# Brylinski-Radon transformation in characteristic $p>0$

Deepam Patel  Department of Mathematics, Purdue University, 150 N. University
Street, West Lafayette, IN 47907, U.S.A<EMAIL_ADDRESS>and  K.V.
Shuddhodan  Institut des Hautes Études Scientifiques, Université Paris-Scalay,
CNRS, Laboratoire Alexandre Grothendieck, Le Bois-Marie 35 rte de Chartres,
91440 Bures-sur-Yvette, France<EMAIL_ADDRESS>
###### Abstract.

In this article, we characterize the image of the Brylinski-Radon transform in
characteristic $p>0$ via Beilinson’s theory of singular supports. We also
provide an alternate proof of Brylinski’s results over ${\mathbb{C}}$, which
also works for sheaves with finite coefficients. Along the way, we also obtain
a microlocal criterion for the descent of perverse sheaves which could be of
independent interest.

KVS was supported by the CARMIN project fellowship.

###### Contents

  1. 1 Introduction
    1. 1.1 Summary of results:
      1. 1.1.1 Singular supports of étale sheaves:
      2. 1.1.2 Main Result:
    2. 1.2 Comparison with previous work
    3. 1.3 Idea of the proof
      1. 1.3.1 Proof of Theorem 1.1, (1):
      2. 1.3.2 Proof of Theorem 1.1, (2):
      3. 1.3.3 Proof of Theorem 1.1, (3):
  2. 2 Background and some preliminary observations
    1. 2.1 Recollection of singular support
    2. 2.2 Relative Brylinski-Radon transform
      1. 2.2.1 The incidence correspondence as a Grassmannian bundle:
      2. 2.2.2 Brylinski-Radon transform
  3. 3 Proof of Theorem 1: Preliminary Results
    1. 3.1 A criterion for descent of perverse sheaves.
    2. 3.2 A key proposition
  4. 4 Proof of Theorem 1.1, (1)
    1. 4.1 Some preliminary observations
    2. 4.2 An application of Artin vanishing
    3. 4.3 Proof of 1.1, (1) and Corollaries
  5. 5 Proof of Theorem 1.1, (2) and (3)
    1. 5.1 Proof of Theorem 1.1, (2) and corollaries
    2. 5.2 Proof of Theorem 1.1, (3)

##  1\. Introduction

In [3], Brylinski introduced topological (and geometric) versions of the
classical Radon transforms and proved some fundamental properties for these
transforms. The theory has had numerous applications including to Lefschetz
theory [3, I.III], [7, Chapter IV]. More recently the Radon transform was
crucially used by Beilinson [1] and Saito [10] respectively in the
construction of singular support and characteristic cycle for constructible
sheaves in the algebraic setting over arbitrary perfect fields. The main
result of this article is to use the theory of singular supports to
characterize the image of the Radon transform, generalizing the work of
Brylinski to arbitrary base fields (and finite coefficents). In particular, we
answer a question raised by Brylinski [3, Section 5.11].

###  1.1. Summary of results:

####  1.1.1. Singular supports of étale sheaves:

Let $k$ be an algebraically closed field of char. $p\geq 0$, $\ell\neq p$ a
fixed prime, $\Lambda={\mathbb{Z}}/l^{n}{\mathbb{Z}}$, and ${\rm
D}^{b}_{ctf}(X,\Lambda)$ denote the derived category of bounded constructible
étale sheaves of $\Lambda$-modules with finite tor-dimension on $X$. In the
rest of the article, we denote this category simply by ${\rm D}^{b}_{c}(X)$.
Given $K\in{\rm D}^{b}_{c}(X)$, $K(n)$ will denote the usual Tate twist of
$K$. If $X$ is smooth and $K\in{\rm D}^{b}_{c}(X)$, then Beilinson [1] defined
the singular support $SS(K)\subset{\rm T}^{*}X$ (see 2.1 for a brief summary
about singular supports). This is a closed conical subset of ${\rm T}^{*}X$,
and for $K\neq 0$, $SS(K)$ is equidimensional of dimension equal to $\dim(X)$.
Moreover when $\text{char}(k)=0$, $SS(K)$ is Lagrangian111A closed conical
subset of ${\rm T}^{*}X$ is said to be Lagrangian if the smooth locus of the
closed subset is both isotropic and involutive with respect to the natural
symplectic structure on ${\rm T}^{*}X$. [10, Proposition 2.2.7]. However this
fails in positive characteristic [1, Example 1.3].

####  1.1.2. Main Result:

Let ${\mathcal{P}}(X)\subset{\rm D}^{b}_{c}(X)$ denote the abelian category of
perverse sheaves (w.r.t middle perversity). In the following, given an object
${\rm K}\in{\rm D}^{b}_{c}(X)$, let
${{}^{p}\mathcal{H}}^{i}(K)\in{\mathcal{P}}(X)$ denote the $i$-th perverse
cohomology sheaf. If $X$ is smooth222In this article, varieties smooth over
$k$ shall be assumed to be connected. over $k$ of dimension $n$, let
${\mathcal{C}}(X)\subset{\mathcal{P}}(X)$ denote the full Serre subcategory of
locally constant perverse sheaves (i.e. complexes of the form
${\mathscr{L}}[n]$ where ${\mathscr{L}}$ is a locally constant constructible
sheaf on $X$), and ${\mathcal{A}}(X)$ the corresponding quotient category. One
can realize ${\mathcal{A}}(X)$ as the heart of the induced perverse
t-structure on a localized triangulated category ${\rm D}^{b}_{c}(X)_{T}$
obtained by localizing ${\rm D}^{b}_{c}(X)$ along the multiplicative set of
morphisms $f$ such that $\text{ker}({{}^{p}\mathcal{H}}^{i}(f))$ and
$\text{coker}({{}^{p}\mathcal{H}}^{i}(f))$ are locally constant perverse
sheaves for all $i$. As above, let
${{}^{p}\mathcal{H}}_{T}^{i}(K)\in{\mathcal{A}}(X)$ denote the $i$-th
‘perverse cohomology’ sheaf of the image of $K$ in ${\rm D}^{b}_{c}(X)_{T}$.

We now recall the Brylinski-Radon transform. Let $\mathbb{P}$ denote
projective space of dimension $n\geq 2$ over $k$, and $Y:={\bf G}(d)$ denote
the denote the Grassmanian of $d$-planes (where $1\leq d\leq n-1$) in
$\mathbb{P}$. Consider the incidence correspondence $Q\subset\mathbb{P}\times
Y$. The Brylinski-Radon transform is defined as follows. Consider the diagram:

$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\textstyle{\mathbb{P}}$$\textstyle{Y,}$

where $p_{i}$ are the natural projections. Given ${\mathcal{K}}\in{\rm
D}^{b}_{c}({\mathbb{P}}^{n})$, let
$\mathcal{R}({\mathcal{K}}):=p_{2,*}p_{1}^{\dagger}{\mathcal{K}}\in{\rm
D}^{b}_{c}(Y)$333Give a smooth morphism $f:X\to S$ of relative dimension $d$
with geometrically connected fibers, we set $f^{\dagger}:=f^{*}[d]$. Also, by
$f^{\dagger}{\mathcal{P}}(S)$, we mean the full subcategory ${\mathcal{P}}(X)$
consisting of perverse sheaves of the form $f^{\dagger}M$ [2, Proposition
4.2.5], where $M$ is a perverse sheaf on $S$.. Similarly, we set
$\check{{\mathcal{R}}}({\mathcal{K}}):=p_{1,*}p_{2}^{\dagger}({\mathcal{K}})$.

Let $C\subset{\rm T}^{*}{\mathbb{P}}$ be a closed conical subset. The
Brylinski-Radon transform of $C$ is defined to be $p_{2\circ}p_{1}^{\circ}(C)$
(see Section 2.1 for the notation). A closed conical subset of ${\rm T}^{*}Y$
is said to be in the image of the Brylinski-Radon transform if it is contained
in the Brylinski-Radon transform of a closed conical subset of ${\rm
T}^{*}{\mathbb{P}}$.

It follows from [1, Theorem 1.4, (ii)] that perverse sheaves whose singular
support is in the image of the Brylinski-Radon transform form an abelian
subcategory of ${\mathcal{P}}(Y)$ (denoted below by
${\mathcal{P}}(Y)_{\text{Rad}}$) which is stable under extensions. Let
${\mathcal{A}}(Y)_{\text{Rad}}$ be the full abelian subcategory of
${\mathcal{A}}(Y)$, consisting of objects which are images of
${\mathcal{K}}\in{\mathcal{P}}(Y)_{\text{Rad}}$. It is easy to see that both
${\mathcal{R}}$ and $\check{{\mathcal{R}}}$ naturally induce functors between
${\rm D}^{b}_{c}({\mathbb{P}})_{T}$ and ${\rm D}^{b}_{c}(Y)_{T}$. We are now
ready to state the main result of this article.

######  Theorem 1.1.

With notation as above:

  1. (1)

${\mathcal{R}}$ is $t$-exact for the perverse $t$-structures on ${\rm
D}^{b}_{c}({\mathbb{P}})_{T}$ and ${\rm D}^{b}_{c}(Y)_{T}$.

  2. (2)

The functor
${{}^{p}\mathcal{H}}_{T}^{d(n-d-1)}\circ\check{{\mathcal{R}}}(d(n-d))\circ{{}^{p}\mathcal{H}}_{T}^{0}\circ{\mathcal{R}}$
is naturally equivalent to the identity functor on
${\mathcal{A}}({\mathbb{P}})$.

  3. (3)

The functor ${{}^{p}\mathcal{H}}^{0}_{T}\circ{\mathcal{R}}$ induces an
equivalence of categories between ${\mathcal{A}}({\mathbb{P}})$ and
${\mathcal{A}}(Y)_{\text{Rad}}$.

###  1.2. Comparison with previous work

  1. (1)

If $k={\mathbb{C}}$, and one considers constructible sheaves in the classical
topology with ${\mathbb{C}}$ coefficients, then this is one of the main
results of Brylinski [3, Théorème 5.5]. The problem of a characteristic $p$
analogue of Brylinski’s theorem was already posed as a question by Brylinski
[3, 5.11]. The results of this article answer this question in the affirmative
albeit, with appropriate modifications to account for wild ramification.

  2. (2)

If $\text{char}(k)=0$, [10, Proposition 2.2.7] and [3, Lemme 5.6, (d)] imply
that one can alternatively describe ${\mathcal{P}}(Y)_{\text{Rad}}$ as those
perverse sheaves who singular support is contained in
$p_{2\circ}p_{1}^{\circ}{\rm T}^{*}{\mathbb{P}}$. In particular, the statement
of Theorem 1.1 is consistent with the analogous statement proved by Brylinski
over the complex numbers.

  3. (3)

If $d=n-1$, then the aforementioned theorem gives an equivalence of categories
between ${\mathcal{A}}({\mathbb{P}})$ and
${\mathcal{A}}(\check{{\mathbb{P}}})$. In this setting, Brylinski [3,
Corollaire 9.15] also proves the result over an algebraic closure of a finite
field as an application of the Deligne-Fourier transform in characteristic
$p>0$.

###  1.3. Idea of the proof

In this section we briefly describe the ideas underlying the proof of Theorem
1.1.

####  1.3.1. Proof of Theorem 1.1, (1):

The proof is an easy application of Artin vanishing and is along the lines of
the proof in [7, Chapter IV, Corollary 2.3], where the case of $n=d-1$ is
handled. The proof in [3, Théorème 5.5, (1)] is in comparison microlocal in
nature and does not carry over when the coefficients are finite.

####  1.3.2. Proof of Theorem 1.1, (2):

The essential point here is to understand the pushforward of the constant
sheaf along the map $Q\times_{Y}Q\to{\mathbb{P}}\times{\mathbb{P}}$. This map
is smooth outside the diagonal ${\mathbb{P}}$, however the fibers of the map
are Grassmannians. This allows us to compute
${\mathcal{R}}^{\vee}\circ{\mathcal{R}}$ in the localized category ${\rm
D}^{b}_{c}({\mathbb{P}})_{T}$ (see (14)) and deduce Theorem 1.1, (2). We do so
without recourse to the decomposition theorem which is technically important
for us since we allows finite coefficients. As a corollary of the proof we are
able to show that ${{}^{p}\mathcal{H}}^{0}\circ{\mathcal{R}}$ is fully
faithful and induces isomorphism on $\text{Ext}^{1}$ (see Corollary 5.3).

####  1.3.3. Proof of Theorem 1.1, (3):

The proof of Theorem 1.1, (3) constitutes the technical heart of the paper.
The first step is to prove a microlocal criterion for the descent of perverse
sheaves. More precisely we prove the following statement which generalizes a
result of Laumon444We thank Ahmed Abbes for pointing out the connection of our
result with Laumon’s work. [8, Proposition 5.4.1]. Let $k$ be a perfect field
and $S/k$ a smooth variety. Let $f:X\to S$ a proper and smooth morphism with
geometrically connected and simply connected fibers.

######  Proposition 1.2.

Then a non-zero perverse sheaf $L$ on $X$ is of the form $f^{\dagger}M$ iff
$SS(K)\subset f^{\circ}\Lambda$, for a closed conical subset
$\Lambda\subset{\rm T}^{*}S$ of dimension equal to $\text{dim}(S)$. Moreover
when $\text{char}(k)=0$, it suffices to assume that $SS(K)\subset
f^{\circ}{\rm T}^{*}S$.

Using this descent criterion and an inductive argument (see Proposition 3.7)
we are able to show that simple objects in ${\mathcal{A}}(Y)_{\text{Rad}}$ are
in the image of the Radon transform. The inductive nature of our method
naturally leads us to consider relative versions of Brylinski-Radon transforms
and we develop the necessary background in Section 2.2. The base case (i.e.
$n-d=1$) for the induction follows from the work of Laumon [9, Proposition
5.7]. Finally using the isomorphism on $\text{Ext}^{1}$ (Corollary 5.3) we
deduce Theorem 1.1, (3).

We would like to note that our proof of Theorem 1.1 also applies to
$\ell$-adic étale sheaves using the notion of singular support for $\ell$-adic
sheaves as described in [11, Section 5.5]. It also works when $k={\mathbb{C}}$
and one considers algebraically constructible sheaves in the analytic topology
with Kashiwara-Schapira’s [6, Chapter V] definition of singular supports.

Acknowledgements:

We would like to thank Ahmed Abbes for his interest and encouragement during
the course of this project. KVS would like to thank Ofer Gabber and Ankit Rai
for useful conversations. In particular, he is thankful to Ofer Gabber for
presenting a counterexample to an optimistic form of Corollary 3.3, ultimately
resulting in the formulation of Proposition 7. KVS would also like to thank
Hiroki Kato for patiently answering his questions about sensitivity of
vanishing cycles to test functions in positive characteristics.

##  2\. Background and some preliminary observations

###  2.1. Recollection of singular support

Let $X$ be a smooth variety over a perfect base field $k$. Let $C\subset{\rm
T}^{*}X$ denote a closed conical subset, and $h:U\rightarrow X$ a morphism
with $U$ smooth. Then $h$ is said to be $C$-transversal if for all geometric
points $u$ of $U$,

$ker(dh_{u})\cap C_{h(u)}\setminus\\{0\\}=\emptyset.$

Note $C$-transversality implies that $dh|_{C\times_{X}U}$ is finite and
Beilinson defines $h^{\circ}(C)$ to be its image in ${\rm T}^{*}U$, also a
closed conical subset [1, Section 1.2]. In particular, $h^{\circ}$ always
makes sense when $h$ is a smooth morphism (since such morphisms are
automatically $C$-transversal for any $C$). This will be the only relevant
case for us. Similarly, for any closed conical subset $C\subset{\rm T}^{*}U$
whose base is proper over $X$, Beilinson defines $h_{\circ}(C)$ to be the
image of $dh^{-1}(C)$ under the natural projection ${\rm
T}^{*}X\times_{X}U\to{\rm T}^{*}X$. This is a closed conical subset of ${\rm
T}^{*}X$.

For any sheaf $K\in{\rm D}^{b}_{c}(X)$, Beilinson defines the singular support
$SS(K)\subset{\rm T}^{*}(X)$. We recall some properties of $SS(K)$ which will
be used in the following.

  1. (1)

For $K\neq 0$, $SS(K)$ is a equidimensional closed conical subset of ${\rm
T}^{*}(X)$ of dimension equal to $\dim(X)$ [1, Theorem 1.3 (ii)] .

  2. (2)

Given an $SS(K)$-transversal morphism $h:U\rightarrow X$, $SS(h^{*}K)\subset
h^{\circ}(SS(K))$ [1, Lemma 2.2 (i)]. Moreover, one has equality if $h$ is a
smooth morphism [1, Theorem 1.4, (i))].

  3. (3)

Suppose $f\colon X\to Y$ is a proper morphism of smooth varieties, then for
any sheaf $K$ on $X$, $SS(f_{*}K)\subset f_{0}(SS(K))$ [1, Lemma 2.2 (ii))].

  4. (4)

$SS(K)$ is the zero section (denoted below by $0_{{\rm T}^{*}X}$) iff
$\mathcal{H}^{i}(K)$ are locally constant for all $i$ and atleast one of them
is non-zero [1, Lemma 2.1 (iii)].

  5. (5)

For any sheaf $K$ one has $SS(K)=\bigcup_{\alpha}SS(K_{\alpha})$, where
$K_{\alpha}$ runs over the various Jordan-Holder components of
${{}^{p}\mathcal{H}}^{i}(K)$ for every $i$ [1, Theorem 1.4, (ii)].

We record the following standard lemma for use below.

######  Lemma 2.1.

Let $X\xrightarrow{f}Y\xrightarrow{g}Z$ be smooth proper morphisms of smooth
varieties over an algebraically closed field $k$.

  1. (1)

Given a conic $\Lambda\subset{\rm T}^{*}X$, $(g_{\circ}\circ
f_{\circ})(\Lambda)=(g\circ f)_{\circ}(\Lambda)$.

  2. (2)

Given a conic $\Lambda\subset{\rm T}^{*}Z$, $(f^{\circ}\circ
g^{\circ})(\Lambda)=(g\circ f)^{\circ}(\Lambda)$.

  3. (3)

Given a conic $\Lambda\subset{\rm T}^{*}Y$, $(f_{\circ}\circ
f^{\circ})(\Lambda)=\Lambda$.

  4. (4)

Consider a commutative square:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{W}$

where all morphisms and varieties are smooth proper. Then, given
$\Lambda\subset{\rm T}^{*}Z$, one has $((g^{\prime})^{\circ}\circ
f^{\prime}_{\circ})(\Lambda)=(f_{\circ}\circ g^{\circ})(\Lambda)$

###### Proof.

The first three parts of the lemma are immediate from the definition. Using
(3) we can reduce (4) to the case when the diagram is cartesian in which case
the lemma is clear. ∎

###  2.2. Relative Brylinski-Radon transform

In what follows, we shall fix a base scheme $S$ which is assumed to be smooth
over an algebraically closed field $k$.

Let ${\mathscr{E}}$ be a vector bundle over $S$ of rank $n+1\geq 2$. Let
$0\leq d\leq n-1$ be an integer. We denote by ${\bf G}(d,{\mathscr{E}})$ the
Grassmannian bundle parametrizing locally free quotients of
${\mathscr{E}}^{\vee}$ of rank $d+1$. In particular, given an $S$-scheme
$\pi:T\rightarrow S$, ${\bf G}(d,{\mathscr{E}})(T)$ consists of equivalence
classes of quotients $\pi^{*}{\mathscr{E}}^{\vee}\rightarrow{\mathscr{F}}$
where ${\mathscr{F}}$ is locally free of rank $d+1$. We denote by
$\pi_{d,{\mathscr{E}}}$ the canonical morphism from ${\bf G}(d,{\mathscr{E}})$
to $S$. It is a proper and smooth morphism of relative dimension $(d+1)(n-d)$.

######  Remark 2.2.

Note that we may identify ${\bf G}(d,{\mathscr{E}})$ with ${\bf
G}(n-d-1,{\mathscr{E}}^{\vee})$ by passing to duals. Below, when working over
$S={\rm Spec}(k)$ (where $k$ is algebraically closed), we denote by ${\bf
G}(d,n)$555We use the convention that ${\bf G}(d,n)=\emptyset$ if $d$ is
negative. the Grassmanian of $d+1$-planes in $V=k^{n+1}$. We shall also
sometimes identify the latter with the $d$-planes in ${\mathbb{P}}^{n}$.

The following decomposition theorem is well-known, and is recorded here for
future use.

######  Lemma 2.3.

For any $K\in{\rm D}^{b}_{c}(S)$, there exists a functorial (in $K$)
isomorphism

(1) $\bigoplus_{i=0}^{n}K\otimes
R^{2i}\pi_{d,{\mathscr{E}}}\Lambda[-2i](i)\simeq\pi_{d,{\mathscr{E}}*}\pi_{d,{\mathscr{E}}}^{*}K$

###### Proof.

Using the projection formula we may assume that $K=\Lambda$. In this case the
result is a consequence of proper base change and [4, Théorème 1.5] owing to
the cohomology of Grassmannian satisfying hard Lefschetz (even with torsion
coefficients).

∎

####  2.2.1. The incidence correspondence as a Grassmannian bundle:

Given a pair of integers $0\leq d_{1}<d_{2}\leq n-1$, we denote by
$Q_{d_{1},d_{2},{\mathscr{E}}}\subset{\bf
G}(d_{1},{\mathscr{E}})\times_{S}{\bf G}(d_{2},{\mathscr{E}})$ the incidence
correspondence. More precisely, given a test scheme $T$ as above, recall that
an element of ${\bf G}(d_{1},{\mathscr{E}})\times_{S}{\bf
G}(d_{2},{\mathscr{E}})(T)$ is given by a tuple (upto equivalence)
$(\pi^{*}{\mathscr{E}}^{\vee}\rightarrow{\mathscr{F}}_{1},\pi^{*}{\mathscr{E}}^{\vee}\rightarrow{\mathscr{F}}_{2})$
where ${\mathscr{F}}_{i}$ is a rank $d_{i}+1$ quotient. With this notation,
$Q_{d_{1},d_{2},{\mathscr{E}}}(T)$ consists of tuples such that there is a
surjection ${\mathscr{F}}_{2}\rightarrow{\mathscr{F}}_{1}$ compatible with the
maps $\pi^{*}{\mathscr{E}}^{\vee}\rightarrow{\mathcal{F}}_{i}$. Note that if
such a surjection exists, it is unique. Moreover, this is a closed subscheme
of ${\bf G}(d_{1},{\mathscr{E}})\times_{S}{\bf G}(d_{2},{\mathscr{E}})$.

Let
$0\rightarrow{\mathscr{S}}_{n-d,{\mathscr{E}}^{\vee}}\rightarrow\pi^{*}_{d,{\mathscr{E}}}{\mathscr{E}}^{\vee}\rightarrow{\mathscr{Q}}_{d+1,{\mathscr{E}}^{\vee}}\rightarrow
0$ denote the universal exact sequence on ${\bf G}(d,{\mathscr{E}})$. Here
${\mathscr{S}}_{n-d,{\mathscr{E}}^{\vee}}$ (resp.
${\mathscr{Q}}_{d+1,{\mathscr{E}}^{\vee}}$) is the universal sub-bundle of
rank $n-d$ (resp. quotient of rank $d+1$). With this notation, one can
identify $Q_{d_{1},d_{2},{\mathscr{E}}}(T)$ as the rank $n-d_{2}$ quotients of
$\pi_{T}^{*}({\mathscr{S}}_{n-d_{1},{\mathscr{E}}^{\vee}}^{\vee})$, and in
particular, we may view $Q_{d_{1},d_{2},{\mathscr{E}}}\rightarrow{\bf
G}(d_{1},{\mathscr{E}})$ as the Grasmmannian bundle ${\bf
G}(n-d_{2}-1,{\mathscr{S}}_{n-d_{1},{\mathscr{E}}^{\vee}}^{\vee})$. By the
aforementioned remark, we may also view this as the Grassmannian bundle ${\bf
G}(d_{2}-d_{1}-1,{\mathscr{S}}_{n-d_{1},{\mathscr{E}}^{\vee}})$. In a similar
manner, we may view the incidence correspondence as a Grassmannian bundle
${\bf G}(d_{1},{\mathscr{Q}}_{d_{2}+1,{\mathcal{E}}^{\vee}}^{\vee})$ over
${\bf G}(d_{2},{\mathscr{E}})$.

We denote by $p_{d_{1},d_{2},{\mathscr{E}}}$ (resp.
$p^{\vee}_{d_{1},d_{2},{\mathscr{E}}}$, resp.
$\pi_{d_{1},d_{2},{\mathscr{E}}}$) the induced map from
$Q_{d_{1},d_{2},{\mathscr{E}}}$ to ${\bf G}(d_{1},{\mathscr{E}})$ (resp. ${\bf
G}(d_{2},{\mathscr{E}})$, resp. $S$). As noted above
$p_{d_{1},d_{2},{\mathscr{E}}}$ (resp. $p^{\vee}_{d_{1},d_{2},{\mathscr{E}}}$)
is a Grassmannian bundle parametrizing locally free quotients of rank
$d_{2}-d_{1}$ (resp. of rank $d_{1}+1$) of a vector bundle of rank $n-d_{1}$
(resp. of rank $d_{2}+1$) on ${\bf G}(d_{1},{\mathscr{E}})$ (resp. ${\bf
G}(d_{2},{\mathscr{E}})$). Thus $p_{d_{1},d_{2},{\mathscr{E}}}$ (resp.
$p^{\vee}_{d_{1},d_{2},{\mathscr{E}}}$) is proper and smooth of relative
dimension $(d_{2}-d_{1})(n-d_{2})$ (resp. $(d_{1}+1)(d_{2}-d_{1})$).

####  2.2.2. Brylinski-Radon transform

We define functors ${\mathcal{R}}_{d_{1},d_{2},{\mathscr{E}}}:{\rm
D}^{b}_{c}({\bf G}(d_{1},{\mathscr{E}}))\rightarrow{\rm D}^{b}_{c}({\bf
G}(d_{2},{\mathscr{E}}))$ and
$\check{{\mathcal{R}}}_{d_{1},d_{2},{\mathscr{E}}}:{\rm D}^{b}_{c}({\bf
G}(d_{2},{\mathscr{E}}))\rightarrow{\rm D}^{b}_{c}({\bf
G}(d_{1},{\mathscr{E}}))$ as follows,

(2)
${\mathcal{R}}_{d_{1},d_{2},{\mathscr{E}}}(K):=p^{\vee}_{d_{1},d_{2},{\mathscr{E}}*}p^{\dagger}_{d_{1},d_{2},{\mathscr{E}}}K$

and

(3)
$\check{{\mathcal{R}}}_{d_{1},d_{2},{\mathscr{E}}}(L):=p_{d_{1},d_{2},{\mathscr{E}}*}p^{\vee\dagger}_{d_{1},d_{2},{\mathscr{E}}}L.$

Finally, we make explicit a condition on closed conical subsets of ${\rm
T}^{*}{\bf G}(d,{\mathscr{E}})$ (resp. ${\rm
T}^{*}Q_{d_{1},d_{2},{\mathscr{E}}}$) which will be important in the
following666See Example 3.4 for a motivation to consider the condition
($\ast$)..

######  Definition 2.4.

We will say that a closed conical subset $C\subset{\rm T}^{*}{\bf
G}(d,{\mathscr{E}})$ (resp. ${\rm T}^{*}Q_{d_{1},d_{2},{\mathscr{E}}}$) is
regular over $S$ (or just regular if $S$ is clear from context) if the
following condition is satisfied:

  1. ($\ast$)

Every irreducible component $\Lambda$ of $C$ contained in
$\pi_{d,{\mathscr{E}}}^{\circ}{\rm T}^{*}S$ (resp.
$\pi_{d_{1},d_{2},{\mathscr{E}}}^{\circ}{\rm T}^{*}S$) is of the form
$\pi_{d,{\mathscr{E}}}^{\circ}\Lambda^{\prime}$ (resp.
$\pi_{d_{1},d_{2},{\mathscr{E}}}^{\circ}\Lambda^{\prime}$) for an irreducible
closed conical subset $\Lambda^{\prime}\subset{\rm T}^{*}S$.

Note that condition ($\ast$) above is trivially satisfied when $S={\rm
Spec}(k)$ and $C$ is of pure dimension of dimension equal to $\text{dim}({\bf
G}(d,{\mathscr{E}}))$ (resp. $\text{dim}(Q_{d_{1},d_{2},{\mathscr{E}}})$).

Let $C\subset{\rm T}^{*}{\mathbb{P}}({\mathscr{E}})$ be a closed conical
subset. We denote by

(4)
$\text{Rad}_{0,d,{\mathscr{E}}}(C):=(p^{\vee}_{0,d,{\mathscr{E}}})_{\circ}(p_{0,d,{\mathscr{E}}})^{\circ}(C),$

the Radon transform of $C$ with respect to $R_{0,d,{\mathscr{E}}}$. This is a
closed conical subset of ${\rm T}^{*}{\bf G}(d,{\mathscr{E}})$.

Let $q_{0,d,{\mathscr{E}}}$ and $q_{0,d,{\mathscr{E}}}^{\vee}$ denote the
morphism from ${\rm
T}^{*}_{Q_{0,d,{\mathscr{E}}}}({\mathbb{P}}({\mathscr{E}})\times_{S}{\bf
G}(d,{\mathscr{E}}))$ to ${\rm T}^{*}{\mathbb{P}}({\mathscr{E}})$ and ${\rm
T}^{*}{\bf G}(d,{\mathscr{E}})$ respectively. We need the following, which is
the relative version of [3, Lemme 5.6], and follows from it.

######  Lemma 2.5.

Let $\dot{q}_{0,d,{\mathscr{E}}}$ and $\dot{q}_{0,d,{\mathscr{E}}}^{\vee}$
respectively be the induced morphisms from ${\rm
T}^{*}_{Q_{0,d,{\mathscr{E}}}}({\mathbb{P}}({\mathscr{E}})\times_{S}{\bf
G}(d,{\mathscr{E}}))\backslash\pi_{0,d,{\mathscr{E}}}^{\circ}{\rm T}^{*}S$ to
${\rm
T}^{*}{\mathbb{P}}({\mathscr{E}})\backslash\pi_{0,{\mathscr{E}}}^{\circ}{\rm
T}^{*}S$ and ${\rm T}^{*}{\bf
G}(d,{\mathscr{E}})\backslash\pi_{d,{\mathscr{E}}}^{\circ}{\rm T}^{*}S$. Then

  1. (a)

$\dot{q}_{0,d,{\mathscr{E}}}$ is smooth and proper of relative dimension
$d(n-d-1)$.

  2. (b)

$\dot{q}_{0,d,{\mathscr{E}}}^{\vee}$ is a closed immersion.

As a consequence we have the following.

######  Corollary 2.6.

Let $C\subset{\rm T}^{*}Q_{0,d,{\mathscr{E}}}$ be a closed conical subset.
Suppose
$C=p_{0,d,{\mathscr{E}}}^{\circ}C_{1}=p_{0,d,{\mathscr{E}}}^{\vee\circ}C_{2}$
for closed conical subsets $C_{1}$ and $C_{2}$ in ${\rm
T}^{*}{\mathbb{P}}({\mathscr{E}})$ and ${\rm T}^{*}{\bf G}(d,{\mathscr{E}})$
respectively. Then $C\subset\pi_{0,d,{\mathscr{E}}}^{\circ}{\rm T}^{*}S$.

######  Remark 2.7.

Note that by Remark 2.2 the above corollary is also true for correspondences
between ${\bf G}(d,{\mathscr{E}})$ and ${\bf G}(n-1,{\mathscr{E}})$ with
$d<n-1$.

######  Corollary 2.8.

Let $M$ be perverse sheaf on $Q_{d,n-1,{\mathscr{E}}}$ (with $d<n-1$) that
belongs to both $p_{d,n-1}^{\dagger}{\mathcal{P}}({\bf G}(d,{\mathscr{E}}))$
and $p_{d,n-1}^{\vee\dagger}{\mathcal{P}}({\bf G}(n-1,{\mathscr{E}}))$. Then
$SS(M)\subset\pi_{Q_{d,n-1,{\mathscr{E}}}}^{\circ}{\rm T}^{*}S$.

###### Proof.

The corollary is an immediate consequence of the remark above and Section 2.1,
(2). ∎

We also note the following corollary.

######  Corollary 2.9.

Let $C\subset{\rm T}^{*}{\mathbb{P}}({\mathscr{E}})$ be a closed conical
subset regular over $S$. Then $\text{Rad}_{0,d,{\mathscr{E}}}(C)$ is also
regular over $S$.

###### Proof.

Let $\Lambda\subset C$ be an irreducible component of the form
$\pi_{0,{\mathscr{E}}}^{\circ}\Lambda^{\prime}$. Then Lemma 2.1 implies that
$\text{Rad}_{0,d,{\mathscr{E}}}(\Lambda)=\pi_{d,{\mathscr{E}}}^{\circ}\Lambda^{\prime}$.
On the other hand if $\Lambda$ is not contained in
$\pi_{0,{\mathscr{E}}}^{\circ}{\rm T}^{*}S$, then Lemma 2.5 implies that
$\text{Rad}_{0,d,{\mathscr{E}}}(\Lambda)$ is an irreducible component of
$\text{Rad}_{0,d,{\mathscr{E}}}(C)$ and is not contained in
$\pi_{d,{\mathscr{E}}}^{\circ}{\rm T}^{*}S$. ∎

##  3\. Proof of Theorem 1: Preliminary Results

In this section, we collect some results which will be used in the following
for the proof of part (3) of Theorem 1.1.

###  3.1. A criterion for descent of perverse sheaves.

As before, let $k$ be an algebraically closed field, $S/k$ be a smooth variety
and let $f\colon X\to S$ be a smooth morphism whose fibres are connected of
dimension $d$. In general, it is hard to characterise the subcategory
$f^{\dagger}{\mathcal{P}}(S)$ of ${\mathcal{P}}(X)$. If, in addition to the
above assumptions, $f$ is proper and the fibres of $f$ are simply connected,
then we have the following descent criterion.

######  Proposition 3.1.

777Our proof also works when $k$ is only assumed to be perfect, provided $f$
is geometrically connected.

A (non-zero) simple perverse sheaf $K\in{\mathcal{P}}(X)$ is in the essential
image of $f^{\dagger}$ iff $SS(K)\subseteq f^{\circ}\Lambda$, for some closed
conical subset $\Lambda\subset T^{*}S$ of dimension equal to $\text{dim}(S)$.
Moreover, when $\text{char}(k)=0$ it suffices to assume that $SS(K)\subset
f^{\circ}T^{*}S$.

###### Proof.

Since $f$ is smooth, the necessity results from the preservation of singular
supports under pullback (see Section 2.1, (2)). Suppose now that $K$ is a
(non-zero) simple perverse sheaf on $X$ such that as in $SS(K)\subseteq
f^{\circ}\Lambda$, with $\Lambda$ as in the proposition. Since $K$ is simple,
there exists a triple $(X^{\prime},U,{\mathscr{L}})$ consisting of an
irreducible closed subset $X^{\prime}\xhookrightarrow{i}X$, a non-empty smooth
(over $k$) open subset $U\xhookrightarrow{j}X^{\prime}$ and a non-zero
irreducible local system ${\mathscr{L}}$ on $U$ such that
$K=i_{*}j_{!*}{\mathscr{L}}[\text{dim}(X^{\prime})]$ [2, Théorème 4.3.1,
(ii)]. Note that $f^{\circ}$ preserves irreducible components since $f$ is
smooth. As a consequence, by removing any extra components (if necessary), we
may assume that $SS(K)=f^{0}\Lambda$.

Claim 1: It is sufficient to prove the theorem after replacing $S$ by an open
dense subset $S^{\prime}\xhookrightarrow{j^{\prime}}S$, $X$ by
$X_{S^{\prime}}:=X\times_{S}S^{\prime}$, and $K$ by $K|_{X_{S^{\prime}}}$
provided $K|_{X_{S^{\prime}}}$ is non-zero.
Proof: Let $j^{\prime\prime}:X_{S^{\prime}}\hookrightarrow X$ denote the
resulting open immersion. First note that the resulting map
$f^{\prime}:X_{S^{\prime}}\rightarrow S^{\prime}$ satisfies the hypotheses of
the theorem, and
$SS(K|_{X_{S^{\prime}}})=SS(K)|_{X_{S^{\prime}}}=(f^{\prime})^{\circ}(\Lambda|_{S^{\prime}})$
(Section 2.1, (2)). If $M$ is a simple perverse sheaf on $S^{\prime}$ such
that $(f^{\prime})^{\dagger}M=K|_{X_{S^{\prime}}}$, then
$f^{\dagger}j^{\prime}_{!*}(M)=j^{\prime\prime}_{!*}((f^{\prime})^{\dagger}M)=j^{\prime\prime}_{!*}(K|_{X_{S^{\prime}}})=K$.
Here the first equality follows from the fact that intermediate extensions
commute with pull back along smooth morphisms [2, Lemme 4.2.6.1], and the last
follows from the fact that $K$ is a simple perverse sheaf.

Claim 2: We may assume that the base $S^{\prime}$ of $\Lambda$ is smooth,
$X^{\prime}=X\times_{S}S^{\prime}$, and $SS(K)=f^{\circ}(\Lambda)$.
Proof: Let $S^{\prime}$ be the base of $\Lambda$. Since the base of $SS(K)$
equals the support of $K$ [1, Lemma 2.3 (iii)] we have
$X^{\prime}=f^{-1}(S^{\prime})$. Let $Z$ denote the singluar locus of
$S^{\prime}$. Since $k$ is algebraically closed, this is a strict closed
subset of $S^{\prime}$. In particular, $S\setminus Z$ is open, and by the
previous claim, we may base change everything to $S\setminus Z$.

Claim 3: Let $\Lambda^{\prime}$ be an irreducible component of
$f^{\circ}\Lambda$ which is not equal to ${\rm T}^{*}_{X^{\prime}}X$, the
conormal bundle of $X^{\prime}$ in $X$. Then the base of $\Lambda^{\prime}$
does not dominate $S^{\prime}$. In particular, the union of the bases of the
components of $SS(K)$ not equal to ${\rm T}^{*}_{X^{\prime}}X$ (denoted by
$X^{\prime\prime}$ below) cannot dominate $S^{\prime}$ under $f$.
Proof: Let $Z\subset X^{\prime}$ be the base of $\Lambda^{\prime}$. We claim
that $Z$ does not dominate $S^{\prime}$ under $f$. First note that, if $Z\neq
X^{\prime}$, then it does not dominate $S^{\prime}$. We’re reduced to showing
that if $Z=X^{\prime}$, then $\Lambda^{\prime}={\rm T}^{*}_{X^{\prime}}X$.
Since $X^{\prime}$ is smooth and
$K=i_{*}j_{!*}{\mathscr{L}}[\text{dim}(X^{\prime})]$,
$SS(K)=i_{0}SS(j_{!*}{\mathscr{L}}[\text{dim}(X^{\prime})])$ (Combine [1,
Lemma 2.5 (i)] and [1, Theorem 1.5]). Note that $i_{0}$ preserves bases of
irreducible components, and there exists a unique component of
$SS(j_{!*}{\mathscr{L}}[\text{dim}(X^{\prime})])$ whose base equals
$X^{\prime}$ (namely the zero section). It follows that there is a unique
component of $SS(K)$ whose base is $X^{\prime}$ (namely ${\rm
T}^{*}_{X^{\prime}}X$).

Note that $X^{\prime\prime}=f^{-1}(f(X^{\prime\prime}))\subsetneq X$. Let
$U^{\prime}=X^{\prime}\backslash X^{\prime\prime}$, then
$f|_{U^{\prime}}\colon U^{\prime}\to S^{\prime}\backslash f(X^{\prime\prime})$
is a proper morphism with connected and simply connected fibres. Thus by [5,
Exposé X, Corollaire 2.2] there exists a local system ${\mathscr{M}}$ on
$S^{\prime}\backslash f(X^{\prime\prime})$ such that
$f|_{U^{\prime}}^{*}{\mathscr{M}}={\mathscr{K}}$. Thus by uniqueness
$K=f^{*}(i_{S^{\prime}*}j_{U^{\prime}!*}({\mathscr{M}}[\text{dim}(S)])$, here
$i_{S^{\prime}}$ (resp. $j_{U^{\prime}}$) are the immersions from $S^{\prime}$
(resp. $U^{\prime}$) into $S$ (resp. $S^{\prime}$).

Now suppose $\text{char}(k)=0$, then every irreducible component (say
$\tilde{\Lambda}$) of $SS(K)$ is Lagrangian [10, Proposition 2.2.7] and
further the smooth locus of $\tilde{\Lambda}$ is the conormal to the smooth
locus in the intersection of $\tilde{\Lambda}$ with the zero section of ${\rm
T}^{*}X$ ([1], Exercise in Section 1.3). Such a component $\tilde{\Lambda}$ is
in $f^{0}{\rm T}^{*}S$ iff it is the inverse image of a closed conical subset
of ${\rm T}^{*}S$.

∎

######  Remark 3.2.

It follows from the proof of Proposition 7 that even in positive
characteristic, as long as the components of the singular support are
conormals (and not just Lagrangians!), the apparently weaker assumption
$SS(K)\subset f^{\circ}T^{*}S$ suffices.

While the following corollary will not be used in what follows, we record it
here since it may be of independent interest.

######  Corollary 3.3.

Let $f\colon X\to S$ and $K$ be as in Proposition 7. Then $K$ is lisse iff
${{}^{p}\mathcal{H}}^{d}f_{*}K$ is lisse.

We continue using the notation from Proposition 7. We record below an example
which shows that if $\text{char}(k)>0$, it is in general not sufficient to
assume $SS(K)\subset f^{0}T^{*}S$.

######  Example 3.4.

Let $k$ be a perfect field of characteristic $p>0$. Let
$S={\mathbb{A}}^{1}_{s}$,
$X={\mathbb{A}}^{1}_{s}\times{\mathbb{P}}^{1}_{[t:t^{\prime}]}$888We use
subscripts to denote a choice of a coordinate system, and $f\colon X\to S$ the
projection map. Let $\tilde{X}:=Z(t^{\prime
p}(x^{p^{2}}-x)-(s+x^{p})t^{p})\subseteq{\mathbb{A}}^{1}_{x}\times{\mathbb{A}}^{1}_{s}\times{\mathbb{P}}^{1}_{[t:t^{\prime}]}$
and denote by $\pi:\tilde{X}\to X$ the induced map. We denote by
$\tilde{X}_{t\neq 0}$ (resp. $X_{t\neq 0}$) and $\tilde{X}_{t^{\prime}\neq 0}$
(resp. $X_{t^{\prime}\neq 0}$) the open cover of $\tilde{X}$ (resp. $X$)
obtained from the usual cover on ${\mathbb{P}}^{1}_{[t:t^{\prime}]}$.

Note that $\tilde{X}$ is a smooth surface over $k$ and that $\pi$ is finite
étale of rank $p^{2}$ over $X_{t^{\prime}\neq 0}$. Over the line
$t^{\prime}=0$, it is a totally ramified cover of ${\mathbb{A}}^{1}_{s}$. Thus
$\pi$ is finite. and we denote by $K=\pi_{*}(\Lambda[2])$ and thus by Section
2.1, (3), $SS(K)\subseteq\pi_{\circ}(0_{{\rm T}^{*}\tilde{X}})$.

It follows from the definition of $\pi_{\circ}$ that $\pi_{\circ}(0_{{\rm
T}^{*}\tilde{X}})=0_{T^{*}X}\cup\Lambda$. Here $\Lambda$ is $f^{\circ}{\rm
T}^{*}{\mathbb{A}}^{1}_{s}|_{t^{\prime}=0}$. By proper base change $K$ is not
a lisse perverse sheaf, hence $SS(K)=0_{{\rm T}^{*}X}\cup\Lambda$. Moreover,
$K$ is not the pullback of a perverse sheaf from ${\mathbb{A}}^{1}_{s}$, since
if that were the case then its restriction to $s=0$ would have to be trivial
by proper base change. This in turn implies that the finite étale cover
$\tilde{X}_{t^{\prime}\neq 0}\to X_{t^{\prime}\neq 0}$ is trivial restricted
to $s=0$, which is not the case by the choice of the Artin-Schrier cover.

###  3.2. A key proposition

In this section, we prove a key proposition which will be used in the proof of
Theorem 1.1, (3). Recall we have a base scheme $S$ smooth over $k$ ( assumed
to be algebraically closed) and a vector bundle ${\mathscr{E}}$ on $S$ of rank
$n+1$. We continue using the notations from Section 2.2. However, for ease of
exposition, we drop ${\mathscr{E}}$ from the notation. In particular we shall
denote ${\bf G}(0,{\mathscr{E}})$ by ${\mathbb{P}}$, ${\bf
G}(d,{\mathscr{E}})$ by ${\bf G}(d)$ and ${\bf G}(n-1,{\mathscr{E}})$ by ${\bf
G}(n-1)$.

Below, we shall makes use of the following commutative diagram in order to
facilitate an inductive argument.

(5) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
159.37549pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
39.37967pt\raise-40.19998pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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101.75801pt\raise-40.19998pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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148.96054pt\raise-40.19998pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\bf
G}(d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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0.0pt\hbox{$\textstyle{Q_{0,n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
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84.55548pt\raise-80.6633pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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43.56479pt\raise-119.27661pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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101.75801pt\raise-119.27661pt\hbox{\hbox{\kern 0.0pt\raise
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0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{S}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

In diagram (5), the bottom, front and right hand side faces are the
correspondences described in Section 2.2. We define
$Q_{0,d,n-1}:=Q_{0,d}\times_{{\bf G}(d)}Q_{d,n-1}$. This induces a morphism
from $Q_{0,d,n-1}$ to ${\mathbb{P}}\times_{S}{\bf G}(n-1)$, which by
construction factors through $Q_{0,n-1}$ (denoted in the diagram (5) by the
dotted arrow). We have the following lemma which follows from the description
of the incidence correspondence as a Grassmannian bundle in Section 2.2.1.

######  Lemma 3.5.

There exists isomorphisms (as ${\bf G}(n-1)$-schemes)
$Q_{0,n-1}\simeq{\mathbb{P}}({\mathscr{Q}}^{\vee}_{n,{\mathscr{E}}^{\vee}})$,
$Q_{d,n-1}\simeq{\bf G}(d,{\mathscr{Q}}^{\vee}_{n,{\mathscr{E}}^{\vee}})$ and
$Q_{0,d,n-1}\simeq Q_{0,d,{\mathscr{Q}}^{\vee}_{n,{\mathscr{E}}^{\vee}}}$ such
that commutative square

(6) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
15.37967pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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46.3803pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{Q_{d,n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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0.0pt\hbox{$\textstyle{Q_{0,n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
39.37967pt\raise-40.12997pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
39.37967pt\raise-40.12997pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\bf
G}(n-1)}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

in diagram (5) is the one induced by the correspondence
$Q_{0,d,{\mathscr{Q}}^{\vee}_{n,{\mathscr{E}}^{\vee}}}\subset{\mathbb{P}}({\mathscr{Q}}^{\vee}_{n,{\mathscr{E}}^{\vee}})\times_{{\bf
G}(n-1)}{\bf G}(d,{\mathscr{Q}}^{\vee}_{n,{\mathscr{E}}^{\vee}})$.

###### Proof.

Note that $Q_{0,d,n-1}=Q_{d,n-1}\times_{({\bf G}(d)\times_{S}{\bf
G}(n-1))}(Q_{0,d}\times_{S}{\bf G}(n-1))$. Thus in order to prove the lemma,
it suffices to show that projective sub-bundle of ${\mathbb{P}}\times_{S}{\bf
G}(n-1)$ defined by $Q_{0,n-1}$ induces the Grassmannian sub-bundle
$Q_{d,n-1}$ of ${\bf G}(d)\times_{S}{\bf G}(n-1)$. But this follows from the
description in Section 2.2.1.

More precisely using the notations from the section, $Q_{0,n-1}$ is the
projective bundle (over ${\bf G}(n-1)$) defined by the sub-bundle
${\mathscr{Q}}^{\vee}_{n,{\mathscr{E}}^{\vee}}$ of
$\pi_{n-1,{\mathscr{E}}}^{*}{\mathscr{E}}$ and $Q_{d,n-1}$ is the Grassmannian
bundle ${\bf G}(d,{\mathscr{Q}}^{\vee}_{n,{\mathscr{E}}^{\vee}})$. ∎

In what follows we denote the vector bundle
${\mathscr{Q}}^{\vee}_{n,{\mathscr{E}}^{\vee}}$ on ${\bf G}(n-1)$ by
${\mathscr{F}}$. In particular there is a Radon transform (denoted by
${\mathcal{R}}_{0,d,{\mathscr{F}}}$) from ${\rm D}^{b}_{c}(Q_{0,n-1})$ to
${\rm D}^{b}_{c}(Q_{d,n-1})$. The following lemma is an immediate consequence
of proper base change applied to the cartesian square at the top of Diagram
(5).

######  Lemma 3.6.

For any perverse sheaf $K$ on ${\mathbb{P}}$ we have
${\mathcal{R}}^{i}_{0,d,{\mathscr{F}}}(p_{0,n-1}^{\dagger}K)\cong
p^{\dagger}_{d,n-1}{\mathcal{R}}^{i}_{0,d}(K)$999Here and in the rest of this
article by ${\mathcal{R}}^{i}_{d_{1},d_{2},{\mathscr{E}}}$ we mean
${{}^{p}\mathcal{H}}^{i}\circ{\mathcal{R}}_{d_{1},d_{2},{\mathscr{E}}}$. We
use a similar convention for
${\mathcal{R}}^{\vee}_{d_{1},d_{2},{\mathscr{E}}}$. in
${\mathcal{P}}(Q_{d,n-1})$.

Below, for $X$ smooth over $k$ and $\Lambda\subset{\rm T}^{*}X$ is a conical
subset, then ${\mathcal{P}}(X,\Lambda)$ is the full subcategory of the
category of perverse sheaves $K$ such that $SS(K)\subset\Lambda$. Note that
this is is a Serre subcategory (see Section 2.1, (5)).

Let $C\subset{\rm T}^{*}{\mathbb{P}}$ be a closed conical subset
equidimensional of dimension equal to $\text{dim}({\mathbb{P}})$. For the rest
of this section, we assume that closed conical subsets are regular over the
base $S$ (see Definition 2.4).

######  Proposition 3.7.

With notation as above, any simple perverse sheaf $L$ in ${\mathcal{P}}({\bf
G}(d),\text{Rad}_{0,d}(C))$ is either in $\pi_{d}^{\dagger}({\mathcal{P}}(S))$
or there exists a simple perverse sheaf $K$ on ${\mathbb{P}}$ and a
(decreasing) filtration $F^{\cdot}{\mathcal{R}}^{0}_{0,d}K$ on
${\mathcal{R}}^{0}_{0,d}K$ such that

  1. (a)

$SS(K)\subseteq C$.

  2. (b)

$F^{i}R^{0}_{0,d}K=R^{0}_{0,d}K$ for $i\leq 0$.

  3. (c)

$F^{i}R^{0}_{0,d}K=0$ for $i\geq 3$.

  4. (d)

$\text{Gr}^{i}_{F}({\mathcal{R}}^{0}_{0,d}K)$ belongs to
$\pi_{d}^{\dagger}{\mathcal{P}}(S)$ for $i=0,2$ and
$\text{Gr}^{1}_{F}({\mathcal{R}}^{0}_{0,d}K)=L$.

###### Proof.

We may assume $L$ does not belong to $\pi_{d}^{\dagger}({\mathcal{P}}(S))$. We
prove the claim by descending induction on $n-d$ (over varying choices of
$(S,{\mathscr{E}})$). Suppose $n-d=1$ and hence ${\bf
G}(d)={\mathbb{P}}({\mathscr{E}}^{\vee})$. Then (b)-(d) follow immediately
from [9, Corollaire 5.8, (i)]. Moreover, (a) follows from the fact that $K$ is
in fact a sub-quotient of $\check{{\mathcal{R}}}_{0,n-1}(L)$.

Now suppose the Proposition has been verified for $n-d=r\geq 1$ and for all
possible choices of $(S,{\mathcal{E}})$. We shall now prove it for $n-d=r+1$
by induction via Diagram (5). By the induction hypothesis, we may assume that
the Proposition has been verified for ${\mathcal{R}}_{0,d,{\mathscr{F}}}$.

It follows from [2, Corollaire 4.2.6.2] that
$L_{{\mathscr{F}}}:=p_{d,n-1}^{\dagger}L$ is simple and by Section 2.1, (2)
that $SS(L_{{\mathscr{F}}})=p_{d,n-1}^{\circ}SS(L)$. Thus by Lemma 2.1,
$SS(L_{{\mathscr{F}}})$ is contained in the Radon transform of
$p_{0,n-1}^{0}C$ with respect to $R_{0,d,{\mathscr{F}}}$. Moreover by
Corollary 2.8 it follows that $L_{{\mathscr{F}}}$ is not in the essential
image of $p^{\vee\dagger}_{d,n-1}{\mathcal{P}}({\bf G}(n-1))$. Now by
induction hypothesis there exists a simple perverse sheaf $K_{{\mathscr{F}}}$
on $Q_{0,n-1}$ with and a filtration
$F^{\cdot}_{{\mathscr{F}}}{\mathcal{R}}^{0}_{0,d,{\mathscr{F}}}$ such that

  1. (a’)

$SS(K_{{\mathscr{F}}})\subseteq p_{0,n-1}^{0}C$.

  2. (b’)

$F_{{\mathscr{F}}}^{i}R_{0,d,{\mathscr{F}}}^{0}K_{{\mathscr{F}}}=R_{0,d,{\mathscr{F}}}^{0}K_{{\mathscr{F}}}$
for $i\leq 0$.

  3. (c’)

$F_{{\mathscr{F}}}^{i}R_{0,d,{\mathscr{F}}}^{0}K_{{\mathscr{F}}}=0$ for $i\geq
3$.

  4. (d’)

$\text{Gr}^{i}_{F_{{\mathscr{F}}}}(R_{0,d,{\mathscr{F}}}^{0}K_{{\mathscr{F}}})$
belongs to $p_{d,n-1}^{\vee\dagger}{\mathcal{P}}({\bf G}(n-1))$ for $i=0,2$
and
$\text{Gr}^{1}_{F_{{\mathscr{F}}}}(R_{0,d,{\mathscr{F}}}^{0}K_{{\mathscr{F}}})=L_{{\mathscr{F}}}$.

Now using Proposition 7, (a’) above implies that $K_{{\mathscr{F}}}$ descends
to a simple perverse sheaf $K$ on ${\mathbb{P}}$ such $SS(K)\subseteq C$.
Moreover by Lemma 3.6, $R_{0,d,{\mathscr{F}}}^{0}K_{{\mathscr{F}}}$ is in the
essential image of $p_{d,n-1}^{\dagger}{\mathcal{P}}({\bf G}(d))$. Thus by [2,
Section 4.2.6] so are
$\text{Gr}^{i}_{F_{{\mathscr{F}}}}(R_{0,d,{\mathscr{F}}}^{0}K_{{\mathscr{F}}})$
for all $i$. Thus by Corollary 2.8 and Proposition 7,
$\text{Gr}^{i}_{F_{{\mathscr{F}}}}(R_{0,d,{\mathscr{F}}}^{0}K_{{\mathscr{F}}})$
for $i=0,2$ belongs to $(\pi_{d}\circ p_{d,n-1})^{\dagger}{\mathcal{P}}(S)$.
Hence the result.

∎

##  4\. Proof of Theorem 1.1, (1)

In the rest of this article we work over $S={\rm Spec}(k)$, with
${\mathscr{E}}$ a vector space over $k$ of dimension $n+1$ (which we
henceforth ignore from the notation) and use the following notation.

######  Notations 4.1.

We will only consider the Brylinksi-Radon transform between ${\mathbb{P}}$ to
${\bf G}(d)$.

  1. (1)

We will denote ${\bf G}(d)$ by $Y$ and the incidence correspondence $Q_{0,d}$
by $Q$. The projections from $Q$ to ${\mathbb{P}}$ (resp. $Y$) are denoted by
$p_{1}$ (resp. $p_{2}$).

  2. (2)

The morphism from ${\mathbb{P}}$ (resp. $Y$) to ${\rm Spec}(k)$ are denoted by
$\pi_{{\mathbb{P}}}$ (resp. $\pi_{Y}$).

  3. (3)

The Brylinski-Radon transforms are denoted by ${\mathcal{R}}$ and
$\check{{\mathcal{R}}}$.

  4. (4)

Let $E$ be the complement of the incidence variety $Q\subset{\mathbb{P}}\times
Y$. Let $p_{1}^{\circ}$ and $p_{2}^{\circ}$ be the projections to
${\mathbb{P}}$ and $Y$ respectively from $E$.

  5. (5)

In what follows we will need the modified Brylinski-Radon transform defined as
${\mathcal{R}}_{!}K:=p^{\circ}_{2!}p_{1}^{\circ\dagger}K$.

  6. (6)

For a complex $K$ on ${\mathbb{P}}$, by ${\mathscr{K}}$ we mean the complex
$\pi_{{\mathbb{P}}*}K$ on ${\rm Spec}(k)$. Similarly, for complexes $K$ on
$Y$.

  7. (7)

We will use ${\mathcal{R}}^{i}(K)$ (resp. ${\mathcal{R}}_{!}^{i}(K)$,
${\mathscr{K}}^{i}$) to denote the $i^{\mathrm{th}}$ perverse cohomology of
${\mathcal{R}}(K)$ (resp. ${\mathcal{R}}_{!}(K)$, ${\mathscr{K}}$).

###  4.1. Some preliminary observations

The next two lemmas are immediate consequences of the smoothness and
properness of $p_{1}$ and $p_{2}$, and we state them without a proof.

######  Lemma 4.2.

For any sheaf $K\in{\rm D}^{b}_{c}({\mathbb{P}})$ and $L\in{\rm
D}^{b}_{c}(Y)$, $D({\mathcal{R}}(K))\simeq{\mathcal{R}}(DK)(d(n-d))$ and
$D(\check{{\mathcal{R}}}(L))=\check{{\mathcal{R}}}(DL)(d)$ 101010Here $D$ is
the Verdier duality functor..

######  Lemma 4.3.

The functors
$(\check{{\mathcal{R}}}[\delta](d(n-d)),{\mathcal{R}},\check{{\mathcal{R}}}[-\delta](d))$111111In
what follows we set $\delta:=d(n-d-1)$ form an adjoint triple.

The following result is due to Brylinski [3, 5.3.1 (i), (ii)]. Again, while
this is proved in loc. cit. in the complex analytic setting, the same proof
goes through in our setting.

######  Proposition 4.4.

Let ${\mathcal{R}}$ and $\check{{\mathcal{R}}}$ be as before. Then
${\mathcal{R}}$ and $\check{{\mathcal{R}}}$ preserve the localizing set $T$
(see Section 1.1), and in particular one has induced functors
${\mathcal{R}}:{\rm D}^{b}_{c}({\mathbb{P}})_{T}\rightarrow{\rm
D}^{b}_{c}(Y)_{T}$ and $\check{{\mathcal{R}}}:{\rm
D}^{b}_{c}(Y)_{T}\rightarrow{\rm D}^{b}_{c}({\mathbb{P}}^{n})_{T}.$

###  4.2. An application of Artin vanishing

We now record the following easy consequence of Artin vanishing which is used
in the proof of Theorem 1.1, (1).

######  Lemma 4.5.

Let $X/k$ be a base scheme. Let $U$ be the complement in
${\mathbb{P}}^{n}_{X}$ of a linear subspace121212A linear subspace of
${\mathbb{P}}^{n}_{X}$ is a closed subscheme, which Zariski locally over $X$
isomorphic to ${\mathbb{P}}^{d}_{X}\subset{\mathbb{P}}^{n}_{X}$ embedded
linearly. $Z$ of relative dimension $d$, and let $\pi$ be the map from $U$ to
$X$. Then $\pi_{*}$ maps ${}^{p}{\rm D}^{\leq 0}(U)$ to ${}^{p}{\rm D}^{\leq
n-d-1}(X)$.

###### Proof.

The proof is via a repeated application of Artin vanishing in the form of
right t-exactness (for the perverse t-structure) of affine morphisms [2,
Théorème 4.4.1]. After replacing $X$ with a suitable Zariski open we can
consider a chain of linear subspaces $Z_{0}\subsetneq Z_{1}\subsetneq\cdots
Z_{n-d-1}$ of $Z$ such that $Z_{0}={\mathbb{P}}^{d}_{X}$ and
$\text{dim}(Z_{i})=d+i$. Let $U_{i}:={\mathbb{P}}^{n}_{X}\backslash Z_{i}$ be
the corresponding open subscheme. Let $\pi_{i}$ be the map from $U_{i}$ onto
$X$, and we identify $\pi_{0}$ with $\pi$.

We prove the lemma by descending induction on $i$. For $i=n-d-1$ the lemma is
an immediate consequence of Artin vanishing [2, Théorème 4.1.1]. Assuming that
the lemma has been verified up to some $i\leq n-d-1$, we prove it for $i-1$.
Let $j$ (resp. $l$ ) be the inclusion of $U_{i}$ (resp. $Z_{i}\backslash
Z_{i-1}$) inside $U_{i-1}$. Let $K$ be a sheaf on $U_{i-1}$ in ${}^{p}{\rm
D}^{\leq 0}(U_{i-1})$. By induction hypothesis
$\pi_{*}(j_{*}j^{*}K)\in~{}^{p}{\rm D}^{\leq n-d-1-i}(X)$. Thus it suffices to
show $\pi_{*}(l_{*}l^{!}K)\in~{}^{p}{\rm D}^{\leq n-d-i}(X)$.

By construction $Z_{i}\backslash Z_{i-1}$ is at once affine over $X$ and a
complete intersection of codimension $n-d-i$ in $U_{i-1}$, and thus [2,
Corollaire 4.1.10, (ii)] implies the result.

∎

The following corollary will be used below to describe the image of the
Brylinski-Radon transform.

######  Corollary 4.6.

With notation as above, $p^{\circ}_{2!}$ maps ${}^{p}{\rm D}^{\geq 0}(E)$ to
${}^{p}{\rm D}^{\geq-(n-d-1)}(Y)$.

###  4.3. Proof of 1.1, (1) and Corollaries

In fact, we prove the following more refined version of Theorem 1.1, part (1).

######  Theorem 4.7.

Let $K$ be a sheaf on ${\mathbb{P}}$.

  1. (1)

If $K$ is upper semi-perverse then for any $i<0$, we have
${\mathcal{R}}^{i}(K)\simeq\pi_{Y}^{\dagger}{\mathscr{K}}^{i-n+d}$.

  2. (2)

If $K$ is perverse, ${\mathcal{R}}^{i}(K)$ are constant for any $i\neq 0$.
Also the perverse sheaves ${\mathcal{R}}^{i}_{!}(K)$ are constant for $i\neq
n-d+1$.

  3. (3)

Consequently ${\mathcal{R}}$ is $t$-exact for the perverse $t$-structures on
${\rm D}^{b}_{c}({\mathbb{P}})_{T}$ and ${\rm D}^{b}_{c}(Y)_{T}$ (see Section
1.1).

###### Proof.

By definition of ${\mathcal{R}}$ (and ${\mathcal{R}}_{!}$) and proper base
change, we have a triangle on $Y$

(7) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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105.95769pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{\pi_{Y}^{*}{\mathcal{K}}[(d+1)(n-d)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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216.45439pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Now, by Corollary 4.6 and [2, Section 4.2.4], one has that for any
$K\in~{}^{p}{\rm D}^{\geq 0}({\mathbb{P}})$, ${\mathcal{R}}_{!}K\in~{}^{p}{\rm
D}^{\geq-(n-d-1)}(Y)$. Taking the long exact sequence of perverse cohomologies
associated to the triangle (7) gives us (1).

If $K$ is perverse, then applying the first part to $DK$ and using Lemma 4.2
we deduce (2). The constancy of ${\mathcal{R}}_{!}^{i}(K)$ for $i\neq n-d+1$
then follows from the fact that constant sheaves form a Serre subcategory. The
$t$-exactness of ${\mathcal{R}}$ is now clear.

∎

We get the following corollaries by combining Lemma 4.3 and Theorem 4.7.

######  Corollary 4.8.

The functor $\check{{\mathcal{R}}}[-\delta](d)$ (resp.
$\check{{\mathcal{R}}}[\delta](d(n-d))$) is left $t$-exact (resp. right
$t$-exact) for the perverse $t$-structures on ${\rm D}^{b}_{c}(Y)_{T}$ and
${\rm D}^{b}_{c}({\mathbb{P}})_{T}$.

######  Corollary 4.9.

$(\check{{\mathcal{R}}}^{\delta}(d(n-d)),{\mathcal{R}}^{0},\check{{\mathcal{R}}}^{-\delta}(d))$131313We
denote ${{}^{p}\mathcal{H}}^{i}_{T}\circ R$ by ${\mathcal{R}}^{i}$ and a
similar notation for $\check{{\mathcal{R}}}^{i}$. form an adjoint triple
between ${\mathcal{A}}({\mathbb{P}})$ and ${\mathcal{A}}(Y)$. Moreover
$\check{{\mathcal{R}}}^{-\delta}(d)$ (resp.
$\check{{\mathcal{R}}}^{\delta}(d(n-d))$) is left t-exact (resp. right
t-exact).

##  5\. Proof of Theorem 1.1, (2) and (3)

In this section, we prove Theorem 1.1, (2) and (3).

###  5.1. Proof of Theorem 1.1, (2) and corollaries

Consider the following diagram of schemes, where the central square is
cartesian by definition:

(8) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-38.77774pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\hbox{$\textstyle{\mathbb{P}}$}}}}}}}{\hbox{\kern
34.35556pt\raise-76.58327pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Y}$}}}}}}}{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{P}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Let $\pi:Q\times_{Y}Q\to{\mathbb{P}}\times{\mathbb{P}}$ denote the morphism
induced by $p_{1}$ on each factor. Let
$s_{1}:{\mathbb{P}}\times{\mathbb{P}}\rightarrow{\mathbb{P}}$ (resp.
$s_{2}:{\mathbb{P}}\times{\mathbb{P}}\rightarrow{\mathbb{P}}$) be the
projection onto the first (resp. second) factor. An application of proper base
change along the central cartesian square in diagram (8) and the projection
formula gives a natural (in ${\mathcal{K}}$) isomorphism:

(9)
$\check{{\mathcal{R}}}\circ{\mathcal{R}}(K)=s_{2*}\left(s_{1}^{*}K\otimes_{\Lambda}\pi_{*}\Lambda[\delta_{+}]\right).$

Let $\Delta:{\mathbb{P}}\hookrightarrow{\mathbb{P}}\times{\mathbb{P}}$ denote
the diagonal embedding, let $U$ be the complement of the diagonal embedding,
and let $j\colon U\hookrightarrow{\mathbb{P}}\times{\mathbb{P}}$ be the
corresponding open immersion. One has the resulting diagram with cartesian
squares:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
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35.4116pt\raise-38.22218pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{{\mathbb{P}}\times{\mathbb{P}}}$}}}}}}}{\hbox{\kern
92.71704pt\raise-38.22218pt\hbox{\hbox{\kern 0.0pt\raise
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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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We note that $\pi_{U}$ is a Grassmann bundle with fibers ${\bf G}(d-2,n-2)$.
Consider the natural closed immersion $Q\times_{Y}Q\rightarrow
Q\times{\mathbb{P}}$, which on closed points maps $(x,y,L)$ to $(x,L,y)$. Here
$x,y$ are closed points of ${\mathbb{P}}$ and $L\subset{\mathbb{P}}$ is a
$d$-plane containing them. The above commutative diagram factors as:

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where all the squares are Cartesian.

Note that $\tilde{\pi}$ is a Grassmannian bundle with fibers ${\bf
G}(d-1,n-1)$ and is identity along the second projection. Let $Z:=V\setminus
W=Q\times{\mathbb{P}}\setminus Q\times_{Y}Q$, and $\pi_{Z}:Z\rightarrow U$
denote the resulting morphism. We have an exact triangle on $U$

(10) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Since $\pi_{U}$ is a Grassmannian bundle, Lemma 2.3 implies that
$\pi_{U*}\Lambda$ is formal151515A sheaf is said to be formal if it is
isomorphic to a shifted direct sum of its cohomology sheaves and its
cohomology sheaves are locally constant. Since $U$ is simply connected [5,
Exposé X, Corollaire 3.3], they are in fact constant. Let
${\mathscr{M}}_{d-2,n-2}:=\oplus_{i}\underline{M^{i}_{d-2,n-2}}[-i]$161616For
any $\Lambda$-module $M$, by $\underline{M}$ we mean the constant local system
on ${\mathbb{P}}\times{\mathbb{P}}$ with values in $M$., here
$M_{d-2,n-2}^{i}:=H^{0}(U,R^{i}\pi_{U*}\Lambda)$. The restriction of
${\mathscr{M}}_{d-2,n-2}$ to $U$ is isomorphic to $\pi_{U*}\Lambda$171717The
choice of ${\mathscr{M}}_{d-2,n-2}$ is not unique in as much as the choice of
the decomposition in Lemma 2.3, but this non-uniqueness does not play a role
in what follows.. We also denote by
${\mathscr{M}}_{d-1,n-1}:=\tilde{\pi}_{*}\Lambda$. We have exact triangles,

(11) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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(12) $\textstyle{{\mathscr{M}}_{d-1,n-1}\otimes
j_{!}\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{M}}_{d-1,n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{M}}_{d-1,n-1}\otimes\Delta_{*}\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{+1}$

and

(13) $\textstyle{{\mathscr{M}}_{d-2,n-2}\otimes
j_{!}\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{M}}_{d-2,n-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{M}}_{d-2,n-2}\otimes\Delta_{*}\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{+1}$

in ${\rm D}^{b}_{c}({\mathbb{P}}\times{\mathbb{P}})$. Now note that for any
sheaf $K$ on ${\mathbb{P}}$ and any constant sheaf (i.e. the cohomology
sheaves are constant) $L$ on ${\mathbb{P}}\times{\mathbb{P}}$, the sheaf
$s_{2*}(s_{1}^{*}K\otimes L)$ is also constant. Thus combining triangles
(10)-(13) and Equation (9) we get a functorial (in $K$) exact triangle in the
localized category ${\rm D}^{b}_{c}({\mathbb{P}})_{T}$,

(14) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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3.0pt\raise
0.0pt\hbox{$\textstyle{K\otimes\Delta^{*}{\mathscr{M}}_{d-2,n-2}[\delta_{+}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
231.20341pt\raise 5.54723pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.9639pt\hbox{$\scriptstyle{+1}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 250.67564pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
250.67564pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

######  Claim 5.1.

  1. (a)

For any perverse sheaf $K$ on ${\mathbb{P}}$, there exists a natural
isomorphism
$\check{{\mathcal{R}}}^{i}({\mathcal{R}}(K))\simeq\check{{\mathcal{R}}}^{i}({\mathcal{R}}^{0}(K))$
in ${\mathcal{A}}({\mathbb{P}})$ (and hence in
$D^{b}_{c}({\mathbb{P}},\Lambda)_{T}$).

  2. (b)

For any perverse sheaf $K$ on ${\mathbb{P}}$, there exists functorial (in $K$)
isomorphisms in ${\mathcal{P}}({\mathbb{P}})$ (and hence in
${\mathcal{A}}({\mathbb{P}})$)

${{}^{p}\mathcal{H}}^{i}(K\otimes\Delta^{*}{\mathscr{M}}_{d-1,n-1}[\delta_{+}])\simeq
K\otimes\Delta^{*}H^{i+\delta_{+}}({\mathscr{M}}_{d-1,n-1})$

and

${{}^{p}\mathcal{H}}^{i}(K\otimes\Delta^{*}{\mathscr{M}}_{d-2,n-2}[\delta_{+}])\simeq
K\otimes\Delta^{*}H^{i+\delta_{+}}({\mathscr{M}}_{d-2,n-2}).$

  3. (c)

For $i=\delta-1,\delta$, the perverse sheaves
${{}^{p}\mathcal{H}}^{i}(K\otimes\Delta^{*}{\mathscr{M}}_{d-2,n-2}[\delta_{+}])$
vanish. Also
${{}^{p}\mathcal{H}}^{\delta-1}(K\otimes\Delta^{*}{\mathscr{M}}_{d-1,n-1}[\delta_{+}])$
vanishes. Moreover when $n-d>1$,
${{}^{p}\mathcal{H}}^{\delta-2}(K\otimes\Delta^{*}{\mathscr{M}}_{d-2,n-2}[\delta_{+}])$
is also zero.

  4. (d)

For any perverse sheaf $K$ on ${\mathbb{P}}$, there exists a natural (in $K$)
isomorphism in ${\mathcal{P}}({\mathbb{P}})$ (and hence in
${\mathcal{A}}({\mathbb{P}})$),
${{}^{p}\mathcal{H}}^{\delta}(K\otimes\Delta^{*}{\mathscr{M}}_{d-1,n-1}[\delta_{+}])\simeq
K(-d(n-d))$.

###### Proof.

Claim (a) is an immediate consequence of Theorem 4.7. Claims (b) follows from
the formality of ${\mathscr{M}}_{d-1,n-1}$ and ${\mathscr{M}}_{d-2,n-2}$ and
the fact that their cohomology sheaves are local systems.

For claim (c), using (b) it suffices to prove that
$\Delta^{*}H^{i+\delta_{+}}({\mathscr{M}}_{d-2,n-2})$ vanishes for
$\delta-2\leq i\leq\delta$, and that
$\Delta^{*}H^{\delta_{+}+\delta-1}({\mathscr{M}}_{d-1,n-1})=0$. In either case
note that the cohomology sheaves of ${\mathscr{M}}_{d-2,n-2}$ and
${\mathscr{M}}_{d-1,n-1}$ are constant local systems and hence by their
definitions it suffices to show that $R^{i+\delta_{+}}\pi_{U*}\Lambda$ for
$\delta-2\leq i\leq\delta$ and $R^{\delta_{+}+\delta-1}\tilde{\pi}_{*}\Lambda$
vanish. But these follow immediately from the fact that $\pi_{U}$ is a ${\bf
G}(d-2,n-2)$ bundle181818We require $n-d>1$, to ensure that $\text{dim}({\bf
G}(d-2,n-2))<d(n-d)-1$. and that $\tilde{\pi}$ is a ${\bf G}(d-1,n-1)$ bundle.

For claim (d) arguing as above we conclude that
${{}^{p}\mathcal{H}}^{\delta}(K\otimes\Delta^{*}{\mathscr{M}}_{d-1,n-1}[\delta_{+}])\simeq
K\otimes\Delta^{*}R^{2d(n-d)}\tilde{\pi}_{*}\Lambda\simeq K(-d(n-d))$.

∎

Combining claims (a)-(d) above shows that there exists a natural isomorphism

$\check{{\mathcal{R}}}^{\delta}(d(n-d))\circ{\mathcal{R}}^{0}(K)\simeq K$

in ${\mathcal{A}}({\mathbb{P}})$, and therefore complete the Proof of Theorem
1.1 (2). It is also easy to see this map is the co-unit of the adjunction in
Corollary 4.9. Finally, combining Lemma 4.2 and Corollary 4.9 we obtain the
following.

######  Corollary 5.2.

The unit of the adjunction
$K\to\check{{\mathcal{R}}}^{-\delta}(d)\circ{\mathcal{R}}^{0}(K)$ is an
isomorphism in ${\mathcal{A}}({\mathbb{P}})$.

We also have the following corollary of the method of the proof.

######  Corollary 5.3.

We have ${\rm Ext}^{i}_{{\mathcal{A}}({\mathbb{P}})}(K_{1},K_{2})\simeq{\rm
Ext}^{i}_{{\mathcal{A}}(Y)}({\mathcal{R}}^{0}(K_{1}),{\mathcal{R}}^{0}(K_{2}))$
for $i=0,1$.

###### Proof.

The isomorphism for $i=0$ is an immediate consequence of Theorem 1.1, (2) and
the adjunction between $\check{{\mathcal{R}}}^{\delta}(d)$ and
${\mathcal{R}}^{0}$ (Corollary 4.9). We may now assume that $n-d>1$, else the
result follows from the fact that ${\mathcal{R}}^{0}$ induces an equivalence
between ${\mathcal{A}}({\mathbb{P}})$ and
${\mathcal{A}}(\check{{\mathbb{P}}})$ from from Theorem 1.1, (1) and (2).

The triangle (14) and Claim 5.1, (b) and (c) above imply that for
$K\in{\mathcal{A}}({\mathbb{P}})$,

${{}^{p}\mathcal{H}}^{-1}_{T}(\check{{\mathcal{R}}}[\delta]\circ{\mathcal{R}}K(d(n-d)))\simeq
0.$

Since $\check{{\mathcal{R}}}[\delta](d(n-d))$ is right $t$-exact and
${\mathcal{R}}$ is exact, this implies that

(15)
$^{p}_{T}\tau_{\geq-1}\check{{\mathcal{R}}}[\delta]\circ{\mathcal{R}}K(d(n-d))\simeq\check{{\mathcal{R}}}^{\delta}(d(n-d))\circ{\mathcal{R}}^{0}(K),$

which by Theorem 1.1, (2) is isomorphic to $K$ under the co-unit of
adjunction.

We also have

(16) ${\rm
Ext}^{1}_{{\mathcal{A}}(Y)}({\mathcal{R}}^{0}(K_{1}),{\mathcal{R}}^{0}(K_{2}))\simeq\text{Hom}_{{\rm
D}^{b}_{c}({\mathbb{P}})_{T}}(\check{{\mathcal{R}}}[\delta]\circ{\mathcal{R}}K_{1}(d),K_{2}[1])$

and

$\text{Hom}_{{\rm
D}^{b}_{c}({\mathbb{P}})_{T}}(\check{{\mathcal{R}}}[\delta]\circ{\mathcal{R}}K_{1}(d),K_{2}[1])\simeq{\rm
Hom}_{{\rm
D}^{b}_{c}({\mathbb{P}})_{T}}(^{p}_{T}\tau_{\geq-1}\check{{\mathcal{R}}}[\delta]\circ{\mathcal{R}}K_{1}(d(n-d)),K_{2}[1]).$

The first equality being adjunction and the second since $K_{1}$ and $K_{2}$
are perverse, $\check{{\mathcal{R}}}[\delta](d(n-d))$ is right $t$-exact and
${\mathcal{R}}$ is exact. Combining this with (15) gives the necessary
equality.

∎

###  5.2. Proof of Theorem 1.1, (3)

###### Proof.

Thanks to 1.1, (2) and Corollary 5.3, it suffices to show that the simple
objects in ${\mathcal{A}}(Y)_{\text{Rad}}$ are in the image of
${\mathcal{R}}^{0}$. This follows from Proposition 3.7. ∎

Example 3.4 naturally leads to the following question which we have been
unable to answer:

######  Question 5.4.

Does there exist a perverse sheaf on $Y$ with singular support inside
$p_{2\circ}p_{1}^{\circ}{\rm T}^{*}{\mathbb{P}}$ whose image is not in
${\mathcal{A}}(Y)_{\text{Rad}}$, and hence the perverse sheaf is not in the
image of the Radon transform?

Note that the answer to the above question is negative in characteristic $0$
(see Section 1.2, (2)) or when $d=n-1$.

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