## A trick and the decomposition theorem

In this post I’ll talk about a really fun trick Bhargav explained to me yesterday.

Let $K$ be a field with separable closure $C$. Algebraic variety over $K$ means separated $K$-scheme of finite type. Let $\ell$ be a prime invertible in $K$. Perverse sheaf means perverse $\mathbf{Q}_\ell$-sheaf.

If $f:X \to Y$ is a proper map of algebraic varieties over $K$, the decomposition theorem tells you that after base extension to $C$ there is a direct sum decomposition

$Rf_{\ast}IC_{X,\mathbf{Q}_\ell} \simeq \oplus_i IC_{Z_i}(\mathcal{L}_i)[n_i]\,\,\,\,\,\,\,\,(\dagger)$

in $D^b_c(Y_C,\mathbf{Q}_\ell)$. Here $Z_i \subset Y_{C}$ is some finite set of closed subvarieties, and $\mathcal{L}_i$ is a lisse $\mathbf{Q}_\ell$-sheaf on the smooth locus of $Z_i$. (My convention is that $IC_{Z}(\mathcal{L}) = j_{!\ast} (\mathcal{L}[\dim Z])$ where $j:Z^{sm} \to X$ is the natural map, so $IC_{X,\mathbf{Q}_\ell} = IC_{X}(\mathbf{Q}_\ell)$. Some people have different conventions for shifts here.)

The decomposition $(\dagger)$ is non-canonical. In particular, it is not $\mathrm{Aut}(C/K)$-equivariant, so it has no reason to descend to an analogous direct sum decomposition of $Rf_{\ast}IC_{X,\mathbf{Q}_\ell}$ in $D^b_c(Y,\mathbf{Q}_\ell)$. Indeed, typically there is no such decomposition! However, as Bhargav explained to me, one can still descend a certain piece of $(\dagger)$ to $D^b_c(Y,\mathbf{Q}_\ell)$ when $f$ is projective. This turns out to be good enough for some interesting applications.

To present Bhargav’s trick, let me make a definition. (What follows is a slight reinterpretation of what Bhargav told me, all mistakes are entirely due to me.)

Definition. Let $\mathcal{F}$ be a perverse sheaf on a variety $X$. Let $j:U \to X$ be the inclusion of the maximal dense open subvariety such that $j^\ast \mathcal{F}$ is a (shifted) lisse sheaf. Then we define the generic part of $\mathcal{F}$ as the perverse sheaf $\mathcal{F}^{gen} = j_{!\ast} j^{\ast} \mathcal{F}$.

Note that $\mathcal{F}^{gen}$ is zero if and only if $\mathcal{F}$ is supported on a nowhere-dense closed subvariety. Also, in general there is no map between $\mathcal{F}^{gen}$ and $\mathcal{F}$. However, in some cases $\mathcal{F}^{gen}$ is a direct summand of $\mathcal{F}$:

Proposition. Let $\mathcal{F}$ be a perverse sheaf on a $K$-variety $X$, and suppose that the pullback of $\mathcal{F}$ to $X_{C}$ is a direct sum of IC sheaves. Then $\mathcal{F}^{gen}$ is a direct summand of $\mathcal{F}$.

Proof. Let $j:U \to X$ be as in the definition of the generic part of $\mathcal{F}$, with closed complement $Z \subset X$. Our assumptions together with the definition of the generic part guarantee that $\mathcal{F}|X_C \simeq \mathcal{F}^{gen}|X_C \bigoplus \oplus_i IC_{Z_i}(\mathcal{L}_i)$ for some closed subvarieties $Z_i \subset X_C$ contained in $Z_C$.

Now look at the natural maps $\phantom{}^{\mathfrak{p}}j_! j^{\ast} \mathcal{F} \overset{\alpha}{\to} \mathcal{F} \overset{\beta}{\to} \phantom{}^{\mathfrak{p}}j_{\ast} j^{\ast} \mathcal{F}$. Set $\mathcal{G} = \mathrm{im}\,\alpha$ and $\mathcal{H} = \mathrm{im}\,\beta$. Since $\phantom{}^{\mathfrak{p}}j_! j^{\ast} \mathcal{F}$ does not admit any nonzero quotient supported on $Z$, the composite map $\mathcal{G}|X_C \hookrightarrow \mathcal{F}|X_C \to \oplus_i IC_{Z_i}(\mathcal{L}_i)$ is zero.  Thus $\alpha$ factors over an inclusion $\mathcal{G}|X_C \subset \mathcal{F}^{gen}|X_C$. Moreover, $\mathcal{G}$ has the same generic part as $\mathcal{F}$. This is enough to imply that $\mathcal{G} = \mathcal{F}^{gen}$, so we have a natural inclusion $\mathcal{F}^{gen} \simeq \mathcal{G} \subset \mathcal{F}$. A dual argument shows that $\beta$ factors over a surjection $\mathcal{F} \twoheadrightarrow \mathcal{H} \simeq \mathcal{F}^{gen}$. It is now easy to see that the composite map $\mathcal{F}^{gen} \hookrightarrow \mathcal{F} \twoheadrightarrow \mathcal{F}^{gen}$ is an isomorphism, so $\mathcal{F}^{gen}$ is a direct summand of $\mathcal{F}$. $\square$

Corollary 0. Let $f:X \to Y$ be a projective map of $K$-varieties. Then $\phantom{}^{\mathfrak{p}}\mathcal{H}^0(Rf_{\ast}IC_{X,\mathbf{Q}_\ell})^{gen}$ is a direct summand of $Rf_{\ast}IC_{X,\mathbf{Q}_\ell}$.

Proof. The decomposition theorem and the relative hard Lefschetz theorem give a decomposition $Rf_{\ast}IC_{X,\mathbf{Q}_\ell} \simeq \oplus \phantom{}^{\mathfrak{p}}\mathcal{H}^i(Rf_{\ast}IC_{X,\mathbf{Q}_\ell})[-i]$ in $D^b_c(Y,\mathbf{Q}_\ell)$. Then $\phantom{}^{\mathfrak{p}}\mathcal{H}^0(Rf_{\ast}IC_{X,\mathbf{Q}_\ell})$ is a direct sum of IC sheaves after pullback to $Y_C$, so we can apply the previous proposition. $\square$

Corollary 1. Let $f:X \to Y$ be a projective alteration of $K$-varieties with $X$ smooth. Then $IC_{Y,\mathbf{Q}_{\ell}}$ is a direct summand of $Rf_{\ast}\mathbf{Q}_{\ell}[\dim X]$.

Proof. Check that $IC_{Y,\mathbf{Q}_{\ell}}$ is a direct summand of $\phantom{}^{\mathfrak{p}}\mathcal{H}^0(Rf_{\ast}\mathbf{Q}_{\ell}[\dim X])^{gen}$ by playing with trace maps. Now apply the previous corollary. $\square$

Corollary 2. Let $K/\mathbf{Q}_p$ be a finite extension. Then for any $K$-variety $X$, the $p$-adic intersection cohomology $IH^{\ast}(X_{\overline{K}},\mathbf{Q}_p)$ is a de Rham $G_K$-representation.

Proof. Let $X' \to X$ be a resolution of singularities. The previous corollary shows that $IH^{\ast}(X_{\overline{K}},\mathbf{Q}_p)$ is a direct summand of $H^{\ast}(X'_{\overline{K}},\mathbf{Q}_p)$ compatibly with the $G_K$-actions. Since $H^{\ast}(X'_{\overline{K}},\mathbf{Q}_p)$ is de Rham and the de Rham condition is stable under passing to summands, we get the result. $\square$

Note that we can’t prove this corollary by applying the decomposition theorem directly out of the box: the decomposition theorem does immediately give you a split injection $IH^{\ast}(X_{\overline{K}},\mathbf{Q}_p) \to H^{\ast}(X'_{\overline{K}},\mathbf{Q}_p)$, but this map is not guaranteed a priori to be $G_K$-equivariant!

Corollary 3. Let $K$ be a finite extension of $\mathbf{Q}_p$ or $\mathbf{F}_p((t))$. If $H^{\ast}(X_{\overline{K}},\mathbf{Q}_{\ell})$ satisfies the weight-monodromy conjecture for all smooth projective $K$-varieties $X$, then $IH^{\ast}(X_{\overline{K}},\mathbf{Q}_{\ell})$ satisfies the weight-monodromy conjecture for all proper $K$-varieties $X$. In particular, the weight-monodromy conjecture holds for the $\ell$-adic intersection cohomology of all proper $K$-varieties for $K/\mathbf{F}_p((t))$ finite.

Proof. Entirely analogous to the previous proof. $\square$

It would be interesting to know whether Corollary 1 has a “motivic” incarnation. Here I will pretend to understand motives for a minute. Suppose we have an assignment $X \mapsto D_{mot}(X)$ on quasi-projective $K$-varieties, where $D_{mot}(X)$ is a suitable triangulated category of constructible motivic sheaves on $X$ with $\mathbf{Q}$-coefficients. This should come with the formalism of (at least) the four operations $f^{\ast}_{mot}, Rf_{mot\ast}, \otimes, R\mathcal{H}\mathrm{om}$, and with faithful exact $\ell$-adic realization functors $\mathcal{R}_{\ell}: D_{mot}(X) \to D^b_c(X,\mathbf{Q}_{\ell})$ compatible with the four operations. I think this has all been constructed by Ayoub, maybe with some tiny additional hypothesis on $K$? Let $\mathbf{Q}_{X} \in D_{mot}(X)$ denote the symmetric monoidal unit. It then makes sense to ask:

Question. In the setting of Corollary 1, is there an idempotent $e \in \mathrm{End}_{D_{mot}(Y)}(Rf_{mot \ast} \mathbf{Q}_{X}[\dim X])$ such that $\mathcal{R}_{\ell}( e Rf_{mot \ast} \mathbf{Q}_{X}[\dim X]) \simeq IC_{Y,\mathbf{Q}_{\ell}}$ for all $\ell$?

This would imply that the split injections $IH^{\ast}(Y_{\overline{K}},\mathbf{Q}_\ell) \to H^{\ast}(X_{\overline{K}},\mathbf{Q}_\ell)$ provided by Corollary 1 can be chosen “independently of $\ell$”, i.e. that they are the $\ell$-adic realizations of some split injection in $D_{mot}(\mathrm{Spec}\,K)$.

Let $X/k$ be a proper variety over some field, and let $\mathcal{E}$ be a vector bundle on $X$.  The functor of global sections of $\mathcal{E}$, i.e. the functor sending a scheme $f:S \to \mathrm{Spec}\,k$ to the set $H^0(S \times_{k} X, (f \times \mathrm{id})^{\ast} \mathcal{E})$, is (representable by) a nice affine $k$-scheme, namely the scheme $\mathcal{S}(\mathcal{E}) = \mathrm{Spec}(\mathrm{Sym}_k H^0(X,\mathcal{E})^{\vee})$. Let $\mathcal{S}(\mathcal{E})^{\times} \subset \mathcal{S}(\mathcal{E})$ denote the subfunctor corresponding to nowhere-vanishing sections $s \in H^0(S \times_k X, (f \times \mathrm{id})^{\ast} \mathcal{E})$. We’d like this subfunctor to be representable by an open subscheme. How should we prove this?
Let $p: \mathcal{S}(\mathcal{E}) \to \mathrm{Spec}\,k$ be the structure map. The identity map $\mathcal{S}(\mathcal{E}) \to \mathcal{S}(\mathcal{E})$ corresponds to a universal section $s^{\mathrm{univ}} \in H^0(\mathcal{S}(\mathcal{E}) \times_k X, (p \times \mathrm{id})^{\ast}\mathcal{E})$. Let $Z\subset |\mathcal{S}(\mathcal{E}) \times_k X|$ denote the zero locus of $s^{\mathrm{univ}}$. This is a closed subset. But now we observe that the projection $\pi: \mathcal{S}(\mathcal{E}) \times_k X \to \mathcal{S}(\mathcal{E})$ is proper, hence universally closed, and so $|\pi|(Z)$ is a closed subset of $|\mathcal{S}(\mathcal{E})|$.  One then checks directly that $\mathcal{S}(\mathcal{E})^{\times}$ is the open subscheme corresponding to the open subset $|\mathcal{S}(\mathcal{E})| \smallsetminus |\pi|(Z)$, so we win.
I guess this sort of thing is child’s play for an experienced algebraic geometer, and indeed it took Johan about 0.026 seconds to suggest that one should try to argue using the universal section.  I only cared about the above problem, though, as a toy model for the same question in the setting of a vector bundle $\mathcal{E}$ over a relative Fargues-Fontaine curve $\mathcal{X}_S$. In this situation, $\mathcal{S}(\mathcal{E})$ is a diamond over $S^\lozenge$, cf. Theorem 22.5 here, but it turns out the above argument still works after some minor changes.