title:
Flips from 4-folds with isolated complete intersection singularities
  whose downstairs have rational bi-elephants

abstract:
  We shall investigate a flipping contraction g : X -> Y from a 4-fold X with
at most isolated complete intersection singularities. If Y has an
anti-bi-canonical divisor (=bi-elephant) with only rational singularities, then
g carries an inductive structure chained up by blow-ups (La Torre Pendente),
and in particular the flip exists. This naturally contains Miles Reid's
`Pagoda' as an anti-canonical divisor (=elephant) and its proper transforms.


title: Lengths of Periods and Seshadri Constants of Abelian Varieties

abstract:
  The purpose of this note is to point out an elementary but somewhat
surprising connection between the work of Buser and Sarnak on lengths of
periods of abelian varieties and the Seshadri constants measuring the local
positivity of theta divisors. The link is established via symplectic blowing
up, in the spirit of McDuff and Polterovich. As an application of the main
inequality, we get a simple new proof of a statement of Buser-Sarnak type to
the effect that the Jacobian of a curve has a period of unusually small length.
We also deduce a lower bound on the Seshadri constant of a very general
p.p.a.v. which differs from the maximum possible by a factor of less than four.


title: Syzygies of Abelian and Bielliptic Surfaces in P^4

abstract:
  So far only six families of smooth irregular surfaces are known to exist in
P^4 (up to pullbacks by suitable finite covers of P^4). These are the elliptic
quintic scrolls, the minimal abelian and bielliptic surfaces (of degree 10),
two different families of non-minimal abelian surfaces of degree 15, and one
family of non-minimal bielliptic surfaces of degree 15.
  The main purpose of the paper is to describe the structure of the
Hartshorne-Rao modules and the syzygies for each of these smooth irregular
surfaces in P^4, providing at the same time a unified construction method (via
syzygies) for these families of surfaces.


title: Composition law and Nodal genus-2 curves in P^2

abstract:
  Recently, there has been great interest in the application of composition
laws to problems in enumerative geometry. Using the moduli space of stable
maps, we compute the number of irreducible, reduced, nodal, degree-$d$
genus-$2$ plane curves whose normalization has a fixed complex structure and
which pass through $3d - 2$ general points in $\Bbb P^2$.


title: On a relative version of the Krichever correspondence

abstract:
  For a given base scheme, a correspondence is established between a class of
sheaves on curves over this base scheme and certain points of infinite
Grassmannians. This equivalence extends to a characterization of commutative
algebras of ordinary differential operators with coefficients in the ring of
formal power series over a given $k$-algebra. Our construction generalizes the
approach of M.Mulase, which gives the above connection in the case that the
base scheme is one closed point.


title: Recent developments in toric geometry

abstract:
  This paper will appear in the Proceedings of the 1995 Santa Cruz Summer
Institute. The paper is a survey of recent developments in the theory of toric
varieties, including new constructions of toric varieties and relations to
symplectic geometry, combinatorics and mirror symmetry.


title: Positivite et discretion des points algebriques des courbes

abstract:
  We prove the discreteness of algebraic points (with respect to the Neron-Tate
height) on a curve of genus greater than one embedded in his jacobian. This
result was conjectured by Bogomolov. We also prove the positivity of the self
intersection of the admissible dualizing sheaf.


title: Gale Duality and Free Resolutions of Ideals of Points

abstract:
  What is the shape of the free resolution of the ideal of a general set of
points in P^r? This question is central to the programme of connecting the
geometry of point sets in projective space with the structure of the free
resolutions of their ideals. There is a lower bound for the resolution
computable from the (known) Hilbert function, and it seemed natural to
conjecture that this lower bound would be achieved. This is the ``Minimal
Resolution Conjecture'' (Lorenzini [1987], [1993]). Hirschowitz and Simpson
[1994] showed that the conjecture holds when the number of points is large
compared with r, but three examples (with r = 6,7,8) discovered computationally
by Schreyer in 1993 show that the conjecture fails in general. We describe a
novel structure inside the free resolution of a set of points which accounts
for the observed failures and provides a counterexample in P^r for every r\geq
6, r\neq 9. The geometry behind our construction occurs not in P^r but in a
different projective space, in which there is a related set of points, the
``Gale transform'' (or ``associated set'', in the sense of Coble.)


title: Non-Hermitian Yang-Mills connections

abstract:
  We study Yang-Mills connections on holomorphic bundles over complex K\"ahler
manifolds of arbitrary dimension, in the spirit of Hitchin's and Simpson's
study of flat connections. The space of non-Hermitian Yang-Mills (NHYM)
connections has dimension twice the space of Hermitian Yang-Mills connections,
and is locally isomorphic to the complexification of the space of Hermitian
Yang-Mills connections (which is, by Uhlenbeck and Yau, the same as the space
of stable bundles). Further, we study the NHYM connections over hyperk\"ahler
manifolds. We construct direct and inverse twistor transform from NHYM bundles
on a hyperk\"ahler manifold to holomorphic bundles over its twistor space. We
study the stability and the modular properties of holomorphic bundles over
twistor spaces, and prove that work of Li and Yau, giving the notion of
stability for bundles over non-K\"ahler manifolds, can be applied to the
twistors. We identify locally the following two spaces: the space of stable
holomorphic bundles on a twistor space of a hyperk\"ahler manifold and the
space of rational curves in the twistor space of the ``Mukai dual''
hyperk\"ahler manifold.


title: Quantum cohomology of flag manifolds G/B and quantum Toda lattices

abstract:
  We prove the Givental conjecture that the (equivariant) quantum cohomology of
flag manifolds G/B are governed by the conservation law of Toda lattices. In
addition, we find the quantum differential module structure of the flag
manifolds.


title: A relative notion of algebraic Lie group and applications to $n$-stacks

abstract:
  If $S$ is a scheme of finite type over $k=\cc $, let $\Xx /S$ denote the big
etale site of schemes over $S$. We introduce {\em presentable group sheaves}, a
full subcategory of the category of sheaves of groups on $\Xx /S$ which is
closed under kernel, quotient, and extension. Group sheaves which are
representable by group schemes of finite type over $S$ are presentable;
pullback and finite direct image preserve the notions of presentable group
sheaves; over $S=Spec (k)$ then presentable group sheaves are just group
schemes of finite type over $Spec(k)$; there is a notion of connectedness
extending the usual notion over $Spec(k)$; and a presentable group sheaf $G$
has a Lie algebra object $Lie(G $. If $G$ is a connected presentable group
sheaf then $G/Z(G)$ is determined up to isomorphism by the Lie algebra sheaf
$Lie (G)$.
  We envision the category of presentable group sheaves as a generalisation
relative to an arbitrary base scheme $S$, of the category of algebraic Lie
groups over $Spec (k)$.
  The notion of presentable group sheaf is used in order to define {\em
presentable $n$-stacks} over $\Xx$. Roughly, an $n$-stack is presentable if
there is a surjection from a scheme of finite type to its $\pi_0$ (the actual
condition on $\pi_0$ is slightly more subtle), and if its $\pi_i$ (which are
sheaves on various $\Xx /S$) are presentable group sheaves. The notion of
presentable $n$-stack is closed under homotopy fiber product and truncation.
  We propose the notion of presentable $n$-stack as an answer in characteristic
zero for A. Grothendieck's search for what he called ``schematization of
homotopy types''.


title: A remark on the jet bundles over the projective line

abstract:
  This is a footnote of a recent interesting work of Cohen, Manin and Zagier,
where they, among other things, produce a natural isomorphism between the sheaf
of (n-1)-th order jets of the n-th tensor power of the tangent bundle of a
Riemann surface equipped with a projective structure and the sheaf of
differential operators of order n (on the trivial bundle) with vanishing 0-th
order part. We give a different proof of this result without using the
coordinates, and following the idea of this proof we prove:
  Take a line bundle L with $L^2 = T$ on a Riemann surface equipped with a
projective structure. Then the jet bundle $J^n(L^n)$ has a natural flat
connection with $J^n(L^n) = S^n(J^1(L))$. For any $m >n$ the obvious surjection
$J^m(L^n) \rightarrow J^n(L^n)$ has a canonical splitting. In particular,
taking $m = n+1$, one gets a natural differential operator of order $n+1$ from
$L^n$ to $L^{-n-2}$.


title: Mapping Class Groups and Moduli Spaces of Curves

abstract:
  This is a survey paper that also contains some new results. It will appear in
the proceedings of the AMS summer research institute on Algebraic Geometry at
Santa Cruz.


title: The orbifold fundamental group of Persson-Noether-Horikawa surfaces

abstract:
  The Noether-Horikawa surfaces are the minimal surfaces S with K^2=2p_g-4. For
8 | K^2 they belong to two families of respective type C and N (connected,
resp. non connected branch locus for the canonical map). For 16 | K^2 the two
types are homeomorphic. Ulf Persson constructed surfaces of type N with a
maximally singular canonical model X, whose topology encodes information on the
differentiable structure of S. A similar analysis was done by the first author
for type C. In this paper we study the genus 2 fibration on X and, in
particular, our main result is (X^# being the nonsingular locus of X)
\pi_1(X^#)= Z_4 x Z_4 if 8 | K^2 but 16 does not | K^2 \pi_1(X^#)= Z_4 x Z_2 if
16 | K^2.


title: Plane curves with a big fundamental group of the complement

abstract:
  Let $C \s \pr^2$ be an irreducible plane curve whose dual $C^* \s \pr^{2*}$
is an immersed curve which is neither a conic nor a nodal cubic. The main
result states that the Poincar\'e group $\pi_1(\pr^2 \se C)$ contains a free
group with two generators. If the geometric genus $g$ of $C$ is at least 2,
then a subgroup of $G$ can be mapped epimorphically onto the fundamental group
of the normalization of $C$, and the result follows. To handle the cases
$g=0,1$, we construct universal families of immersed plane curves and their
Picard bundles. This allows us to reduce the consideration to the case of
Pl\"ucker curves. Such a curve $C$ can be regarded as a plane section of the
corresponding discriminant hypersurface (cf. [Zar, DoLib]). Applying
Zariski--Lefschetz type arguments we deduce the result from `the bigness' of
the $d$-th braid group $B_{d,g}$ of the Riemann surface of $C$.


title:
On the algebraic dimension of twistor spaces over the connected sum of
  four complex projective planes

abstract:
  We study the algebraic dimension of twistor spaces of positive type over
$4\bbfP^2$. We show that such a twistor space is Moishezon if and only if its
anticanonical class is not nef. More precisely, we show the equivalence of
being Moishezon with the existence of a smooth rational curve having negative
intersection number with the anticanonical class. Furthermore, we give precise
information on the dimension and base locus of the fundamental linear system
$|{-1/2}K|$. This implies, for example, $\dim|{-1/2}K|\leq a(Z)$. We
characterize those twistor spaces over $4\bbfP^2$, which contain a pencil of
divisors of degree one by the property $\dim|{-1/2}K| = 3$.


title: Conjugacy classes in loop groups and G-bundles on elliptic curves

abstract:
  We added an additional result (theorem 1.6) that strengthenns our main
theorem in the G=GL-case by establishing an equivalence of tensor categories.


title: Contractible Extremal Rays on \overline{M}_{0,n}

abstract:
  We consider the cones of curves and divisors on the moduli space of stable
pointed rational curves,M_n, and on the quotient by the symmetric group, Q_n,
which is a moduli space of pairs. We find generators for contractible extremal
rays of the cone of curves NE_1(M_n), and for the cone of divisors NE^1(Q_n).
This second cone turns out to be simplicial. We give complete descriptions of
NE_1(M_n) and NE_1(Q_n) for small n (< 8 in the first case, < 11 in the
second). We also have results of independent interest on when curves in a
divisor generate the cone of curves of the ambient variety.


title:
Equivalence de la th\'eorie homotopique des $n$-groupo\"{i}des et celle
  des espaces topologiques $n$-tronqu\'es

abstract:
  This paper constitutes a recent work using the constructions of a previous
preprint alg-geom/9512006 to show that the functors geometric realisation and
Poincar\'e $n$-groupoid induce an equivalence between the category of
$n$-grouppoids and the category of $n$-truncated topological spaces, when we
localise both categories by weak equivalence.


title: Chern classes for singular hypersurfaces

abstract:
  We prove a simple formula for MacPherson's Chern class of hypersurfaces in
nonsingular varieties. The result highlights the relation between MacPherson's
class and other definitions of homology Chern classes of singular varieties,
such as Mather's Chern class and a class introduced by W. Fulton.


title: Quantum integrable systems and differential Galois theory

abstract:
  The purpose of this paper is to connect two subjects: the theory of quantum
integrable systems (complete commutative rings of differential operators), and
differential Galois theory. We define quantum completely integrable systems
(QCIS), algebraically integrable QCIS, the differential Galois group of a QCIS.
We show that the differential Galois group is always reductive and that a QCIS
is algebraically integrable if and only if its differential Galois group is
commutative. In particular, we show that a differential operator L in one
variable is algebraic in the sense of Krichever (i.e. finite-zone) if and only
if the differential Galois group of the differential equation Lf=af is
commutative for a generic number a. As a by-product, we obtain a proof of the
Veselov-Chalyh conjecture on the algebraic integrability of the elliptic
Calogero-Moser system.

categories: alg-geom dg-ga math.AG math.DG math.QA q-alg


title:
A lower bound for $K_{X}L$ of quasi-polarized surfaces $(X,L)$ with
  non-negative Kodaira dimension

abstract:
  Let $X$ be a smooth projective surface over the complex number field and let
$L$ be a nef-big divisor on $X$. Here we consider the following conjecture; If
the Kodaira dimension $\kappa(X)\geq 0$, then $K_{X}L\geq 2q(X)-4$, where
$q(X)$ is the irregularity of $X$. In this paper, we prove that this conjecture
is true if (1) the case in which $\kappa(X)=0$ or 1, (2) the case in which
$\kappa(X)=2$ and $h^{0}(L)\geq 2$, or (3) the case in which $\kappa(X)=2$, $X$
is minimal, $h^{0}(L)=1$, and $L$ satisfies some conditions.


title: Torsion Points on an Algebraic Subset of an Affine Torus

abstract:
  Work of Laurent and Sarnak, following a conjecture of Lang, shows that the
number of torsion points of order n on an algebraic subset of an affine complex
torus is polynomial periodic. In this paper, we find bounds on the degree and
period of this number as a function of n. Some examples, including the number
of n torsion points on Fermat curves, are computed to illustrate the methods.


title: Master Spaces for stable pairs

abstract:
  We construct master spaces for oriented torsion free sheaves coupled with
morphisms into a fixed reference sheaf. These spaces are projective varieties
endowed with a natural $\C^*$-action. The fixed point set of this action
contains the moduli space of semistable oriented torsion free sheaves and the
quot scheme associated with the given data.
  In the case of curves with trivial reference sheaf, our master spaces
compactify the moduli spaces constructed by Bertram, Daskalopoulos and
Wentworth. In the 2-dimensional case with trivial rank 1 reference sheaf,
master spaces provide algebraic analoga of compactified moduli spaces of
twisted quaternionic monopoles.


title: Low degree polynomial equations: arithmetic, geometry and topology

abstract:
  These are the notes of my lectures at the 1996 European Congress of
Mathematicians. {} Polynomials appear in mathematics frequently, and we all
know from experience that low degree polynomials are easier to deal with than
high degree ones. It is, however, not clear that there is a well defined class
of "low degree" polynomials. For many questions, polynomials behave well if
their degree is low enough, but the precise bound on the degree depends on the
concrete problem. {} It turns out that there is a collection of basic questions
in arithmetic, algebraic geometry and topology all of which give the same class
of "low degree" polynomials. The aim of this lecture is to explain these
properties and to provide a survey of the known results.


title:
On imaginary plane curves and Spin quotients of complex surfaces by
  complex conjugation

abstract:
  It is proven that for any topological or analytical types of isolated
singular points of plane curves, there exists a non-real irreducible plane
algebraic curve of degree $d$ which goes through $d^2$ real distinct points and
has imaginary singular points of the given types. This result is used to
construct a series of examples of complex algebraic surfaces $X$ defined over
$\R$ whose quotients $Y=X/\conj$ by the complex conjugation $\conj$ are $Spin$
simply connected 4-manifolds with signature $16k$, for arbitrary integer $k>0$.


title:
Torelli groups and Jacobian varieties of non-orientable compact Klein
  surfaces

abstract:
  The Torelli group of a compact non-orientable Klein surface is the subgroup
of the modular group consisting of the mapping classes that act trivially on
the first homology group of the surface. We prove that if a surface has genus
at least $3$, then the Torelli group acts fixed points free on the
Teichm\"{u}ller space of the surface. That gives an embedding of the Torelli
space of a Klein surface in the Torelli space of its complex double. We also
construct real tori associated to Klein surfaces, which we call the Jacobian of
the surface. We prove that this Jacobian is isomorphic to a component of the
real part of the Jacobian of the complex double.


abstract:
  The Hodge de Rham theory of relative Malcev completion is developed in this
paper. In the special case where one takes the corresponding reductive group to
be trivial, one recovers Chen's de Rham theory of the fundamental group and the
corresponding Hodge theory due to Morgan and the author. This work is a
principal technical tool in the author's work on the mapping class groups.


title: Filtered Perverse Complexes

abstract:
  We introduce the notion of filtered perversity of a filtered differential
complex on a complex analytic manifold $X$, without any assumptions of
coherence, with the purpose of studying the connection between the pure Hodge
modules and the \lt-complexes. We show that if a filtered differential complex
$(\cM^\bullet,F_\bullet)$ is filtered perverse then
$\aDR(\cM^\bullet,F_\bullet)$ is isomorphic to a filtered $\cD$-module; a
coherence assumption on the cohomology of $(\cM^\bullet,F_\bullet)$ implies
that, in addition, this $\cD$-module is holonomic. We show the converse: the de
Rham complex of a holonomic Cohen-Macaulay filtered $\cD$-module is filtered
perverse.


title: Embeddings of curves and surfaces

abstract:
  We prove a general embedding theorem for Cohen--Macaulay curves (possibly
nonreduced), and deduce a cheap proof of the standard results on pluricanonical
embeddings of surfaces, assuming vanishing H^1(2K_X)=0.


title: Ample divisors on the blow up of P^3 at points

abstract:
  We give a condition for certain divisors on the blow up of P^3 at points in
general position to be ample. The result extends a theorem of G. Xu on the blow
up of the projective plane.


title: Motives of uniruled 3-folds

abstract:
  We construct projectors in the ring of correspondences of a complex uniruled
3-fold $X$ which lift the Kuenneth components of the diagonal in singular
cohomology and have other properties which were conjectured by J. Murre. Such
Murre decompositions have been already obtained for curves, surfaces, abelian
varieties and varieties with cell decompositions by the work of Manin,
Shermenev, Beauville, Murre et.al.. In particular they define a natural
filtration on the Chow groups of $X$ which was conjectured by Bloch and
Beilinson. To do this we use Mori theory and construct projectors in the
situation of a fiber space over a surface. These projectors may also be used in
more general situations.


title: The Equivariant Chow Ring of SO(4)

abstract:
  The integral equivariant Chow ring of S0(4) is computed via the geometry of
ruled quadric surfaces in P^3.


title: The Chow Ring of the Hilbert Scheme of Rational Normal Curves

abstract:
  Let H(d) be the (open) Hilbert scheme of rational normal curves of degree d
in P^d. A presentation of the integral Chow ring of H(d) is given via
equivariant Chow ring computations. Included also in the paper are algebraic
computations of the integral equivariant Chow rings of the algebraic groups
O(n), SO(2k+1). The results for S0(3)=PGL(2) are needed for the Hilbert scheme
calculation.


title: Projective structures on a Riemann surface

abstract:
  For a compact Riemann surface $X$ of any genus $g$, let $L$denote the line
bundle $K_{X\times X}\otimes {\cal O}_{X\times X}(2\Delta)$ on $X\times X$,
where $K_{X\times X}$ is the canonical bundle of $X\times X$ and $\Delta$ is
the diagonal divisor. We show that $L$ has a canonical trivialisation over the
nonreduced divisor $2\Delta$. Our main result is that the space of projective
structures on $X$ is canonically identified with the space of all
trivialisations of $L$ over $3\Delta$ which restrict to the canonical
trivialisation of $L$ over $2\Delta$ mentioned above. We give a direct
identification of this definition of a projective structure with a definition
of Deligne.We also describe briefly the origin of this work in the study of the
so-called "Sugawara form" of the energy-momentum tensor in a conformal quantum
field theory.


title:
The Braid Monodromy of Plane Algebraic Curves and Hyperplane
  Arrangements

abstract:
  To a plane algebraic curve of degree n, Moishezon associated a braid
monodromy homomorphism from a finitely generated free group to Artin's braid
group B_n. Using Hansen's polynomial covering space theory, we give a new
interpretation of this construction. Next, we provide an explicit description
of the braid monodromy of an arrangement of complex affine hyperplanes, by
means of an associated ``braided wiring diagram.'' The ensuing presentation of
the fundamental group of the complement is shown to be Tietze-I equivalent to
the Randell-Arvola presentation. Work of Libgober then implies that the
complement of a line arrangement is homotopy equivalent to the 2-complex
modeled on either of these presentations. Finally, we prove that the braid
monodromy of a line arrangement determines the intersection lattice. Examples
of Falk then show that the braid monodromy carries more information than the
group of the complement, thereby answering a question of Libgober.


title: The Picard group of the moduli of G-bundles on a curve

abstract:
  Let G be a complex semi-simple group, and X a compact Riemann surface. The
moduli space of principal G-bundles on X, and in particular the holomorphic
line bundles on this space and their global sections, play an important role in
the recent applications of Conformal Field Theory to algebraic geometry. In
this paper we determine the Picard group of this moduli space when G is of
classical or G_2 type (we consider both the coarse moduli space and the moduli
stack).


title: A relation between standard conjectures and their arithmetic analogues

abstract:
  In this paper we resolve the arithmetic analogues of standard conjectures for
an arithmetic variety, which are proposed by Gillet and Soule, into original
standard conjectures and similar conjectures for two cycle class groups. One is
the homologically trivial cycles. This is well-known as conjectures about the
height pairing raised by Beilinson. The other is the cycle class groups
consisting of vertical cycles. We show that the conjectures for the above two
cycle class groups with Grothendieck's standard conjectures imply their
arithmetic analogues.


title: Mirror Symmetry via 3-tori for a class of Calabi-Yau Threefolds

abstract:
  We give an example of the recent proposed mirror construction of Strominger,
Yau and Zaslow in ``Mirror Symmetry is T-duality,'' hep-th/9606040. The paper
first considers mirror symmetry for K3 surfaces in light of this construction.
We then consider the example of mirror symmetry for Calabi-Yau threefolds of
the type considered by Voisin and Borcea, of the form SxE/involution where S is
a K3 surface with involution, and E is an elliptic curve. We show how dualizing
a family of special Lagrangian real 3-tori does actually produce the mirrors in
these examples.


title: Residues and duality for Cousin complexes

abstract:
  We construct a canonical pseudofunctor ^# on the category of finite-type maps
of (say) connected noetherian universally catenary finite-dimensional separated
schemes, taking values in the category of Cousin complexes. This pseudofunctor
is a concrete approximation to the restriction of the Grothendieck Duality
pseudofunctor ^! to the full subcategory of the derived category having
Cohen-Macaulay complexes as objects (a subcategory equivalent to the category
of Cousin complexes, once a codimension function has been fixed). Specifically,
for Cousin complexes M and any scheme map f:X -> Y as above, there is a
functorial derived-category map \gamma: f^# M -> f^! M inducing a functorial
isomorphism in the category of Cousin complexes f^# M \iso E(f^! M) (where E is
the Cousin functor). \gamma itself is an isomorphism if the complex f^! M is
Cohen-Macaulay--which will be so whenever the map f is flat or whenever the
complex M is injective. Also, f^# takes residual (resp. injective) complexes on
Y to residual (resp. injective) complexes on X; and so the pseudofunctor ^#
generalises--and makes canonical--the "variance theory" of residual complexes
developed in Chapter VI of Hartshorne's "Residues and Duality."
  Moreover, we generalise the Residue Theorem of loc.cit., p.369 by defining a
functorial Trace map of graded modules Tr_f(M): f_*f^# M -> M (a sum of local
residues) such that whenever f is proper, Tr_f(M) is a map of complexes and the
pair (f^# M, Tr_f(M)) represents the functor Hom(f_*C, M) of Cousin complexes
C.


title: The Geometry Underlying Mirror Symmetry

abstract:
  The recent result of Strominger, Yau and Zaslow relating mirror symmetry to
the quantum field theory notion of T-duality is reinterpreted as providing a
way of geometrically characterizing which Calabi-Yau manifolds have mirror
partners. The geometric description---that one Calabi-Yau manifold should serve
as a compactified, complexified moduli space for special Lagrangian tori on the
other Calabi-Yau manifold---is rather surprising. We formulate some precise
mathematical conjectures concerning how these moduli spaces are to be
compactified and complexified, as well as a definition of geometric mirror
pairs (in arbitrary dimension) which is independent of those conjectures. We
investigate how this new geometric description ought to be related to the
mathematical statements which have previously been extracted from mirror
symmetry. In particular, we discuss how the moduli spaces of the `mirror'
Calabi-Yau manifolds should be related to one another, and how appropriate
subspaces of the homology groups of those manifolds could be related. We treat
the case of K3 surfaces in some detail.


title: On the general elephant conjecture for Mori conic bundles

abstract:
  Let $f:X\to S$ be an extremal contraction from a threefolds with terminal
singularities onto a surface (so called Mori conic bundle). We study some
particular cases of such contractions: quotients of usual conic bundles and
index two contractions. Assuming Reid's general elephants conjecture we also
obtain a rough classification. We present many examples.


title: Vanishing Theorems and Syzygies for K3 Surfaces and Fano Varieties

abstract:
  In this article we prove some strong vanishing theorems on K3 surfaces. As an
aplication of them, we obtain higher syzygy results for K3 surfaces and Fano
varieties.


title: Residues in intersection homology and L_p-cohomology

abstract:
  Suppose $M^{n+1}$ is a complex manifold and K is a hypersurface with isolated
singularities. Let $\omega$ be a holomorphic form on $M\setminus K$ with the
first order pole on K. The Leray residue of such form gives an element in the
n-th homology of K which is the Alexander dual to $[\omega ]\in
H^{n+1}(M\setminus K)$. It always lifts to the intersection homology if 0 does
not belong to the spectra of a singular points. We assume that singularities
are described by the quasihomogeneous equations in certain coordinate systems.
Suppose that oscillation indicators of the singular points are greater then -1.
Then we find a metric on $K\setminus\Sigma$ in which the residue form is square
integrable (and even L_p-integrable for p>2). Applying the isomorphism of
L_p-cohomology and intersection homology we obtain a particular lift of the
residue class in homology to intersection homology.


title: Lower Bounds for diophantine Approximation

abstract:
  We introduce a subexponential algorithm for geometric solving of multivariate
polynomial equation systems whose bit complexity depends mainly on intrinsic
geometric invariants of the solution set. From this algorithm, we derive a new
procedure for the decision of consistency of polynomial equation systems whose
bit complexity is subexponential, too. As a byproduct, we analyze the division
of a polynomial modulo a reduced complete intersection ideal and from this, we
obtain an intrinsic lower bound for the logarithmic height of diophantine
approximations to a given solution of a zero--dimensional polynomial equation
system. This result represents a multivariate version of Liouville's classical
theorem on approximation of algebraic numbers by rationals. A special feature
of our procedures is their {\em polynomial} character with respect to the
mentioned geometric invariants when instead of bit operations only arithmetic
operations are counted at unit cost. Technically our paper relies on the use of
straight--line programs as a data structure for the encoding of polynomials, on
a new symbolic application of Newton's algorithm to the Implicit Function
Theorem and on a special, basis independent trace formula for affine Gorenstein
algebras.


title: Notes on stable maps and quantum cohomology

abstract:
  These are notes from a jointly taught class at the University of Chicago and
lectures by the first author in Santa Cruz. Topics covered include:
construction of moduli spaces of stable maps, Gromov-Witten invariants, quantum
cohomology, and examples. These notes will appear in the proceedings of the
1995 Santa Cruz conference.


title: Compactified jacobians

abstract:
  Let J be the jacobian of a reduced projective curve C with nodes only. 1) We
give a simple and natural definition for its many compactifications and show
the connection with various other definitions appearing in the literature. 2)
Among all compactifications we choose one canonical, and define a theta divisor
on it. 3) We give two very explicit and simple descriptions of a stratification
of this canonical compactification into homogeneous spaces over J.


title: Log canonical singularities and complete moduli of stable pairs

abstract:
  1) Assuming log Minimal Model Conjecture, we give a construction of a
complete moduli space of stable log pairs of arbitrary dimension generalizing
directly the space M_{g,n} of pointed stable curves. Each stable pair has semi
log canonical singularities. 2) We prove that a stable quasiabelian pair,
defined by author and I.Nakamura as the limit of abelian varieties with theta
divisors, has semi log canonical singularities.


title: On Mumford's construction of degenerating abelian varieties

abstract:
  We prove that a 1-dimnl family of abelian varieties with an ample sheaf
defining principal polarization can be canonically compactified (after a finite
base change) to a projective family with an ample sheaf. We show that the
central fiber (P,L), which we call an SQAV, has a canonical Cartier theta
divisor. We give a combinatorial definition for SQAVs and describe their
geometrical properties, in particular compute cohomologies of L^n, n\ge0.


title: Complete intersections and rational equivalence

abstract:
  Two cycles on a projective variety over an algebraically closed field are
shown to be rationally equivalent if and only if their difference equals a
difference of complete intersections of a certain kind. Some of Bloch's
conjectures for zero-cycles on surfaces can be restated more geometrically,
which leads to several questions.


title: On the Thomae formula for $Z_N$ curves

abstract:
  We shall give an elementary and rigorous proof of the Thomae formula for
${\bf Z}_N$ curves which was discovered by Bershadsky and Radul. Instead of
using the determinant of the Laplacian we use the traditional variational
method which goes back to Riemann, Thomae, Fuchs. In the proof we made explicit
the algebraic expression of the chiral Szeg\"{o} kernels and proves the
vanishing of zero values of derivatives of theta functions with ${\bf Z}_N$
invariant $1/2N$ characteristics.


title: Balanced Varieties

abstract:
  After the work of Bloch and Srinivas on correspondences and algebraic cycles
we begin the study of a birational class of algebraic varieties determined by
the property that a multiple of the diagonal is rationally equivalent to a
cycle supported on proper subschemes.


title: Rank Two Bundles on Algebraic Curves and Decoding of Goppa Codes

abstract:
  We study a connection between two topics: Decoding of Goppa codes arising
from an algebraic curve, and rank two extensions of certain line bundles on the
curve.


title: The genus of curves on the three dimensional quadric

abstract:
  By means of an {\it ad hoc} modification of the so-called
``Castelnuovo-Harris analysis" we derive an upper bound for the genus of
integral curves on the three dimensional nonsingular quadric which lie on an
integral surface of degree $2k$, as a function of $k$ and the degree $d$ of the
curve. In order to obtain this we revisit the Uniform Position Principle to
make its use computation-free. The curves which achieve this bound can be
conveniently characterized.
  Keywords: Castelnuovo-Harris Bound, Uniform Position Principle, Low
Codimension, Linkage


title:
A finiteness theorem for low-codimensional nonsingular subvarieties of
  quadrics

abstract:
  We prove that there are only finitely many families of codimension two
nonsingular subvarieties of quadrics $\Q{n}$ which are not of general type, for
$n=5$ and $n\geq 7$. We prove a similar statement also for the case of higher
codimension. The case $n=6$ has been recently settled by Fania-Ottaviani.
  Keywords: Codimension two, Grassmannians, Lifting, Low codimension, Not of
General Type, Polynomial Bound, Quadrics


title:
Codimension two nonsingular subvarieties of quadrics: scrolls and
  classification in degree $d\leq 10$

abstract:
  Let $X$ be a codimension two nonsingular subvariety of a nonsingular quadric
$\Q{n}$ of dimension $n\geq 5$. We classify such subvarieties when they are
scrolls. We also classify them when the degree $d\leq 10$. Both results were
known when $n=4$.
  Keywords: Classification; liaison; low codimension; low degree; quadric;
scroll; vector bundle


title:
Some adjunction-theoretic properties of codimension two nonsingular
  subvarieties of quadrics

abstract:
  We make precise the structure of the first two reduction morphisms associated
with codimension two nonsingular subvarieties of quadrics $\Q{n}$, $n\geq 5$.
We give a coarse classification of the same class of subvarieties when they are
assumed to be not of log-general-type.
  Keywords: Adjunction Theory, classification, codimension two, conic bundles,
low codimension, non log-general-type, quadric, reduction, special variety


title: Enumerating curves on rational surfaces: the rational fibration method

abstract:
  A new, simple method to approach enumerative questions about rational curves
on rational surfaces is described. Applications include a short proof of
Kontsevich's formula for plane curves and a the solution of the analogous
problem for the Hirzebruch surface F_3.


title:
Parameter spaces for curves on surfaces and enumeration of rational
  curves

abstract:
  We answer some enumerative questions about irreducible rational curves on
Hirzebruch surfaces, by combining an idea of Kontsevich with the study of the
geometry of certain natural parameter spaces. Our formulas generalize
Kontsevich's formula for rational plane curves of given degree.


title: Counting plane curves of any genus

abstract:
  We obtain a recursive formula answering the following question: How many
irreducible, plane curves of degree d and (geometric) genus g pass through
3d-1+g general points in the plane? The formula is proved by studying suitable
degenerations of plane curves.


title:
The intersection form in the cohomology of the moduli space of genus 0
  $n$-pointed curves $H^*(\overline{M}_{0,n})$ and the explicit K\"unneth
  formula in quantum cohomology

abstract:
  We prove a general formula for the intersection form of two arbitrary
monomials in boundary divisors. Furthermore we present a tree basis of the
cohomology of $\overline {M}_{0,n}$. With the help of the intersection form we
determine the Gram matrix for this basis and give a formula for its inverse.
This enables us to calculate the tensor product of the higher order
multiplications arising in quantum cohomology and formal Frobenius manifolds.
In the context of quantum cohomology this establishes the explicit K\"unneth
formula.


title: Holomorphic bundles on O(-k) are algebraic

abstract:
  We show that holomorphic bundles on O(-k) for k > 0 are algebraic. We also
show holomorphic bundles on O(-1) are trivial outside the zero section.


title: Algebraic Cuts

abstract:
  Let $X$ be a projective variety with a torus action, which for simplicity we
assume to have dimension 1. If $X$ is a smooth complex variety, then the
geometric invariant theory quotient $X//G$ can be identifed with the symplectic
reduction $X_r$. Lerman introduced a construction (valid for symplectic
manifolds) called symplectic cutting, which constructs a manifold $X_c$, such
that $X_c$ is the union of $X_r$ and an open subset $X_{>0} \subset X$.
Moreover, there is a natural torus action on $X_c$ such that $X_r$ is a
component of the fixed locus. Using localization for equivariant cohomology,
this construction can be used to study of $X_r$.
  In this note, we give an algebraic version of this construction valid for
projective but possibly singular varieties defined over arbitrary fields. This
construction is useful for studying $X_r$ from the point of view of algebraic
geometry, using the equivariant intersection theory developed by the authors.
At the end of the paper we briefly give an adaptation of Lerman's proof of the
Kalkman residue formula and use it to give some formulas for characteristic
numbers of quotients by a torus.


title:
Intersection theory on moduli spaces of holomorphic bundles of arbitrary
  rank on a Riemann surface

abstract:
  We prove formulas (found by Witten in 1992 using physical methods) for
intersection pairings in the cohomology of the moduli space M(n,d) of stable
holomorphic vector bundles of rank n and degree d (assumed coprime) on a
Riemann surface of genus g greater than or equal to 2. We also use these
formulas for intersection numbers to obtain a proof of the Verlinde formula for
the dimension of the space of holomorphic sections of a line bundle over
M(n,d).


title: Genus one enumerative invariants in P^n with fixed j invariant

abstract:
  We prove recursive formulas for $\tau_d$, the number of degree $d$ elliptic
curves with fixed j-invariant in P^n. We use analysis to relate the classical
invariant $\tau_d$ to the genus one perturbed invariant $RT_{1,d}$ defined
recently by Ruan and Tian (the later invariant can be computed inductively). By
considering a sequence of perturbations converging to zero, we then apply
Taubes' Obstruction Bundle method to compute the difference between the two
invariants.


title:
Monodromy of hypergeometric functions arising from arrangements of
  hyperplanes

abstract:
  Given an arrangement of hyperplanes in $\P^n$, possibly with non-normal
crossings, we give a vanishing lemma for the cohomology of the sheaf of
$q$-forms with logarithmic poles along our arrangement. We give a basis for the
ideal $\cal J$ of relations for the Orlik-Solomon's algebra. Under certain
genericity conditions it was shown by H.~Esnault, V.~Schechtman and E.~Viehweg
that the cohomology of a local system is given by the Aomoto complex. We
generalize this result to a deformation of local systems obtained via a
deformation of our arrangement. We calculate the Gau\ss-Manin connection for
this case. We give a basis for the Gau\ss-Manin bundle for which, with help of
the basis for $\cal J$, we give then a method to calculate a representation of
this connection. From here, with the results of K-T.~Chen or P.~Deligne, one
can calculate the monodromy representation. This gives a generalization of the
hypergeometric functions.


title:
Virtual moduli cycles and Gromov-Witten invariants of general symplectic
  manifolds

abstract: We construct Gromov-Witten invariants of general symplectic manifolds.


title:
Estimates of the number of rational mappings from a fixed variety to
  varieties of general type

abstract:
  First we find effective bounds for the number of dominant rational maps $f:X
\rightarrow Y$ between two fixed smooth projective varieties with ample
canonical bundles. The bounds are of the type $\{A \cdot K_X^n\}^{\{B \cdot
K_X^n\}^2}$, where $n=dimX$, $K_X$ is the canonical bundle of $X$ and $A,B $
are some constants, depending only on $n$. Then we show that for any variety
$X$ there exist numbers $c(X)$ and $C(X)$ with the following properties: For
any threefold $Y$ of general type the number of dominant rational maps $f:X \r
Y$ is bounded above by $c(X)$. The number of threefolds $Y$, modulo birational
equivalence, for which there exist dominant rational maps $f:X \r Y$, is
bounded above by $C(X)$. If, moreover, $X$ is a threefold of general type, we
prove that $c(X)$ and $C(X)$ only depend on the index $r_{X_c}$ of the
canonical model $X_c$ of $X$ and on $K_{X_c}^3$.


title:
The ring of global sections of multiples of a line bundle on a toric
  variety

abstract:
  In this article we prove, in a simple way, that for any complete toric
variety, and for any Cartier divisor, the ring of global sections of multiples
of the line bundle associated to the divisor is finitely generated.


title: Equations of (1,d)-polarized Abelian Surfaces

abstract:
  We study the equations of abelian surfaces embedded in P^{n-1} with a line
bundle of polarization of type (1,n). For n>9, we show that the ideal of a
general abelian surface with this polarization is generated by quadrics, and if
the embedding is Heisenberg invariant, we give a very specific description of
these quadrics. These quadrics are produced using a generalization of the Moore
matrices which yields a uniform description of these ideals. We prove that
these equations generate the ideals using a degeneration argument, degenerating
abelian surfaces to schemes obtained from Stanley-Reisner ideals of certain
triangulations of the torus. Furthermore, we obtain information on how to
compute the moduli spaces of abelian surfaces of type (1,n) with canonical
level structure. In a sequel to this paper, we will study the structure of the
equations for n<=9, and also give detailed geometric descriptions for the
moduli spaces of abelian surfaces of low degree.


title: Algebraically Constructible Functions and Signs of Polynomials

abstract:
  We show that on real algebraic sets algebraically constructible functions
coincide with the finite sums of signs of polynomials. Then we give some
applications.


title:
Polar Varieties and Efficient Real Equation Solving: The Hypersurface
  Case

abstract:
  The objective of this paper is to show how the recently proposed method by
Giusti, Heintz, Morais, Morgenstern, Pardo \cite{gihemorpar} can be applied to
a case of real polynomial equation solving. Our main result concerns the
problem of finding one representative point for each connected component of a
real bounded smooth hypersurface. The algorithm in \cite{gihemorpar} yields a
method for symbolically solving a zero-dimensional polynomial equation system
in the affine (and toric) case. Its main feature is the use of adapted data
structure: Arithmetical networks and straight-line programs. The algorithm
solves any affine zero-dimensional equation system in non-uniform sequential
time that is polynomial in the length of the input description and an
adequately defined {\em affine degree} of the equation system. Replacing the
affine degree of the equation system by a suitably defined {\em real degree} of
certain polar varieties associated to the input equation, which describes the
hypersurface under consideration, and using straight-line program codification
of the input and intermediate results, we obtain a method for the problem
introduced above that is polynomial in the input length and the real degree.


title:
Polar Varieties, Real Equation Solving and Data-Structures: The
  hypersurface case

abstract:
  In this paper we apply for the first time a new method for multivariate
equation solving which was developed in \cite{gh1}, \cite{gh2}, \cite{gh3} for
complex root determination to the {\em real} case. Our main result concerns the
problem of finding at least one representative point for each connected
component of a real compact and smooth hypersurface. The basic algorithm of
\cite{gh1}, \cite{gh2}, \cite{gh3} yields a new method for symbolically solving
zero-dimensional polynomial equation systems over the complex numbers. One
feature of central importance of this algorithm is the use of a
problem--adapted data type represented by the data structures arithmetic
network and straight-line program (arithmetic circuit). The algorithm finds the
complex solutions of any affine zero-dimensional equation system in non-uniform
sequential time that is {\em polynomial} in the length of the input (given in
straight--line program representation) and an adequately defined {\em geometric
degree of the equation system}. Replacing the notion of geometric degree of the
given polynomial equation system by a suitably defined {\em real (or complex)
degree} of certain polar varieties associated to the input equation of the real
hypersurface under consideration, we are able to find for each connected
component of the hypersurface a representative point (this point will be given
in a suitable encoding). The input equation is supposed to be given by a
straight-line program and the (sequential time) complexity of the algorithm is
polynomial in the input length and the degree of the polar varieties mentioned
above.


title: Straight--Line Programs in Geometric Elimination Theory

abstract:
  We present a new method for solving symbolically zero--dimensional polynomial
equation systems in the affine and toric case. The main feature of our method
is the use of problem adapted data structures: arithmetic networks and
straight--line programs. For sequential time complexity measured by network
size we obtain the following result: it is possible to solve any affine or
toric zero--dimensional equation system in non--uniform sequential time which
is polynomial in the length of the input description and the ``geometric
degree" of the equation system. Here, the input is thought to be given by a
straight--line program (or alternatively in sparse representation), and the
length of the input is measured by number of variables, degree of equations and
size of the program (or sparsity of the equations). The geometric degree of the
input system has to be adequately defined. It is always bounded by the
algebraic--combinatoric "B\'ezout number" of the system which is given by the
Hilbert function of a suitable homogeneous ideal. However, in many important
cases, the value of the geometric degree of the system is much smaller than its
B\'ezout number since this geometric degree does not take into account
multiplicities or degrees of extraneous components (which may appear at
infinity in the affine case or may be contained in some coordinate hyperplane
in the toric case). Our method contains a new application of a classic tool to
symbolic computation: we use Newton iteration in order to simplify
straight--line programs occurring in elimination procedures. Our new technique
allows for practical implementations a meaningful characterization of the
intrinsic {\it algebraic complexity} of typic elimination problems and reduces
the still unanswered question of their intrinsic {\it bit complexity} to


title:
One-parameter families containing three-dimensional toric Gorenstein
  singularities

abstract:
  For affine toric varieties, the vector space T1 (containing the infinitesimal
deformations) will be interpreted via Minkowski summands of cross cuts of the
defining polyhedral cone. This result will be applied to study the deformation
theory of (in particular non-isolated) three-dimensional Gorenstein
singularities.


title: Enumerative geometry for real varieties

abstract:
  We discuss the problem of whether a given problem in enumerative geometry can
have all of its solutions be real. In particular, we describe an approach to
problems of this type, and show how this can be used to show some enumerative
problems involving the Schubert calculus on Grassmannians may have all of their
solutions be real. We conclude by describing the work of Fulton and
Ronga-Tognoli-Vust, who (independently) showed that there are 5 real plane
conics such that each of the 3264 conics tangent to all five are real.


title:
Density and completeness of subvarieties of moduli spaces of curves or
  abelian varieties

abstract:
  Let $V$ be a subvariety of codimension $\leq g$ of the moduli space $\cA_g$
of principally polarized abelian varieties of dimension $g$ or of the moduli
space $\tM_g$ of curves of compact type of genus $g$. We prove that the set
$E_1(V)$ of elements of $V$ which map onto an elliptic curve is analytically
dense in $V$. From this we deduce that if $V \subset \cA_g$ is complete, then
$V$ has codimension equal to $g$ and the set of elements of $V$ isogenous to a
product of $g$ elliptic curves is countable and analytically dense in $V$. We
also prove a technical property of the conormal sheaf of $V$ if $V \subset
\tM_g$ (or $\cA_g$) is complete of codimension $g$.


title:
Resolution for Sheaf of Differential Operators on Smooth Free Geometric
  Quotient of Linear Action of Algebraic Group

abstract:
  We obtain a global resolution for the sheaf of differential operators on
smooth geometric quotients of free linear actions of algebraic groups. The
terms of our resolution involve symmetric and alternating powers of vector
bundles easily constructed geometrically from the algebraic group and the
vector space on which it acts.


title:
Counting rational curves with multiple points and Gromov-Witten
  invariants of blow-ups

abstract:
  We study Gromov-Witten invariants on the blow-up of P^n at a point, which is
probably the simplest example of a variety whose moduli spaces of stable maps
do not have the expected dimension. It is shown that many of these invariants
can be interpreted geometrically on P^n as certain numbers of rational curves
having a multiple point of given order at the blown up point. Moreover, all
these invariants can actually be calculated, giving enumerative invariants of
P^n which have not been known before.


title: Algebraic structures on hyperkaehler manifold

abstract:
  Let $M$ be a compact hyperkaehler manifold. The hyperkaehler structure equips
$M$ with a set $R$ of complex structures parametrized by $CP^1$, called "the
set of induced complex structures". It was known previously that induced
complex structures are non-algebraic, except may be a countable set. We prove
that a countable set of induced complex structures is algebraic, and this set
is dense in $R$. A more general version of this theorem was proven by Fujiki.


title: A linear lower bound on the gonality of modular curves

abstract:
  The result in the title is proven, using the Selberg estimate on the leading
eigenvalue of the non-Euclidean Laplacian, and the method of conformal volumes
of Li and Yau.


title: Equivariant resolution of singularities in characteristic 0

abstract:
  A new proof of equivariant resolution of singularities under a finite group
action in characteristic 0 is provided. We assume we know how to resolve
singularities without group action. We first prove equivariant resolution of
toroidal singularities. Then we reduce the general case to the toroidal case.


title: Algebraic (geometric) $n$-stacks

abstract:
  We propose a generalization of Artin's definition of algebraic stack, which
we call {\em geometric $n$-stack}. The main observation is that there is an
inductive structure to the definition whereby the ingredients for the
definition of geometric $n$-stack involve only $n-1$-stacks and so are already
previously defined. We use this inductive structure to obtain some basic
properties. We look at maps from a projective variety into certain such
$n$-stacks, and obtain an interpretation of the Brill-Noether locus as the set
of points of a geometric $n$-stack. At the end we explain how this provides a
context for looking at de Rham theory for higher nonabelian cohomology, how one
can define the Hodge filtration and so on.


title: Moduli of vector-bundles on surfaces

abstract:
  This is a survey paper: we discuss certain recent results, with some
improvements. It will appear in the S. Cruz proceedings.


title: Torsion algebraic cycles and complex cobordism

abstract:
  We show that the cycle map on a variety X, from algebraic cycles modulo
algebraic equivalence to integer cohomology, lifts canonically to a
topologically defined quotient of the complex cobordism ring of X. This more
refined cycle map gives a topological proof that the Griffiths group is nonzero
for some varieties X, without any use of Hodge theory. We also use this more
refined cycle map to give examples of torsion algebraic cycles which map to 0
in Deligne cohomology but are not algebraically equivalent to 0, thus answering
some questions by Colliot-Thelene and Schoen.


title: The Verlinde formula for PGL(p)

abstract:
  Let G be a complex semi-simple group, X a Riemann surface, M_G the moduli
space of principal G-bundles on X. When G is simply-connected, there exists a
closed formula expressing the dimension of the space H^0(M_G,L) for any line
bundle L on M_G (this is usually called among mathematicians, somewhat
incorrectly, the "Verlinde formula").In this paper we find an analogous formula
for G = PGL(r) where r is prime.


title: Equivariant intersection theory

abstract:
  This is a revised and shortened version of our paper "Equivariant
intersection theory" (alg-geom/9603008). In particular, the sections on
Riemann-Roch and localization are omitted. They will appear in separate papers,
at which time alg-geom/9603008 will become obsolete. We have intsead added a
section of examples, and have include a calculation of the integral Chow ring
of the mdouli stack of elliptic curves.


title: Secants of Abelian Varieties, Theta functions, and Soliton Equations

abstract:
  This paper is a survey on relations between secant identities and soliton
equations and applications of soliton equations to problems of algebraic
geometry, i.e., the Riemann-Schottky problem and its analogues. A short
introduction into the analytic theory of theta functions is also given.


title: Non-abelian Seiberg-Witten theory and projectively stable pairs

abstract:
  We introduce the concept of Spin^G-structure in a SO-bundle, where $G\subset
U(V)$ is a compact Lie group containing $-id_V$. We study and classify
$Spin^G(4)$-structures on 4-manifolds, we introduce the G-Monopole equations
associated with a $Spin^G$-structure. On Kaehler surfaces a Kobayashi-Hitchin
correspondence can be proved for the corresponding moduli spaces. Using this
complex geometric interpretation, we determine explicitely a moduli space of
"PU(2)-Monopoles" on $\P^2$, we describe its Uhlenbeck compactification, as
well as the Donaldson- and the abelian locus.


title: Mathematical Aspects of Mirror Symmetry

abstract:
  Lecture notes from 1993 Park City lectures and 1994 Trento lectures. The
focus of these lectures is on giving a mathematical description of the A-model
and B-model correlation functions on a Calabi--Yau manifold, and a precise
mathematical conjecture relating these for a pair of mirror manifolds.


title: Automorphic forms with singularities on Grassmannians

abstract:
  We construct some families of automorphic forms on Grassmannians which have
singularities along smaller sub Grassmannians, using Harvey and Moore's
extension of the Howe (or theta) correspondence to modular forms with poles at
cusps. Some of the applications are as follows. We construct families of
holomorphic automorphic forms which can be written as infinite products, which
give many new examples of generalized Kac-Moody superalgebras. We extend the
Shimura and Maass-Gritsenko correspondences to modular forms with
singularities. We prove some congruences satisfied by the theta functions of
positive definite lattices, and find a sufficient condition for a Lorentzian
lattice to have a reflection group with a finite volume fundamental domain. We
give some examples suggesting that these automorphic forms with singularities
are related to Donaldson polynomials and to mirror symmetry for K3 surfaces.


title:
Crepant Terminalisations and Orbifold Euler Numbers for SL(4)
  Singularities

abstract:
  Let $X$ and $Y$ be two analytic canonical Gorenstein orbifolds. A resolution
of singularities $Y\to X$ is called an Euler resolution if $Y$ and $X$ have the
same orbifold Euler number. If $Y$ is only terminal rather than smooth, it is
called an Euler terminalisation. It is proved that Euler terminalisations exist
for toric varieties in any dimension, for 4-dimensional toroidal varieties, and
for singularities $\C^4/G$ where $G$ belongs to certain classes of $\SL(4)$
subgroups. The method of proof is expected to be applicable to a sizeable
number of finite $\SL(4)$ subgroups and to lead to a generalisation of the
Dixon-Harvey-Vafa-Witten orbifold Euler number conjecture to dimension~4.


title:
Linearization of group stack actions and the Picard group of the moduli
  of $\SL_r/\mu_s$-bundles on a curve

abstract:
  We first study the descent theory of line bundles under a morphism which is
tors or under a group stack and then use this technical result to determine the
exact structure of $\Pic(\M_G)$ where $G=\SL_r/\mu_s$ (we include a minor
modification to explain the genus 0 case).


title: Specialization of integral dependence for modules

abstract:
  We establish the principle of specialization of integral dependence for
submodules of finite colength of free modules, as part of the general
algebraic-geometric theory of the Buchsbaum--Rim multiplicity. Then we apply
the principle to the study of equisingularity of ICIS germs, obtaining results
for such equisingularity conditions as Whitney's Condition A, Thom's Condition
A_f, and Henry, Merle and Sabbah's Condition W_f. Notably, we describe these
conditions for analytic families in terms of various numerical invariants,
which, for the most part, depend only on the members of a family, not on its
total space.


title:
Partial Resolutions of Orbifold Singularities via Moduli Spaces of
  HYM-type Bundles

abstract:
  Let $\Gamma$ be a finite group acting linearly on $\C^n$, freely outside the
origin, and let $N$ be the number of conjugacy classes of $\Gamma$ minus one. A
construction of Kronheimer of moduli spaces $X_\zeta$ of translation-invariant
$\Gamma$-equivariant instantons on $\C^2$ is generalised to $\C^n$. The moduli
spaces $X_\zeta$ depend on a parameter $\zeta\in\Q^N$. The following results
are proved: for $\zeta=0$, $X_0$ is isomorphic to $\C^n/\Gamma$; if $\zeta\neq
0$, the natural maps $X_\zeta\to X_0$ are partial resolutions. The moduli
$X_\zeta$ are furthermore shown to admit K\"ahler metrics which are
Asymptotically Locally Euclidean (ALE). A description of the singularities of
$X_\zeta$ using deformation complexes is given, and is applied in particular to
the case $\Gamma\subset\SU(3)$. It is conjectured that for general $\Gamma$ and
generic $\zeta$ that the singularities of $X_\zeta$ are at most quadratic. When
$\Gamma\subset\SU(3)$ a natural holomorphic 3-form is constructed on the smooth
locus of $X_\zeta$, which is conjectured to be non-vanishing. The morphims
$X_\zeta\to X_0$ are expected to be crepant resolutions and $X_\zeta$ to be
smooth for generic choices of the parameter $\zeta$. Related open problems in
higher-dimensional complex geometry are also mentioned. The paper has a
companion paper which identifies the moduli $X_\zeta$ with representation
moduli of McKay quivers, and describes them completely in the case of abelian
groups.


title: Resolutions of Orbifold Singularities and Flows on the McKay Quiver

abstract:
  Let $\Gamma$ be a finite group acting linearly on $\C^n$, freely outside the
origin. In previous work a generalisation of Kronheimer's construction of
moduli of Hermitian-Yang-Mills bundles with certain invariance properties was
given. This produced varieties $X_\zeta$ (parameterised by $\zeta\in\Q^N$)
which are partial resolutions of $\C^n/\Gamma$. In this article, it is shown
the same $X_\zeta$ can be described as moduli spaces of representations of the
McKay quiver associated to the action of $\Gamma$. It it shown that, for
abelian groups, $X_\zeta$ are toric varieties defined by convex polyhedra which
are the solution sets for a generalisation of the transportation problem on the
McKay quiver. The generalised transportation problem is solved for an arbitrary
quiver to give a description of the extreme points, faces, and tangent cones to
the solution polyhedra in terms of certain distinguished trees in the quiver.
Applied the McKay quiver, this gives an explicit procedure for calculating
$X_\zeta $, its Euler number, and its singularities for any $\zeta$. The
$\zeta$-parameter-space is thus partitioned into a finite disjoint union of
cones inside which the biregular type of $X_\zeta$ remains constant. Finally,
the example $\C^3/\Z_5$ (weights $1,2,3$) is worked out in detail, and figures
of smooth and singular $X_\zeta$ and their corresponding flows are drawn. A
further example of smooth crepant resolution $X_\zeta$ is drawn for the
singularity ${1/11}(1,4,6)$.


title:
Bounds for the Hilbert function of polynomial ideals and for the degrees
  in the Nullstellensatz

abstract:
  We present a new effective Nullstellensatz with bounds for the degrees which
depend not only on the number of variables and on the degrees of the input
polynomials but also on an additional parameter called the {\it geometric
degree of the system of equations}. The obtained bound is polynomial in these
parameters. It is essentially optimal in the general case, and it substantially
improves the existent bounds in some special cases. The proof of this result is
combinatorial, and it relies on global estimations for the Hilbert function of
homogeneous polynomial ideals. In this direction, we obtain a lower bound for
the Hilbert function of an arbitrary homogeneous polynomial ideal, and an upper
bound for the Hilbert function of a generic hypersurface section of an unmixed
radical polynomial ideal.


title: Chow motives of elliptic modular surfaces and threefolds

abstract:
  The main result of this paper is the proof for elliptic modular threefolds of
conjectures on the existence and structure of a filtration on the Chow groups
of smooth projective varieties. In the form we prove them these conjectures
were formulated by the second-named author, but as Jannsen has shown, these
conjectures are equivalent to a conjecture of Beilinson. These conjectures have
previously been proved for surfaces in general, but for elliptic modular
surfaces we obtain more precise results which are then used in proving the
conjectures for elliptic modular threefolds.


title: The Rank Stable Topology of Instantons on $\cpbar$

abstract:
  Let $\M_{k}^{n}$ be the moduli space of based (anti-self-dual) instantons on
$\cpbar$ of charge $k$ and rank $n$. There is a natural inclusion of rank $n$
instantons into rank $n+1$. We show that the direct limit space is homotopy
equivalent to $BU(k)\times BU(k)$. The moduli spaces also have the following
algebro-geometric interpretation: Let $\linf$ be a line in the complex
projective plane and consider the blow-up at a point away from $\linf$. $\M
_{k}^{n}$ can be described as the moduli space of rank $n$ holomorphic bundles
on the blownup projective plane with $c_{1}=0$ and $c_{2}=k$ and with a fixed
holomorphic trivialization on $\linf$.


title: Hilbert-Kunz functions of cubic curves and surfaces

abstract:
  We determine the Hilbert-Kunz function of plane elliptic curves in odd
characteristic, as well as over arbitrary fields the generalized Hilbert-Kunz
functions of nodal cubic curves. Together with results of K. Pardue and P.
Monsky, this completes the list of Hilbert-Kunz functions of plane cubics.
Combining these results with the calculation of the (generalized) Hilbert-Kunz
function of Cayley's cubic surface, it follows that in each degree and over any
field of positive characteristic there are curves resp. surfaces taking on the
minimally possible Hilbert-Kunz multiplicity.


title: Deformations of trianalytic subvarieties of hyperk\"ahler manifolds

abstract:
  Let $M$ be a compact complex manifold equipped with a hyperk\"ahler metric,
and $X$ be a closed complex analytic subvariety of $M$. In alg-geom/9403006, we
proved that $X$ is trianalytic, i. e., complex analytic with respect to all
complex structures induced by the hyperk\"ahler structure, provided that $M$ is
generic in its deformation class. Here we study the complex analytic
deformations of trianalytic subvarieties. We prove that all deformations of $X$
are trianalytic and naturally isomorphic to $X$ as complex analytic varieties.
We show that this isomorphism is compatible with the metric induced from $M$.
Also, we prove that the Douady space of complex analytic deformations of $X$ in
$M$ is equipped with a natural hyperk\"ahler structure.


title: Special points on the product of two modular curves

abstract:
  We prove, assuming the Generalized Riemann Hypothesis for imaginary quadratic
fields, that irreducible curves in the product of two modular curves that
contain infinitely many complex multiplication points are either a Hecke
correspondence or a fibre for one of the two projections. This gives evidence
for a conjecture of Oort that says that irreducible components of the Zariski
closure of a set of CM points in a Shimura variety are sub Shimura varieties.


title: Monomial Resolutions

abstract:
  Call a monomial ideal M "generic" if no variable appears with the same
nonzero exponent in two distinct monomial generators. Using a convex polytope
first studied by Scarf, we obtain a minimal free resolution of M. Any monomial
ideal M can be made generic by deformation of its generating exponents. Thus,
the above construction yields a (usually nonminimal) resolution of M for
arbitrary monomial ideals, bounding the Betti numbers of M in terms of the
Upper Bound Theorem for Convex Polytopes.
  We show that our resolutions are DG-algebras, and consider realizability
questions and irreducible decompositions.


title: Rationality of Moduli Spaces of Parabolic Bundles

abstract:
  The moduli space of parabolic bundles with fixed determinant over a smooth
curve of genus greater than one is proved to be rational whenever one of the
multiplicities associated to the quasi-parabolic structure is equal to one. It
follows that if rank and degree are coprime, the moduli space of vector bundles
is stably rational, and the bound obtained on the level is strong enough to
conclude rationality in many cases.


title:
Moduli Spaces of Parabolic Higgs Bundles and Parabolic K(D) Pairs over
  Smooth Curves: I

abstract:
  This paper concerns the moduli spaces of rank two parabolic Higgs bundles and
parabolic K(D) pairs over a smooth curve. Precisely which parabolic bundles
occur in stable K(D), pairs and stable Higgs bundles is determined. Using Morse
theory, the moduli space of parabolic Higgs bundles is shown to be a
non-compact, connected, simply connected manifold, and a computation of its
Poincar\'e polynomial is given.


title: Universal Einstein-Hermitian Metrics

abstract: Lemma 3.4 is wrong.


title: Arithmetically Cohen--Macaulay curves in P^4 of degree 4 and genus 0

abstract:
  We show that the arithmetically Cohen--Macaulay (ACM) curves of degree 4 and
genus 0 in ${\bold P}^4$ form an irreducible subset of the Hilbert scheme.
Using this, we show that the singular locus of the corresponding component of
the Hilbert scheme has dimension greater than 6. Moreover, we describe the
structures of all ACM curves of Hilb$^{4m+1}({\bold P}^4)$.


title: Prime rings with PI rings of constants

abstract:
  It is shown that if the ring of constants of a restricted differential Lie
algebra with a quasi-Frobenius inner part satisfies a polynomial identity (PI)
then the original prime ring has a generalized polynomial identitiy (GPI). If
additionally the ring of constants is semiprime then the original ring is PI.
The case of a non-quasi-Frobenius inner part is also considered.


title: Equations Defining Toric Varieties

abstract:
  This article will appear in the proceedings of the AMS Summer Institute in
Algebraic Geometry at Santa Cruz, July 1995. The topic is toric ideals, by
which I mean the defining ideals of subvarieties of affine or projective space
which are parametrized by monomials. Numerous open problems are given.


title: The Laumon's resolution of Drinfeld's compactification is small

abstract:
  Let $C$ be a smooth projective curve of genus 0. Let $\FF$ be the variety of
complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple
$\alpha$ of positive integers one can consider the space $\MM\alpha$ of
algebraic maps of degree $\alpha$ from $C$ to $\FF$. This space has drawn much
attention recently in connection with Quantum Cohomology. The space $\MM\alpha$
is smooth but not compact. The problem of compactification of $\MM\alpha$
proved very important. One compactification $\MML\alpha$ (the space of {\em
quasiflags}), was constructed in \cite{L}. However, historically the first and
most economical compactification $\MMD\alpha$ (the space of {\em quasimaps})
was constructed by Drinfeld (early 80-s, unpublished). The latter
compactification is singular, while the former one is smooth. Drinfeld has
conjectured that the natural map $\pi:\MML\alpha\to\MMD\alpha$ is a small
resolution of singularities. In the present note we prove this conjecture. As a
byproduct, we compute the stalks of $IC$ sheaf on $\MMD\alpha$ and, moreover,
the Hodge structure in these stalks. Namely, the Hodge structure is a pure Tate
one, and the generating function for the $IC$ stalks is just the Lusztig's
$q$-analogue of Kostant's partition function (see \cite{Lu}).


title:
Weil-Petersson geometry and determinant bundles on inductive limits of
  moduli spaces

abstract:
  In an earlier paper [Acta Mathematica, v. 176, 1996, 145-169,
alg-geom/9505024 ] the present authors and Dennis Sullivan constructed the
universal direct system of the classical Teichm\"uller spaces of Riemann
surfaces of varying genus. The direct limit, which we called the universal
commensurability Teichm\"uller space, $T_{infty}$, was shown to carry on it a
natural action of the universal commensurability mapping class group,
$MC_{\infty}$. In this paper we identify an interesting cofinal sub-system
corresponding to the tower of finite-sheeted characteristic coverings over any
fixed base surface. Utilizing a certain subgroup inside $MC_{\infty}$,
(associated intimately to the inverse system of characteristic coverings), we
can now descend to an inductive system of moduli spaces, and construct the
direct limit ind-variety $M_{\infty}$.
  Invoking curvature properties of Quillen metrics on determinant bundles, and
naturality under finite coverings of Weil-Petersson forms, we are able to
construct on $M_{\infty}$ the natural sequence of determinant of cohomology
line bundles, as well as the Mumford isomorphisms connecting these.


title: Jack polynomials and Hilbert schemes of points on surfaces

abstract:
  The Jack symmetric polynomials $P_\lambda^{(\alpha)}$ form a class of
symmetric polynomials which are indexed by a partition $\lambda$ and depend
rationally on a parameter $\alpha$. They reduced to the Schur polynomials when
$\alpha=1$, and to other classical families of symmetric polynomials for
several specific parameters.
  It is well-known that Schur polynomials can be realized as certain elements
of homology groups of Grassmann manifolds. The purpose of this paper is to give
a similar geometric realization for Jack polynomials. However, spaces which we
use are totally different. Our spaces are Hilbert schemes of points on a
surface X which is the total space of a line bundle L over the projective line.
The parameter $\alpha$ in Jack polynomials relates to our surface X by $\alpha
= -<C,C>$, where C is the zero section, and <C,C> is the self-intersection
number of C.

categories: alg-geom hep-th math.AG math.QA nlin.SI q-alg solv-int


title: Automorphic Forms and Lorentzian Kac--Moody Algebras. Part I

abstract:
  Using the general method which was applied to prove finiteness of the set of
hyperbolic generalized Cartan matrices of elliptic and parabolic type, we
classify all symmetric (and twisted to symmetric) hyperbolic generalized Cartan
matrices of elliptic type and of rank 3 with a lattice Weyl vector.
  We develop the general theory of reflective lattices T with 2 negative
squares and reflective automorphic forms on homogeneous domains of type IV
defined by T. We consider this theory as mirror symmetric to the theory of
elliptic and parabolic hyperbolic reflection groups and corresponding
hyperbolic root systems. We formulate Arithmetic Mirror Symmetry Conjecture
relating both these theories and prove some statements to support this
Conjecture. This subject is connected with automorphic correction of Lorentzian
Kac--Moody algebras. We define Lie reflective automorphic forms which are the
most beautiful automorphic forms defining automorphic Lorentzian Kac--Moody
algebras, and we formulate finiteness Conjecture for these forms.
  Detailed study of automorphic correction and Lie reflective automorphic forms
for generalized Cartan matrices mentioned above will be given in Part II.


title: On maximal curves

abstract:
  We study arithmetical and geometrical properties of maximal curves, that is,
curves defined over the finite field F_{q^2} whose number of F_{q^2}-rational
points reaches the Hasse-Weil upper bound. Under a hypothesis on non-gaps at a
rational point, we prove that maximal curves are F_{q^2}-isomorphic to y^q + y
= x^m, for some $m \in Z^+$. As a consequence we show that a maximal curve of
genus g=(q-1)^2/4 is F_{q^2}-isomorphic to the curve y^q + y = x^{(q+1)/2}.


title: On framed instanton bundles and their deformations

abstract:
  We consider a compact twistor space P and assume that there is a surface S in
P, which has degree one on twistor fibres and contains a twistor fibre F, e.g.
P a LeBrun twistor space. Similar to Donaldson and Buchdahl we examine the
restriction of an instanton bundle V equipped with a fixed trivialisation along
F to a framed vector bundle over (S,F). First we develope inspired by
Huybrechts and Lehn a suitable deformation theory for vector bundles over an
analytic space framed by a vector bundle over a subspace of arbitrary
codimension. In the second section we describe the restriction as a smooth
natural transformation into a fine moduli space. By considering framed
U(r)-instanton bundles as a real structure on framed instanton bundles over P,
we show that the bijection between isomorphism classes of framed U(r)-instanton
bundles and isomorphism classes of framed vector bundles over (S,F) due to
Buchdahl is actually an isomorphism of moduli spaces.


title: Segre Numbers and Hypersurface Singularities

abstract:
  We define the Segre numbers of an ideal as a generalization of the
multiplicity of an ideal of finite colength. We prove generalizations of
various theorems involving the multiplicity of an ideal such as a principle of
specialization of integral dependence, the Rees-B\"oger theorem, and the
formula for the multiplicity of the product of two ideals. These results are
applied to the study of various equisingularity conditions, such as Verdier's
condition W, and conditions $A_f$ and $W_f$.


title: Resolving mixed Hodge modules on configuration spaces

abstract:
  Given a mixed Hodge module E on a scheme X over the complex numbers, and a
quasi-projective morphism f:X->S, we construct in this paper a natural
resolution of the nth exterior tensor power of E restricted to the nth
configuration space of f. The construction is reminiscent of techniques from
the theory of hyperplane arrangements, and relies on Arnold's calculation of
the cohomology of the configuration space of the complex line. This resolution
is S_n-equivariant. We apply it to the universal elliptic curve with complete
level structure of level N>=3 over the modular curve Y(N), obtaining a formula
for the S_n-equivariant Serre polynomial (Euler characteristic of H^*_c(V,Q) in
the Grothendieck group of the category of mixed Hodge structures) of the moduli
space M_{1,n}. In a sequel to this paper, this is applied in the calculation of
the S_n-equivariant Hodge polynomial of the compactication \bar{M}_{1,n}.


title:
The Second Cohomology with Symplectic Coefficients of the Moduli Space
  of Smooth Projective Curves

abstract:
  Each finite dimensional irreducible rational representation V of the
symplectic group Sp_{2g} determines a generically defined local system \V over
the moduli space M_g of genus g smooth projective curves. We study H^2(M_g;\V)
and the mixed Hodge structure on it. Specifically, we prove that if g>5, then
the natural map IH^2(MS_g;\V)-->H^2(M_g;\V) is an isomorphism where MS_g is the
Satake compactification of M_g. Using the work of Saito we conclude that the
mixed Hodge structure on H^2(M_g;\V) is pure of weight 2+r if \V underlies a
variation of Hodge structure of weight r. We also obtain estimates on the
weight of the mixed Hodge structure on H^2(M_g;\V) for 2<g<6. Results of this
article can be applied in the study of relations in the Torelli group T_g.


title: Bott-Chern Forms and Arithmetic Intersections

abstract:
  Let \E : 0 --> S --> E --> Q --> 0 be a short exact sequence of hermitian
vector bundles with metrics on S and Q induced from that on E. We compute the
Bott-Chern form of \E corresponding to any characteristic class, assuming E is
projectively flat. The result is used to obtain a new presentation of the
Arakelov Chow ring of the arithmetic grassmannian.


title: Arithmetic Intersection Theory on Flag Varieties

abstract:
  Let F be the complete flag variety over Spec(Z) with the tautological
filtration 0 \subset E_1 \subset E_2 \subset ... \subset E_n=E of the trivial
bundle E over F. The trivial hermitian metric on E(\C) induces metrics on the
quotient line bundles L_i(\C). Let \hat{c}_1(L_i) be the first Chern class of
L_i in the arithmetic Chow ring \hat{CH}(F) and x_i = -\hat{c}_1(L_i). Let
h(X_1,...,X_n) be a polynomial with integral coefficients in the ideal
<e_1,...,e_n> generated by the elementary symmetric polynomials e_i. We give an
effective algorithm for computing the arithmetic intersection h(x_1,...,x_n) in
\hat{CH}(F), as the class of a SU(n)-invariant differential form on F(\C). In
particular we show that all the arithmetic Chern numbers one obtains are
rational numbers. The results are true for partial flag varieties and
generalize those of Maillot for grassmannians. An `arithmetic Schubert
calculus' is established for an `invariant arithmetic Chow ring' which
specializes to the Arakelov Chow ring in the grassmannian case.


title: Higher Bott Chern forms and Beilinson's regulator

abstract:
  Let X be a smooth complex algebraic variety. In this paper, we associate, to
each exact n-cube of hermitian vector bundles over X, a differential form,
called higher Bott Chern form, which generalizes the Bott Chern forms
associated to an exact sequence of hermitian vector bundles. With these forms
we construct characteristic classes for higher K-theory and prove that these
classes agree with Beilinson's regulator. Thus we obtain a description of
Beilinson's regulator map in terms of hermitian vector bundles.


title: Projective Moduli for Hitchin Pairs

abstract:
  We give an algebraic geometric compactification of certain moduli spaces of
semistable E-pairs in the sense of Yokogawa. In particular, we obtain a
compactification of the moduli spaces of semistable Higgs pairs on a curve
which were constructed by N. Hitchin.


title:
Birational automorphisms of algebraic varieties with a pencil of cubic
  surfaces

abstract:
  It is proved that on a smooth algebraic variety, fibered into cubic surfaces
over the projective line and sufficiently ``twisted'' over the base, there is
only one pencil of rational surfaces -- that is, this very pencil of cubics. In
particular, this variety is non-rational; moreover, it can not be fibered into
rational curves. The proof is obtained by means of the method of maximal
singularities.


title: On the Complexity of Smooth Projective Toric Varieties

abstract:
  In this paper we answer a question posed by V.V. Batyrev. The question asked
if there exists a complete regular fan with more than quadratically many
primitive collections. We construct a smooth projective toric variety
associated to a complete regular fan $\Delta$ in R^d with $n$ generators where
the number of primitive collections of $\Delta$ is at least exponential in
$n-d$. We also exhibit the connection between the number of primitive
collections of $\Delta$ and the facet complexity of the Gr\"obner fan of the
associated integer program.


title: Dynkin Graphs, Gabri\'{e}lov Graphs and Triangle Singularities

abstract:
  Fourteen kinds of triangle singularities with modality one in Arnold's
classification list are discussed. We consider which kinds of combinations of
rational double points can appear on small deformation fibers of the
singularities. We show that possible combinations of rational double points can
be described by a unique principle from the view point of Dynkin graphs.


title: The quantum cohomology of blow-ups of P^2 and enumerative geometry

abstract:
  We compute the Gromov-Witten invariants of the projective plane blown up in r
general points. These are determined by associativity from r+1 intial values.
Applications are given to the enumeration of rational plane curves with
prescribed multiplicities at fixed general points. We show that the numbers are
enumerative if at least one of the prescribed multiplicities is 1 or 2. In
particular, all the invariants for r<=8 (the Del Pezzo case) are enumerative.


title: On the semi-simplicity of the $U_p$-operator on modular forms

abstract:
  Let $p$ be a prime number and $N$ an integer prime to $p$. We show that the
operator $U_p$ on the space of cuspidal modular forms of level $pN$ and weight
two is semi-simple. It follows from this that the Hecke algebra acting on the
space of weight two forms of level $M$ is reduced if $M$ is cube free. Assuming
Tate's conjecture for cycles on smooth projective varieties over finite fields,
we generalize these results to higher weights. The main point in the proof is
that the crystalline Frobenius of the reduction mod $p$ of the motive
associated to a newform of level prime to $p$ and weight at least two cannot be
a scalar. Assuming Tate's conjecture, it follows that Ramanujan's inequality is
strict. For $N$ prime, we relate the discriminant of the weight two Hecke
algebra to the height of the modular curve $X_0(N)$, for which we get an upper
bound.


title: Andre-Quillen cohomology of monoid algebras

abstract:
  We compute the Andre-Quillen cohomology of an affine toric variety. The best
results are obtained either in the general case for the first three cohomology
groups, or in the case of isolated singularities for all cohomology groups,
respectively.


title: Desingularization of singular hyperkaehler varieties I

abstract:
  Let $M$ be a singular hyperkaehler variety, obtained as a moduli space of
stable holomorphic bundles on a compact hyperkaehler manifold
(alg-geom/9307008). Consider $M$ as a complex variety in one of the complex
structures induced by the hyperkaehler structure. We show that normalization of
$M$ is smooth, hyperkaehler and does not depend on the choice of induced
complex structure.


title: Giambelli-type formula for subbundles of the tangent bundle

abstract:
  Let us consider a generic n-dimensional subbundle V of the tangent bundle TM
on some given manifold M. Given V one can define different degeneracy loci
S_r(CV), r=(r_1<= r_2<= r_3<=...<=r_k) on M consisting of all points x in M for
which the dimension of the subspace V^j(x) in TM(x) spanned by all length <= j
commutators of vector fields tangent to V at x is less than or equal to r_j. We
calculate 'explicitly' the cohomology classes dual to S_r(V) using
determinantal formulas due to W.Fulton and the expression for the Chern classes
of the associated bundle of free Lie algebras in terms of the Chern classes of
V itself.


title: A view on contractions of higher dimensional varieties

abstract:
  In this paper we discuss some recent results about extremal contractions of
complex algebraic varieties. These are proper surjective maps, $\phi:
X\longrightarrow Z$, of normal varieties with connected fibers such that $X$
has mild singularities (in particular $X$ is $Q$-Gorenstein) and the
anticanonical divisor of $X$ is $\phi$ ample. It will appear on the Proceedings
of AMS-SRI on Algebraic Geometry, Santa Cruz 1995.


title: Algebraic Differential Characters

abstract:
  We give a construction of algebraic differential characters, receiving
classes of algebraic bundles with connection, lifitng the Chern-Simons
invariants defined with S. Bloch, the classes in the Chow group and the
analytic secondary invariants if the variety is defined over the field of
complex numbers.


title: Bott-Samelson Varieties and Configuration Spaces

abstract:
  We give a new construction of the Bott-Samelson variety $Z$ as the closure of
a $B$-orbit in a product of flag varieties $(G/B)^l$. This also gives an
embedding of the projective coordinate ring of the variety into the function
ring of a Borel subgroup: $\CC[Z] \subset \CC[B]$.
  In the case of the general linear group $G = GL(n)$, this identifies $Z$ as a
configuration variety of multiple flags subject to certain inclusion
conditions, closely related to the the matrix factorizations of Berenstein,
Fomin and Zelevinsky. As an application, we give a geometric proof of the
theorem of Kraskiewicz and Pragacz that Schubert polynomials are characters of
Schubert modules.
  Our work leads on the one hand to a Demazure character formula for Schubert
polynomials and other generalized Schur functions, and on the other hand to a
Standard Monomial Theory for Bott-Samelson varieties.


title:
Liouville and Carath\'eodory coverings in Riemannian and complex
  geometry

abstract:
  A Riemannian manifold resp. a complex space $X$ is called Liouville if it
carries no nonconstant bounded harmonic resp. holomorphic functions. It is
called Carath\'eodory, or Carath\'eodory hyperbolic, if bounded harmonic resp.
holomorphic functions separate the points of $X$. The problems which we discuss
in this paper arise from the following question:
  When a Galois covering $X$ with Galois group $G$ over a Liouville base $Y$ is
Liouville or, at least, is not Carath\'eodory hyperbolic?


title: Virtual neighborhoods and pseudo-holomorphic curves

abstract:
  We use virtual neighborhood technique to establish GW-invariants, Quantum
cohomology, equivariant GW-invariants, equivariant quantum cohomology and Floer
cohomology for general symplectic manifold. We also establish GW-invariants for
a family of symplectic manifolds. As a consequence, we prove Arnold conjecture
for nondegenerate Hamiltonian symplectomorphisms.


title:
Bornes effectives pour la torsion des courbes elliptiques sur les corps
  de nombres

abstract:
  In this paper we give a detailed proof of a result we announced a year ago.
This result is an effective version of the theorem of Mazur-Kamienny-Merel
concerning uniform bounds for rational torsion points on elliptic curves over
number fields.


title: Separation of Semialgebraic Sets

abstract:
  In this paper we study the problem of deciding whether two disjoint
semialgebraic sets of an algebraic variety over R are separable by a
polynomial. For that we isolate a dense subfamily of Spaces of Orderings, named
Geometric, which suffice to test separation and that reduce the problem to the
study of the behaviour of the semialgebraic sets in their boundary. Then we
derive several characterizations for the generic separation, among which there
is a Geometric Criterion that can be tested algorithmically. Finally we show
how to check recursively whether we can pass from the generic separation to the
separation of the two sets, yielding a decision procedure to solve the problem.


title: Existence of good divisors on Mukai manifolds

abstract:
  A normal projective variety X is called Fano if a multiple of the
anticanonical Weil divisor, -K_X, is an ample Cartier divisor, the index of a
Fano variety is the number i(X):=sup{t: -K_X= tH, for some ample Cartier
divisor H}. Mukai announced, the classification of smooth Fano manifolds X of
index i(X)=n-2, under the assumption that the linear system |H| contains a
smooth divisor. In this paper we prove that this assumption is always
satisfied. Therefore the result of Mukai provide a complete classification of
smooth Fano n-folds of index $i(X)=n-2$, Mukai manifolds.


title:
On symmetric degeneracy loci, spaces of symmetric matrices of constant
  rank and dual varieties

abstract:
  Let X be a nonsingular simply connected projective variety of dimension m, E
a rank n vector bundle on X, and L a line bundle on X. Suppose that $S^2(E^{*})
\otimes L$ is an ample vector bundle and that there is a constant even rank $r
\ge 2$ symmetric bundle map $E \to E^{*} \otimes L$. We prove that $m \le n-r$.
We use this result to solve the constant rank problem for symmetric matrices,
proving that the maximal dimension of a linear subspace of the space of
$m\times m$ symmetric matrices such that each nonzero element has even rank $r
\ge 2$ is $m-r+1$. We explain how this result relates to the study of dual
varieties in projective geometry and give some applications and examples.


title: On the density of ratios of Chern numbers of embedded threefolds

abstract:
  We show that the limit points of (x,y) for all 3-folds in P^5 with the Chern
ratios $x=c_1^3/c_1c_2$, $y=c_3/c_1c_2$ must lie on the line segment $x+y=2$,
$1\le x\le 2$. (Note that the determinantal ones already give x+y=2, $1\le x\le
17/12$.)


title: Differential forms on singular varieties and cyclic homology

abstract:
  A classical result of A. Connes asserts that the Frechet algebra of smooth
functions on a smooth compact manifold X provides, by a purely algebraic
procedure, the de Rham cohomology of X. Namely the procedure uses Hochschild
and cyclic homology of this algebra. In the situation of a Thom-Mather
stratified variety, we construct a Frechet algebra of functions on the regular
part and a module of poles along the singular part. We associate to these
objects a complex of differential forms and an Hochschild complex, on the
regular part, both with poles along the singular part. The de Rham cohomology
of the first complex and the cylic homology of the second one are related to
the intersection homology of the variety, the corresponding perversity is
determined by the orders of poles.


title: Automorphic Forms and Lorentzian Kac-Moody Algebras. Part II

abstract:
  We give variants of lifting construction, which define new classes of modular
forms on the Siegel upper half-space of complex dimension 3 with respect to the
full paramodular groups (defining moduli of Abelian surfaces with arbitrary
polarization). The data for these liftings are Jacobi forms of integral and
half-integral indices. In particular, we get modular forms which are
generalizations of the Dedekind eta-function. Some of these forms define
automorphic corrections of Lorentzian Kac-Moody algebras with hyperbolic
generalized Cartan matrices of rank three classified in Part I of this paper.
We also construct many automorphic forms which give discriminants of moduli of
K3 surfaces with conditions on Picard lattice. These results are important for
Mirror Symmetry and theory of Lorentzian Kac-Moody algebras.


title: Counting curves on surfaces: a guide to new techniques and results

abstract:
  This is a survey describing recents developments in enumerative geometry of
curves on projective varieties. Various methods to arrive at results such as
Kontsevich's formula for plane rational curves, or Caporaso-Harris's formula
for plane curves of any genus, are illustrated on concrete examples It will
appear on the Proceedings of the European Congress of Mathematics, Budapest
1996.


title: Irregular canonical double surfaces

abstract:
  We study minimal surfaces X of general type with $K^2_X=6p_g-14$ and $q(X)>0$
such that $K_X$ is ample, the image of the canonical map is a canonically
embedded surface of general type and the canonical map is not birational. The
main result states that if X satisfies the above assumptions and $q(X)\ge 2$,
then, apart from a finite number of exceptions, X belongs to an infinite series
of examples due to Beauville. The exceptions are described in detail and some
new examples are given.


title: On Kodaira energy and adjoint reduction of polarized threefolds

abstract:
  Here we study and classify polarized threefolds whose Kodaira energies are
less than -1/2. The Kodaira energy of a polarized manifold (M, L), a pair
consisting of a smooth complex algebraic variety M and an ample line bundle L
on it, is defined to be $-\Inf{t\in Q\vert \kappa(K+tL)\ge0}$, where K is the
canonical bundle of M and $\kappa$ denotes the Iitaka dimension of a Q-bundle.
The method is a variant of those in my paper; manuscripta math. 76(1992).


title: Rankin-Cohen Operators for Jacobi and Siegel Forms

abstract:
  For any non-negative integer v we construct explicitly [v/2]+1 independent
covariant bilinear differential operators from J_{k,m} x J_{k',m'} to
J_{k+k'+v,m+m'}. As an application we construct a covariant bilinear
differential operator mapping S_k^{(2)} x S^{(2)}_{k'} to S^{(2)}_{k+k'+v}.
Here J_{k,m} denotes the space of Jacobi forms of weight k and index m and
S^{(2)}_k the space of Siegel modular forms of degree 2 and weight k. The
covariant bilinear differential operators constructed are analogous to
operators already studied in the elliptic case by R. Rankin and H. Cohen and we
call them Rankin-Cohen operators.


title: Sch\'emas de Fano

abstract:
  Let X be a subvariety of $P^n$ defined by equations of degrees $ d
=(d_1,...,d_s)$, over an algebraically closed field k of any characteristic. We
study properties of the Fano scheme $F_r(X)$ that parametrizes linear subspaces
of dimension r contained in X. We prove that $F_r(X)$ is connected and smooth
of the expected dimension for n big enough (this was previously known in
characteristic 0 or for r=1). Using Bott's theorem, we prove a vanishing
theorem for certain bundles on the Grassmannian and use it to calculate the
cohomology groups of $F_r(X)$ in degree $\le \dim X-2r$, and to prove that
$F_r(X)$ is projectively normal in the Grassmannian. Finally, we prove that for
n big enough, the rational Chow group $A_1(F_r(X))$ is of rank 1, and $F_r(X)$
is unirational. All bounds on n are effective.


title: Residue in intersection homology for quasihomogeneous singularities

abstract:
  Suppose M is a complex manifold of dimension $n+1$ and K is a hypersurface in
M. By Poincar\'e duality we define a residue morphism $res:H^{k+1}(M\setminus
K)\longrightarrow H_{2n-k}(K)$ which generalizes the classical Leray residue
morphism to cohomology for smooth K. We assume that K has isolated
quasihomogeneous singularities. Suppose $\omega$ is a holomorphic form of the
type $(n+1,0)$ with the first order pole on K. The purpose of this note is to
give a short, self contained proof of a criterion which tells us when the
residue of $\omega$ lifts to the intersection homology of K.


title:
On the semi-simplicity of the quantum cohomology algebras of complete
  intersections

abstract:
  In this short note, we study the semi-simplicity of the quantum cohomology
algebra H(V, C) of a Fano complete intersection V, and show that the subspace
of H(V, C) generated by the hyperplane sections is semi-simple in the sense of
Dubrovin when the degree of V satisfies certain conditions.


title:
Stationary Phase Integrals, Quantum Toda Lattices, Flag Manifolds and
  the Mirror Conjecture

abstract:
  A generalization of the mirror conjecture is proven for the manifolds of
complete flags in C^n.