\rm an},Y_{2}^{\rm
an}\big{)}.}$$\scriptstyle{f}$$\scriptstyle{f_{*}}$ (4.28)

By means of such changes of coordinates, one can easily derive two distinct
integral representations of the same period. For example, taking
$[\gamma_{1}]\in H_{\bullet}^{\text{s}}\big{(}X_{1}^{\rm an},Y_{1}^{\rm
an}\big{)}$ and $[\omega_{2}]\in H^{\bullet}_{\rm dR}(X_{2},Y_{2})$, we have

$\displaystyle\int_{[\gamma_{1}]}f^{*}[\omega_{2}]=\int_{f_{*}[\gamma_{1}]}[\omega_{2}].$
(4.29)

The corresponding two motivic representations of the same period

$\displaystyle[H^{\bullet}(X_{1},Y_{1}),f^{*}[\omega_{2}],[\gamma_{1}]]^{\rm
m},\qquad[H^{\bullet}(X_{2},Y_{2}),[\omega_{2}],f_{*}[\gamma_{1}]]^{\rm m}$
(4.30)

could a priori be different motivic periods. However, they are identified with
each other by change of variables. Indeed, the period conjecture corresponds
to the statement that, whenever different motivic representations of the same
period arise, they can always be interrelated by the three equivalence
relations in Definition 4.4.

######  Definition 4.6.

Let $X$ be a smooth quasi-projective $\mathbb{Q}$-variety, $Y\subset X$ a
closed subvariety, and $H=H^{\bullet}(X,Y)$ the $H$-system of $X$ relative to
$Y$. Assume that $\\{[\omega_{j}]\\}_{j=1}^{n}$ is a basis of the algebraic de
Rham cohomology $H^{\bullet}_{\rm dR}(X,Y)$, and that
$\\{[\gamma_{i}]\\}_{i=1}^{n}$ is a basis of the Betti homology
$H_{\bullet}^{{\rm B}}(X,Y)$. We denote by $\mathop{\rm per}\nolimits|_{H}$
the period map restricted to the motivic periods in $\mathcal{P}^{\rm m}$ that
are built on the given Hodge structure $H$. Observe that $\mathop{\rm
per}\nolimits|_{H}$ is fully determined by the values that it takes when
evaluated at $[H,[\omega_{j}],[\gamma_{i}]]^{\rm m}$, which are

$\displaystyle\mathop{\rm
per}\nolimits|_{H}([H,[\omega_{j}],[\gamma_{i}]]^{\rm
m})=\int_{\gamma_{i}}\omega_{j}$ (4.31)

for each pair of indices $(i,j)$ with $i,j=1,\dots,n$. We define the period
matrix of $H$ as the $n\times n$-matrix with complex entries
$(p_{ij})_{i,j=1,\dots,n}$ given by

$\displaystyle p_{ij}=\int_{\gamma_{i}}\omega_{j}.$ (4.32)

The period matrix expresses in a different guise the same information
contained in the period map, once it has been restricted to a specific
$H$-system.

######  Remark 4.7.

For a given mixed $H$-system $H=(H_{\rm dR},\,H_{{\rm B}},\,\mathop{\rm
comp})$, there is a canonical choice of bases on $H_{\rm dR}$ and $H^{{\rm
B}}$ which is compatible454545See Brown [15] for details. with the comparison
isomorphism and with the Hodge and weight filtrations in $H$. We often
implicitly assume to work in the canonical bases when writing the period
matrix. Let us denote by $\\{e_{i}\\}_{i=1}^{n}$ and $\\{f_{i}\\}_{i=1}^{n}$
the canonical bases on the de Rham and Betti realisations of $H$,
respectively. For $i=1,\dots,n$, the action of the comparison isomorphism on
the $i$-th element of the canonical basis of $H_{\rm dR}$ is given by

$\displaystyle\mathop{\rm comp}\colon\ H_{\rm
dR}\otimes_{\mathbb{Q}}\mathbb{C}$ $\displaystyle\xlongrightarrow{\sim}H_{{\rm
B}}\otimes_{\mathbb{Q}}\mathbb{C},$ (4.33) $\displaystyle e_{i}\otimes
2\pi{\rm i}$ $\displaystyle\longmapsto f_{i}^{\vee}\otimes 2\pi{\rm i},$
(4.34)

where $f_{i}^{\vee}$ denotes the standard vector dual basis element of $f_{i}$
on $H_{{\rm B}}$. For $i,j=1,\dots,n$, the pairing map gives

$\displaystyle H_{\rm dR}\times H^{{\rm B}}$
$\displaystyle\longrightarrow\mathbb{C},$ (4.35) $\displaystyle(e_{j},f_{i})$
$\displaystyle\longmapsto\int_{f_{i}}e_{j}=p_{ij}.$ (4.36)

For $i=1,\dots,n$, we define the vector dual $e_{i}^{\vee}$ of the basis
element $e_{i}$ to be $e_{i}^{\vee}=f_{i}$. Observe that, since we cannot
easily make sense of a notion of de Rham homology, the dual of a basis of
$H_{\rm dR}$ is defined to be a basis of $H^{{\rm B}}$.

######  Example 4.8.

Let $H=H^{1}(\mathbb{G}_{m},\\{1,z\\})$ with
$z\in\mathbb{Q}\backslash\\{0,1\\}$. As shown in Examples 3.7 and 3.14, a
basis of the Betti homology group $H_{1}^{{\rm
B}}(\mathbb{G}_{m},\\{1,z\\})\simeq
H^{\text{s}}_{1}(\mathbb{C}^{*},\\{1,z\\})$ is given by $[\gamma_{1}]$, where
$\gamma_{1}$ is a continuous oriented map from $1$ to $z$ which does not
encircle the origin, and $[\gamma_{2}]$, where $\gamma_{2}$ is a
counterclockwise cycle encircling the origin. A basis of the algebraic de Rham
cohomology group $H^{1}_{\rm dR}(\mathbb{G}_{m},\\{1,z\\})$ is given by
$[\omega_{1}]=\big{[}\frac{{\rm d}x}{z-1}\big{]}$ and
$[\omega_{2}]=\big{[}\frac{{\rm d}x}{x}\big{]}$. Such a choice of bases is
indeed canonical, and the period matrix of $H$ is

$\displaystyle\begin{pmatrix}1&\log(z)\\\ 0&2\pi{\rm i}\end{pmatrix}\\!.$
(4.37)

###  4.4 Examples

####  4.4.1 Motivic $\boldsymbol{2\pi{\rm i}}$

The period $2\pi{\rm i}$ is given by the contour integral

$\displaystyle 2\pi{\rm i}=\oint\displaylimits_{\gamma_{2}}\frac{{\rm
d}x}{x},$ (4.38)

where $\gamma_{2}$ is a counterclockwise cycle encircling the origin in the
punctured complex plane $\mathbb{C}^{*}$. As observed in Example 3.5, the
complex manifold $\mathbb{C}^{*}$ is isomorphic to the topological space of
complex points $\mathbb{G}_{m}^{\rm an}$ underlying the algebraic variety
$\mathbb{G}_{m}$ over $\mathbb{Q}$. As shown in Examples 3.2 and 3.11, we have
that

$\displaystyle H_{1}^{{\rm B}}(\mathbb{G}_{m})=\mathbb{Q}[\gamma_{2}],\qquad
H^{1}_{\rm dR}(\mathbb{G}_{m})=\mathbb{Q}\bigg{[}\frac{{\rm d}x}{x}\bigg{]}.$
(4.39)

Recalling that $H^{1}(\mathbb{G}_{m})=\big{(}H^{1}_{\rm
dR}(\mathbb{G}_{m}),H^{1}_{{\rm B}}(\mathbb{G}_{m}),{\rm comp}\big{)}$, a
motivic version of $2\pi{\rm i}$ is

$\displaystyle(2\pi{\rm i})^{\rm
m}=\bigg{[}H^{1}(\mathbb{G}_{m}),\bigg{[}\frac{{\rm
d}x}{x}\bigg{]},[\gamma_{2}]\bigg{]}^{\rm m},$ (4.40)

which is alternatively represented by the pairing

$\displaystyle H^{1}_{\rm dR}(\mathbb{G}_{m})\times H_{1}^{{\rm
B}}(\mathbb{G}_{m})$ $\displaystyle\longrightarrow\mathbb{C},$ (4.41)
$\displaystyle\bigg{(}\bigg{[}\frac{{\rm d}x}{x}\bigg{]},[\gamma_{2}]\bigg{)}$
$\displaystyle\longmapsto\oint_{\gamma_{2}}\frac{{\rm d}x}{x}=2\pi{\rm i}.$
(4.42)

A second integral representation of $2\pi{\rm i}$ is given by

$\displaystyle 2\pi{\rm i}=\int_{\mathbb{P}^{1}(\mathbb{C})}\frac{{\rm
d}z\wedge{\rm d}\bar{z}}{(1+z\bar{z})^{2}},$ (4.43)

where $\frac{{\rm d}z\wedge{\rm d}\bar{z}}{(1+z\bar{z})^{2}}$ is a closed
algebraic $2$-form over the projective manifold $\mathbb{P}^{1,\text{an}}$.
Because $\mathbb{P}^{1,\text{an}}$ is compact and Kähler, Theorem 3.23
applies, giving the Hodge decomposition

$\displaystyle H^{2}_{\rm
dR}(\mathbb{P}^{1})\otimes_{\mathbb{Q}}\mathbb{C}=\bigoplus_{p+q=2}H^{p,q}_{{\rm
D}}\big{(}\mathbb{P}^{1},\mathbb{C}\big{)}$ (4.44)

which implies that the pure $H$-system $H^{2}(\mathbb{P}^{1})$ has weight 2.
Recalling that the differential forms in $H^{p,q}_{{\rm D}}$ contain $p$
copies of the differential ${\rm d}z$ and $q$ copies of the anti-holomorphic
differential ${\rm d}\bar{z}$, we have that $\big{[}\frac{{\rm d}z\wedge{\rm
d}\bar{z}}{(1+z\bar{z})^{2}}\big{]}\in H^{1,1}_{{\rm
D}}\big{(}\mathbb{P}^{1},\mathbb{C}\big{)}$. Therefore, the integral (4.43)
corresponds to the motivic period464646Note that we are here using the
intuitive definition of algebraic de Rham cohomology of non-affine varieties
given in Section 3.2.1. Although $\frac{{\rm d}z\wedge{\rm
d}\bar{z}}{(1+z\bar{z})^{2}}$ is not a global algebraic $2$-form on
$\mathbb{P}^{1}(\mathbb{C})$, and indeed there are no non-zero global
algebraic $2$-forms on $\mathbb{P}^{1}(\mathbb{C})$ for dimension reasons, one
can still rigorously make sense of $2\pi{\rm i}$ as a period of
$H^{2}(\mathbb{P}^{1})$ via the Čeck construction mentioned in Section 3.2.1,
that is, choosing a Zariski open affine covering of $\mathbb{P}^{1}$ and
computing the algebraic de Rham cohomology as a hypercohomology of sheaves.

$\displaystyle(2\pi{\rm i})^{\rm
m}=\bigg{[}H^{2}(\mathbb{P}^{1}),\bigg{[}\frac{{\rm d}z\wedge{\rm
d}\bar{z}}{(1+z\bar{z})^{2}}\bigg{]},\left[\mathbb{P}^{1,\text{an}}\right]\bigg{]}^{\rm
m}.$ (4.45)

######  Remark 4.9.

The two apparently different motivic periods in (4.40) and (4.45) are the
same, thus preserving the period conjecture. To show this, define

$\displaystyle
A=\mathbb{P}^{1,\text{an}}\backslash\\{\infty\\}\cong\mathbb{C}\subset\mathbb{P}^{1,\text{an}},\qquad
B=\mathbb{P}^{1,\text{an}}\backslash\\{0\\}\cong\mathbb{C}\subset\mathbb{P}^{1,\text{an}}$
(4.46)

which satisfy the relations

$\displaystyle A\cap B\simeq\mathbb{C}^{*}\simeq\mathbb{G}_{m}^{\rm an},\qquad
A\cup B=\mathbb{P}^{1,\text{an}}.$ (4.47)

By the Mayer–Vietoris theorem applied to the singular homology groups, the
following long exact sequence holds red!0 red!70!black

${0}$${H^{\text{s}}_{0}(A\cup B)}$${H^{\text{s}}_{0}(A)\oplus
H^{\text{s}}_{0}(B)}$${\underbrace{H^{\text{s}}_{1}(A)\oplus
H^{\text{s}}_{1}(B)}_{\simeq 0}}$${H^{\text{s}}_{1}(A\cup
B)}$${H^{\text{s}}_{0}(A\cap B)}$${H^{\text{s}}_{1}(A\cap
B)}$${H^{\text{s}}_{2}(A\cup B)}$${\underbrace{H^{\text{s}}_{2}(A)\oplus
H^{\text{s}}_{2}(B)}_{\simeq 0}.}$ (4.51)

Hence, the step $H^{\text{s}}_{1}(A\cap B)\rightarrow H^{\text{s}}_{2}(A\cup
B)$ is an isomorphism, giving

$\displaystyle H^{\text{s}}_{1}(\mathbb{G}_{m}^{\rm an})\simeq
H^{\text{s}}_{2}\big{(}\mathbb{P}^{1,\text{an}}\big{)}.$ (4.52)

Similarly, one can prove that the whole $H$-systems $H^{1}(\mathbb{G}_{m})$
and $H^{2}(\mathbb{P}^{1})$ are isomorphic and that the change of coordinates
occurring between them relates the cohomology classes $\big{[}\frac{{\rm
d}z\wedge{\rm d}\bar{z}}{(1+z\bar{z})^{2}}\big{]}$ and $\big{[}\frac{{\rm
d}x}{x}\big{]}$ and the homology classes $[\gamma_{0}]$ and
$\left[\mathbb{P}^{1,\text{an}}\right]$ via pull-back and push-forward maps,
respectively.

####  4.4.2 Motivic $\boldsymbol{\log(z)}$

Recall the integral representation of $\log(z)$,
$z\in\mathbb{Q}\backslash\\{0,1\\}$, given by

$\displaystyle\log(z)=\int_{1}^{z}\frac{{\rm d}x}{x}.$ (4.53)

As in the case of $2\pi{\rm i}$, this is an integral over the punctured
complex plane $\mathbb{C}^{*}=\mathbb{G}_{m}^{\rm an}$. However, contrary to
the case of $2\pi{\rm i}$, where the integration path $\gamma_{2}$ is closed,
integral (4.53) is performed on an open path, precisely any continuous
oriented path $\gamma_{1}\subset\mathbb{C}^{*}$ starting at $1$ and ending at
$z$, which does not encircle the origin. The integration path being open
requires us to work in the framework of relative homology. Let
$\mathbb{G}_{m}$ be the ambient variety. Then, $\mathbb{C}^{*}$ is the
underlying topological space, and $\\{1,z\\}\subset\mathbb{C}^{*}$ with
$z\in\mathbb{Q}\backslash\\{0,1\\}$ is a simple normal crossing divisor. As
shown in Examples 3.7 and 3.14, we have

$\displaystyle H_{1}^{{\rm
B}}(\mathbb{G}_{m},\\{1,z\\})=\mathbb{Q}[\gamma_{1},\gamma_{2}],\qquad
H^{1}_{\rm dR}(\mathbb{G}_{m},\\{1,z\\})=\mathbb{Q}\bigg{[}\frac{{\rm
d}x}{z-1},\frac{{\rm d}x}{x}\bigg{]}.$ (4.54)

Observe that we can write $\big{[}\big{(}\frac{{\rm
d}x}{z-1},0,0\big{)},\big{(}\frac{{\rm
d}x}{x},0,0\big{)}\big{]}=\big{[}\frac{{\rm d}x}{z-1},\frac{{\rm
d}x}{x}\big{]}$ as a consequence of Proposition 3.13. Setting as usual
$H^{1}(\mathbb{G}_{m},\\{1,z\\})=\big{(}H^{1}_{\rm
dR}(\mathbb{G}_{m},\\{1,z\\}),H^{1}_{{\rm B}}(\mathbb{G}_{m},\\{1,z\\}),{\rm
comp}\big{)}$, a motivic version of $\log(z)$ is

$\displaystyle\log(z)^{\rm
m}=\bigg{[}H^{1}(\mathbb{G}_{m},\\{1,z\\}),\bigg{[}\frac{{\rm
d}x}{x}\bigg{]},[\gamma_{1}]\bigg{]}^{\rm m}$ (4.55)

which is alternatively represented by the pairing

$\displaystyle H^{1}_{\rm dR}(\mathbb{G}_{m},\\{1,z\\})\times H_{1}^{{\rm
B}}(\mathbb{G}_{m},\\{1,z\\})$ $\displaystyle\longrightarrow\mathbb{C},$
(4.56) $\displaystyle\bigg{(}\bigg{[}\frac{{\rm
d}x}{x}\bigg{]},[\gamma_{1}]\bigg{)}$
$\displaystyle\longmapsto\int_{\gamma_{1}}\frac{{\rm d}x}{x}=\log(z).$ (4.57)

####  4.4.3 Elementary relations

Elementary relations among periods are often simply recast in the formalism of
motivic periods. In fact, de Rham and Betti systems of realisations
conjecturally capture all algebraic relations among periods.

######  Example 4.10.

For $a,b\in\mathbb{Q}\backslash\\{0,1\\}$, such that $ab\neq 1$, consider the
following injective morphisms of pairs of $\mathbb{Q}$-spaces:

$\displaystyle(\mathbb{G}_{m},\\{1,a\\})\hookrightarrow(\mathbb{G}_{m},\\{b,ab\\})\hookrightarrow(\mathbb{G}_{m},\\{1,b,ab\\}),$
(4.58)
$\displaystyle(\mathbb{G}_{m},\\{1,b\\})\hookrightarrow(\mathbb{G}_{m},\\{1,b,ab\\}),$
(4.59)
$\displaystyle(\mathbb{G}_{m},\\{1,ab\\})\hookrightarrow(\mathbb{G}_{m},\\{1,b,ab\\}).$
(4.60)

Since the differential form $\frac{{\rm d}x}{x}$ is invariant under rescaling
of $x$, we have the motivic representations

$\displaystyle\log(a)^{\rm
m}=\bigg{[}H^{1}(\mathbb{G}_{m},\\{1,b,ab\\}),\bigg{[}\frac{{\rm
d}x}{x}\bigg{]},[b,ab]\bigg{]}^{\rm m},$ (4.61) $\displaystyle\log(b)^{\rm
m}=\bigg{[}H^{1}(\mathbb{G}_{m},\\{1,b,ab\\}),\bigg{[}\frac{{\rm
d}x}{x}\bigg{]},[1,b]\bigg{]}^{\rm m},$ (4.62) $\displaystyle\log(ab)^{\rm
m}=\bigg{[}H^{1}(\mathbb{G}_{m},\\{1,b,ab\\}),\bigg{[}\frac{{\rm
d}x}{x}\bigg{]},[1,ab]\bigg{]}^{\rm m},$ (4.63)

where $[z,w]$ denotes the class of a continuous oriented map in
$\mathbb{C}^{*}$ which goes from $z$ to $w$ and it does not encircle the
origin in the Betti homology group $H_{1}^{{\rm
B}}(\mathbb{G}_{m},\\{1,b,ab\\})$ for $z,w\in\\{1,b,ab\\}$. Additivity of the
Betti homology classes $[b,ab]\cup[1,b]=[1,ab]$ implies that motivic
logarithms satisfy the expected relation

$\displaystyle\log(ab)^{\rm m}=\log(a)^{\rm m}+\log(b)^{\rm m}.$ (4.64)

######  Example 4.11.

Consider $H=H^{1}(\mathbb{G}_{m},\\{1,z\\})$ for
$z\in\mathbb{Q}\backslash\\{0,1\\}$. Let $\gamma$ be the union of the paths
$\gamma_{1}$ and $\gamma_{2}$ in the punctured complex plane, as shown in Fig.
11.

$\gamma$$\gamma_{2}$$0$$1$$\gamma_{1}$$z$

Figure 11: The paths $\gamma_{1}$, $\gamma_{2}$, and $\gamma$ in
$\mathbb{C}^{*}$.

The period obtained by integrating $\omega_{2}$ along $\gamma$ is

$\displaystyle\int_{\gamma}\omega_{2}=\int_{\gamma_{1}\cup\gamma_{2}}\omega_{2}=\int_{\gamma_{1}}\omega_{2}+\int_{\gamma_{2}}\omega_{2}=\log(z)+2\pi{\rm
i},$ (4.65)

which translates into the following relation among motivic periods

$\displaystyle\begin{split}(\log(z)+2\pi{\rm i})^{\rm
m}&=\left[H,[\omega_{2}],[\gamma]\right]^{\rm
m}=\left[H,[\omega_{2}],[\gamma_{1}\cup\gamma_{2}]\right]^{\rm m}\\\
&=\left[H,[\omega_{2}],[\gamma_{1}]\right]^{\rm
m}+\left[H,[\omega_{2}],[\gamma_{2}]\right]^{\rm m}\\\ &=\log(z)^{\rm
m}+(2\pi{\rm i})^{\rm m},\end{split}$ (4.66)

where we have used the additivity of the Betti homology classes and the
injective morphism $\mathbb{G}_{m}\hookrightarrow(\mathbb{G}_{m},\\{1,z\\})$.
Because $(\log(z)+2\pi{\rm i})^{\rm m}\in\mathop{\rm
per}\nolimits^{-1}(\log(z)+2\pi{\rm i})$ and $\log(z)^{\rm m}+(2\pi{\rm
i})^{\rm m}\in\mathop{\rm per}\nolimits^{-1}(\log(z))+\mathop{\rm
per}\nolimits^{-1}(2\pi{\rm i})$, it follows that

$\displaystyle\mathop{\rm per}\nolimits^{-1}(\log(z)+2\pi{\rm
i})\cap\big{(}\mathop{\rm per}\nolimits^{-1}(\log(z))+\mathop{\rm
per}\nolimits^{-1}(2\pi{\rm i})\big{)}\neq\varnothing.$ (4.67)

Note that $\mathop{\rm per}\nolimits^{-1}(\log(z)+2\pi{\rm i})=\mathop{\rm
per}\nolimits^{-1}(\log(z))+\mathop{\rm per}\nolimits^{-1}(2\pi{\rm i})$ only
holds conjecturally.

Moreover, many new functional equations among motivic periods are found by
means of the more abstract, and yet more powerful formalism that we discuss in
Section 5. By the period conjecture, new relations among motivic periods
automatically translates into new algebraic relations among the corresponding
numbers.

##  5 Feynman motives

###  5.1 Singularities and the blow up

Multiple zeta values and convergent Feynman integrals are periods by means of
the integral representations (2.26) and (2.20), respectively. In both cases,
singularities of the integrand can be contained in the domain of integration,
a feature that does not occur in the examples of $2\pi{\rm i}$ and $\log(z)$.
Whenever singularities are present, they have to be taken care of with
particular attention.

######  Example 5.1.

The period $\zeta(2)$ is given by the integral

$\displaystyle\zeta(2)=\int\displaylimits_{1\geq x_{1}\geq x_{2}\geq
0}\frac{{\rm d}x_{1}}{x_{1}}\wedge\frac{{\rm d}x_{2}}{1-x_{2}}$ (5.1)

over the complex manifold $\mathbb{C}^{2}$. The domain of integration is the
simplex

$\displaystyle\sigma=\\{(x_{1},x_{2})\in\mathbb{C}^{2}\,|\,1\geq x_{1}\geq
x_{2}\geq 0\\}$ (5.2)

and the integrand is the differential $2$-form

$\displaystyle\omega=\frac{{\rm d}x_{1}}{x_{1}}\wedge\frac{{\rm
d}x_{2}}{1-x_{2}}.$ (5.3)

Observing that $\mathbb{C}^{2}$ is isomorphic to the topological space of
complex points $\mathbb{A}^{2}(\mathbb{C})$, underlying the affine474747For
any positive integer $n$, the $n$-dimensional affine variety over $\mathbb{Q}$
is defined as $\mathbb{A}^{n}=\mathop{\rm Spec}\mathbb{Q}[x_{1},\dots,x_{n}]$.
For any field extension $\mathbb{K}\supseteq\mathbb{Q}$, the space of
$\mathbb{K}$-points of $\mathbb{A}^{n}$ is
$\mathbb{A}^{n}(\mathbb{K})=\mathbb{K}^{n}$. The multiplicative group
$\mathbb{G}_{m}=\mathop{\rm Spec}\mathbb{Q}[x,\frac{1}{x}]$ satisfies
$\mathbb{G}_{m}=\mathop{\rm
Spec}\mathbb{Q}[x_{1},x_{2}]/(1-x_{1}x_{2})=\mathbb{A}^{1}\backslash\\{0\\}\subset\mathbb{A}^{2}$,
that is, $\mathbb{G}_{m}$ is a hyperbola in $\mathbb{A}^{2}$.
$\mathbb{Q}$-algebraic variety $\mathbb{A}^{2}=\mathop{\rm
Spec}\mathbb{Q}[x_{1},x_{2}]$, we may try to build $\zeta(2)^{\rm m}$ as we
did for the examples in Section 4.4. Consider the lines $l_{0}=\\{x_{1}=0\\}$
and $l_{1}=\\{x_{2}=1\\}$ in the affine plane $\mathbb{A}^{2}$. Since
$L=l_{0}\cup l_{1}$ is the locus of singular points of $\omega$, the latter is
an algebraic $2$-form on $X=\mathbb{A}^{2}\backslash L$. Thus, $[\omega]$ is a
class in the second algebraic de Rham cohomology group of $X$ and,
consequently, we may want to consider the integral (5.1) as a period of $X$
relative to some divisor containing the boundary of $\sigma$. In an attempt to
do so, define the simple normal crossing divisor

$\displaystyle
D=\\{x_{1}=x_{2}\\}\cup\\{x_{1}=1\\}\cup\\{x_{2}=0\\}\subset\mathbb{A}^{2}$
(5.4)

containing $\partial\sigma$. Note that $D$ is not in $X$ because $D\cap
L\neq\varnothing$. However, the divisor $D\backslash(D\cap L)\subset X$ does
no longer contain $\partial\sigma$. The problem arises from the fact that
$\sigma$ itself is not contained in $X$, intersecting the singular locus $L$
in two points

$\displaystyle p=(0,0)=\sigma\cap l_{0}=D\cap l_{0},\qquad q=(1,1)=\sigma\cap
l_{1}=D\cap l_{1}.$ (5.5)

Removing the singular points $p$, $q$ from $D$ and considering the second
relative $H$-system $H^{2}(X,D$ $\backslash(D\cap L))$ does not solve the
mentioned technical issue, because $[\sigma]$ is not a class in $H_{2}^{{\rm
B}}(X,D\backslash(D\cap L))$. See Fig. 12.

$D$$q$$p$$l_{0}$$l_{1}$$\sigma$

Figure 12: Construction of $\zeta(2)^{\rm m}$ in the affine plane
$\mathbb{A}^{2}$.

The example of $\zeta(2)$ shows how direct removal of singular points
explicitly fails and motivates a more articulated geometric construction,
called blow up, which proves to be successful in the case of $\zeta(2)$ and
many more examples. Graphically, we may illustrate the procedure as the
removal of a whole region of space centred at the singularity and the
corresponding reshaping of the integration domain. See Fig. 13 for a
qualitative representation of how the blow up of the two singular points
$p,q\in\mathbb{A}^{2}$ acts on $\sigma$ in the case of $\zeta(2)$.

(a)

(b)

Figure 13: Qualitative illustration of the blow up of the singular points of
$\zeta(2)$.

###  5.2 Motivic multiple zeta values

Consider $\zeta(2)$ again. The blow up of the affine plane $\mathbb{A}^{2}$
along the singular points $p$, $q$ is defined as the closed subvariety

$\displaystyle Y=\mathop{\rm
Blow}_{p,q}\big{(}\mathbb{A}^{2}\big{)}\subset\mathbb{A}^{2}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$
(5.6)

given by the equations

$\displaystyle
x_{1}\alpha_{1}=x_{2}\beta_{1},\qquad(x_{1}-1)\alpha_{2}=(x_{2}-1)\beta_{2},$
(5.7)

where $[\alpha_{i}:\beta_{i}]$, $i=1,2$, are homogeneous coordinates on the
two copies of $\mathbb{P}^{1}$. The projection of $Y$ onto the first factor in
$\mathbb{A}^{2}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ is the proper
surjective map

$\displaystyle\pi\colon\ Y$ $\displaystyle\longrightarrow\mathbb{A}^{2},$
(5.8)
$\displaystyle(x_{1},x_{2})\times[\alpha_{1}:\beta_{1}]\times[\alpha_{2}:\beta_{2}]$
$\displaystyle\longmapsto(x_{1},x_{2}).$ (5.9)

Let us write $\pi^{-1}$ to denote the inverse image operator484848Observe that
$\pi^{-1}$ is not a map defined on the affine plane $\mathbb{A}^{2}$ because
$\pi$ is not invertible. under the projection $\pi$. The inverse images of the
singular points $p,q\in\mathbb{A}^{2}$ are the projective lines
$E_{p},E_{q}\subset Y$, called exceptional divisors. Precisely, we have

$\displaystyle\pi^{-1}(p)=\pi^{-1}(0,0)=(0,0)\times\mathbb{P}^{1}\times[1:1]=E_{p},$
(5.10)
$\displaystyle\pi^{-1}(q)=\pi^{-1}(1,1)=(1,1)\times[1:1]\times\mathbb{P}^{1}=E_{q}.$
(5.11)

Moreover, the restriction of $\pi$ to the complement in $Y$ of the exceptional
divisors $E_{p}$, $E_{q}$

$\displaystyle\pi|_{Y\backslash(E_{p}\cup E_{q})}\colon\ Y\backslash(E_{p}\cup
E_{q})$ $\displaystyle\longrightarrow\mathbb{A}^{2}\backslash\\{p,q\\},$
(5.12) $\displaystyle(x_{1},x_{2})\times[1:1]\times[1:1]$
$\displaystyle\longmapsto(x_{1},x_{2})$ (5.13)

is an isomorphism. For any closed subset $C\subset\mathbb{A}^{2}$, the inverse
image $\pi^{-1}(C)$ is called total transform of $C$. The strict transform of
$C$, denoted $\hat{C}$, is instead the closed subset of $Y$ obtained by first
removing the points $p,q$ if they belong to $C$, then taking the inverse image
under $\pi$, and finally taking the Zariski closure, that is

$\displaystyle\hat{C}=\overline{\pi^{-1}(C\backslash\\{p,q\\})}\subseteq\pi^{-1}(C).$
(5.14)

It follows that the strict transforms of $l_{0},l_{1}$ are the affine lines

$\displaystyle
L_{0}=\hat{l}_{0}=\big{\\{}(0,x_{2})\times[1:0]\times[1-x_{2}:1]\,|\,x_{2}\in\mathbb{A}^{1}\big{\\}},$
(5.15) $\displaystyle
L_{1}=\hat{l}_{1}=\big{\\{}(x_{1},1)\times[1:x_{1}]\times[0:1]\,|\,x_{1}\in\mathbb{A}^{1}\big{\\}}$
(5.16)

and their total transforms are

$\displaystyle\pi^{-1}(l_{0})=L_{0}\cup E_{p},\qquad\pi^{-1}(l_{1})=L_{1}\cup
E_{q}.$ (5.17)

We observe that $L_{0}$, $E_{p}$ and $L_{1},E_{q}$ intersect in only one point
each. Precisely

$\displaystyle L_{0}\cap E_{p}=\\{(0,0)\times[1:0]\times[1:1]\\},$ (5.18)
$\displaystyle L_{1}\cap E_{q}=\\{(1,1)\times[1:1]\times[0:1]\\}.$ (5.19)

Moreover, we have

$\displaystyle L_{1}\cap E_{p}=\varnothing=L_{0}\cap E_{q},$ (5.20)
$\displaystyle L_{1}\cap L_{0}=\\{(0,1)\times[1:0]\times[0:1]\\}.$ (5.21)

In a similar way to (5.14), but taking the closure in the analytic topology,
we define the strict transform $\hat{\sigma}$ of the domain of integration.
Observing that the closed points of $E_{p}$ can be interpreted as lines
passing through $p$, and analogously that the closed points of $E_{q}$ can be
interpreted as lines passing through $q$, we obtain

$\displaystyle\hat{\sigma}\cap E_{p}=\\{(0,0)\times[t:1]\times[1:1]\,|\,0\leq
t\leq 1\\},$ (5.22) $\displaystyle\hat{\sigma}\cap
E_{q}=\\{(1,1)\times[1:1]\times[1:t]\,|\,0\leq t\leq 1\\}$ (5.23)

which, combined with (5.19), imply that

$\displaystyle\hat{\sigma}\cap L_{0}=\varnothing,\qquad\hat{\sigma}\cap
L_{1}=\varnothing.$ (5.24)

See Fig. 14 for a graphical representation of the blow up.

$L_{0}$$L_{1}$$\hat{\sigma}$$E_{q}$$E_{p}$

Figure 14: The strict transform of $\sigma$ in the blow up $Y$.

While the inverse image $\pi^{-1}$ is applied to the ambient variety, giving
the reshaped domain $\hat{\sigma}$, the differential form $\omega$ is replaced
by its pull-back $\pi^{*}(\omega)$, denoted by $\hat{\omega}$. Let us now show
that the pull-back $\hat{\omega}$ is only singular494949In principle,
$\hat{\omega}$ might have singularities along the total transform of
$l_{0}\cup l_{1}$, i.e., $L_{0}\cup L_{1}\cup E_{p}\cup E_{q}$. However, in
the case of $\zeta(2)$, it turns out that $\hat{\omega}$ has no singularities
along the exceptional divisors. More generally, this condition determines
whether the blow up prescription turns out to be successful or not for a given
period. on the strict transform $L=L_{0}\cup L_{1}$. We use local coordinates
on the blow up $Y$. In particular, consider a patch of $Y$ around the point
$L_{0}\cap E_{p}$ as shown in Fig. 15.

$L_{0}$$E_{p}$

Figure 15: Local patch of $Y$ around the intersection of $L_{0}$ and $E_{p}$.

Here, a local system of coordinates is explicitly given by

$\displaystyle t=\frac{x_{1}}{x_{2}}=\frac{\beta_{1}}{\alpha_{1}},\qquad
s=x_{2},$ (5.25)

where $L_{0}$ and $E_{p}$ have equations $t=0$ and $s=0$, respectively.
Applying this change of variables to $\hat{\omega}$, we have

$\displaystyle\hat{\omega}=\frac{{\rm d}(st)}{st}\wedge\frac{{\rm
d}s}{1-s}=\frac{{\rm d}s}{s}\wedge\frac{{\rm d}s}{1-s}+\frac{{\rm
d}t}{t}\wedge\frac{{\rm d}s}{1-s}=\frac{{\rm d}t}{t}\wedge\frac{{\rm
d}s}{1-s}.$ (5.26)

It follows that $\hat{\omega}$ is singular along the strict transform $L_{0}$,
while it is smooth along the exceptional divisor $E_{p}$, because it has no
pole at $s=0$. Analogously, we find that $\hat{\omega}$ is singular along
$L_{1}$, but not along $E_{q}$. Then, the singular locus of $\hat{\omega}$ is
$L$. Observe that the complement $Y\backslash L$ is the closed affine
subvariety of $\mathbb{A}^{2}\times\mathbb{A}^{1}\times\mathbb{A}^{1}$ given
by the equations

$\displaystyle x_{1}t=x_{2},\qquad x_{1}-1=(x_{2}-1)s,$ (5.27)

where $t$, $s$ are affine coordinates on the two copies of $\mathbb{A}^{1}$.
Therefore, the differential form $\hat{\omega}$ determines a class in
$H^{2}_{\rm dR}(Y\backslash L)$. Moreover, it follows from (5.24) that, moving
from the original affine plane $\mathbb{A}^{2}$ to the blow up $Y$, the
singular locus of the differential form $\hat{\omega}$ and the domain of
integration $\hat{\sigma}$ are disjoint. As usual, we may want to consider the
integral (5.1) as a period of $Y\backslash L$ relative to some divisor
containing the boundary of $\hat{\sigma}$. The blow up construction is thus
successful for the period $\zeta(2)$ if $[\hat{\sigma}]$ turns out to be a
class in the given relative Betti homology group. To see this, recall that
$\partial\sigma$ is contained in the union $D$ of the affine lines

$\displaystyle m_{1}=\\{x_{1}=x_{2}\\},\qquad m_{2}=\\{x_{1}=1\\},\qquad
m_{3}=\\{x_{2}=0\\}.$ (5.28)

Thus, we naturally consider the normal crossing divisor $M\subset Y$ defined
by

$\displaystyle M=\pi^{-1}(D)=\pi^{-1}(m_{1}\cup m_{2}\cup m_{3})=E_{p}\cup
E_{q}\cup M_{1}\cup M_{2}\cup M_{3},$ (5.29)

where $M_{i}=\hat{m_{i}}$ denotes the strict transform of $m_{i}$ for
$i=1,2,3$. Note that $L\cap M$ is the union of the points $L_{0}\cap E_{p}$
and $L_{1}\cap E_{q}$ expressed in (5.19). Therefore, $\hat{\sigma}$ is
contained in $Y\backslash L$ and $\partial\hat{\sigma}$ is contained in
$M\backslash(M\cap L)\subset Y\backslash L$, implying

$\displaystyle[\hat{\sigma}]\in H_{2}^{{\rm B}}(Y\backslash
L,M\backslash(M\cap L)).$ (5.30)

Besides, the restriction of $\hat{\omega}$ to every irreducible component
$M_{i}$, $i=1,2,3$, of $M$ gives zero, implying

$\displaystyle[\hat{\omega}]\in H^{2}_{\rm dR}(Y\backslash L,M\backslash(M\cap
L)).$ (5.31)

Setting $H=H^{2}(Y\backslash L,M\backslash(M\cap L))$, the resulting motivic
version of $\zeta(2)$ is

$\displaystyle\zeta(2)^{\rm
m}=\left[H,[\hat{\omega}],[\hat{\sigma}]\right]^{\rm m}.$ (5.32)

Indeed, the pairing of $[\hat{\sigma}]$ and $[\hat{\omega}]$ yields

$\displaystyle\int_{\hat{\sigma}}\hat{\omega}=\int_{\hat{\sigma}}\pi^{*}(\omega)=\int_{\pi_{*}(\hat{\sigma})}\omega=\int_{\sigma}\omega=\zeta(2)$
(5.33)

by the equivalence relation under change of variables in $\mathcal{P}^{\rm
m}$. Moreover, the whole canonical period matrix of $H$ is

$\displaystyle\begin{pmatrix}1&\zeta(2)\\\\[4.30554pt] 0&(2\pi{\rm
i})^{2}\end{pmatrix}\\!.$ (5.34)

###  5.3 Motivic Feynman integrals

In an attempt to overcome singularity issues, the blow up procedure can be
similarly applied to generic MZVs and other families of periods, such as
convergent Feynman integrals. For an exposition of the general computation of
the $H$-system of a blow up we refer to Voisin [69].