Brylinski-Radon transformation in characteristic $p>0$

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].

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:

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.

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.

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).

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.

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.

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. $(d_{1}+1)(d_{2}-d_{1})$).

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,

and

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 ($\ast$)..

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

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.

As a consequence we have the following.

We also note the following corollary.