Hodge numbers are not derived invariants in positive characteristic

Abstract

We study a pair of Calabi–Yau threefolds X and M, fibered in non-principally polarized Abelian surfaces and their duals, and an equivalence $D^b(X) \cong D^b(M)$, building on work of Gross, Popescu, Bak, and Schnell. Over the complex numbers, X is simply connected while $\pi _1(M) = (\mathbf {Z}/3)^2$. In characteristic 3, we find that X and M have different Hodge numbers, which would be impossible in characteristic 0. In an appendix, we give a streamlined proof of Abuaf’s result that the ring $\mathrm{H}^{*}({\mathscr {O}})$ is a derived invariant of complex threefolds and fourfolds. A second appendix by Alexander Petrov gives a family of higher-dimensional examples to show that $h^{0,3}$ is not a derived invariant in any positive characteristic.

Appendices

Appendix A. A result of Abuaf

We give a simplified account of the proof of [1, Thm. 1.3(4)].

Theorem A.1

(Abuaf) Let X and Y be smooth complex projective varieties of dimension $\le 4$. If $D^b(X) \cong D^b(Y)$ then $\mathrm{H}^{*}({\mathscr {O}}_X) \cong \mathrm{H}^{*}({\mathscr {O}}_Y)$ as algebras. In particular $h^{0,j}(X) = h^{0,j}(Y)$ for all j.

This will follow from two preparatory results:

Proposition A.2

Let X and Y be as in the statement of Theorem A.1, and let $\Phi :D^b(X) \rightarrow D^b(Y)$ be an equivalence. Then there is a line bundle L on X such that ${{\,\mathrm{rank}\,}}(\Phi L) \ne 0$.

Lemma A.3

Let X be a smooth complex projective variety, and let

$$\begin{aligned} {{\,\mathrm{Hdg}\,}}^{*}(X) := \textstyle \bigoplus _p \mathrm{H}^{2p}(X,\mathbf {Q}) \cap \mathrm{H}^{p,p}(X) \end{aligned}$$

be the ring of Hodge classes, endowed with with the Euler pairing

$$\begin{aligned} \chi (v,w) = \int _X (v_0 - v_2 + v_4 - \cdots ) \cup (w_0 + w_2 + w_4 + \cdots ) \cup {{\,\mathrm{td}\,}}(T_X), \end{aligned}$$

which makes $\chi (E,F) = \chi ({{\,\mathrm{ch}\,}}(E),{{\,\mathrm{ch}\,}}(F))$.

If $\dim X \le 3$, then Chern characters of line bundles span ${{\,\mathrm{Hdg}\,}}^{*}(X)$.

If $\dim X = 4$, then any non-zero $v \in {{\,\mathrm{Hdg}\,}}^{*}(X)$ that is (left or right) $\chi$-orthogonal to all Chern characters of line bundles satisfies $\chi (v,v) > 0$.

Proof of Lemma A.3

We omit the cases $\dim X \le 1$.

If $\dim X = n \ge 2$ then by the Lefschetz theorem on (1, 1)-classes we can choose irreducible divisors $D_1, \dotsc , D_k \subset X$ such that their cohomology classes $[D_i]$ span ${{\,\mathrm{Hdg}\,}}^2(X)$. Suppose without loss of generality that $D_1$ is ample, so $[D_1]^n = d \cdot [\text {pt}]$ for some $d > 0$. Observe that

$$\begin{aligned} {{\,\mathrm{ch}\,}}({\mathscr {O}}_{D_i}) = {{\,\mathrm{ch}\,}}({\mathscr {O}}_X) - {{\,\mathrm{ch}\,}}({\mathscr {O}}_X(-D_i)) = 0 + [D_i] + \cdots , \end{aligned}$$

and that any product of ${{\,\mathrm{ch}\,}}({\mathscr {O}}_{D_i})$s is a linear combination of Chern characters of line bundles.

If $\dim X = 2$ then ${{\,\mathrm{Hdg}\,}}^{*}(X)$ is spanned by

$$\begin{aligned} {{\,\mathrm{ch}\,}}({\mathscr {O}}_X)&= 1 + \cdots \\ {{\,\mathrm{ch}\,}}({\mathscr {O}}_{D_i})&= 0 + [D_i] + \cdots \\ {{\,\mathrm{ch}\,}}({\mathscr {O}}_{D_1})^2&= 0 + 0 + d\cdot [\text {pt}]. \end{aligned}$$

If $\dim X = 3$ then ${{\,\mathrm{Hdg}\,}}^4$ is spanned by $[D_1].[D_i]$ by the hard Lefschetz theorem, so ${{\,\mathrm{Hdg}\,}}^{*}$ is spanned by

$$\begin{aligned} {{\,\mathrm{ch}\,}}({\mathscr {O}}_X)&= 1 + \cdots \\ {{\,\mathrm{ch}\,}}({\mathscr {O}}_{D_i})&= 0 + [D_i] + \cdots \\ {{\,\mathrm{ch}\,}}({\mathscr {O}}_{D_1}) . {{\,\mathrm{ch}\,}}({\mathscr {O}}_{D_i})&= 0 + 0 + [D_1].[D_i] + \cdots \\ {{\,\mathrm{ch}\,}}({\mathscr {O}}_{D_1})^3&= 0 + 0 + 0 + d\cdot [\text {pt}]. \end{aligned}$$

If $\dim X = 4$ then in a similar way we can span

$$\begin{aligned} \mathrm{H}^0(X,\mathbf {Q}) \oplus \mathrm{H}^{1,1}(X,\mathbf {Q}) \oplus [D_1]. \mathrm{H}^{1,1}(X,\mathbf {Q}) \oplus \mathrm{H}^{3,3}(X,\mathbf {Q}) \oplus \mathrm{H}^4(X,\mathbf {Q}). \end{aligned}$$

If v is (left or right) $\chi$-orthogonal to this then $v \in \mathrm{H}^{2,2}_{\mathrm{prim}}(X,\mathbf {Q})$, so if $v \ne 0$ then $\chi (v,v) = v_2.v_2 > 0$ by the Hodge–Riemann bilinear relations.

Proof of Proposition A.2

We have

$$\begin{aligned} {{\,\mathrm{rank}\,}}(\Phi L) = \chi (\Phi L, {\mathscr {O}}_y) = \chi (L, \Phi ^{-1} {\mathscr {O}}_y), \end{aligned}$$

where ${\mathscr {O}}_y$ is the skyscraper sheaf of some point $y \in Y$. Suppose this rank is zero for all $L \in {{\,\mathrm{Pic}\,}}(X)$, and let $v={{\,\mathrm{ch}\,}}(\Phi ^{-1} {\mathscr {O}}_y) \in {{\,\mathrm{Hdg}\,}}^{*}(X)$. The Euler pairing on ${{\,\mathrm{Hdg}\,}}^{*}(X)$ is non-degenerate, so if $\dim X \le 3$ then by Lemma A.3 we have $v = 0$; but this contradicts the fact that $\chi (\Phi ^{-1} {\mathscr {O}}_Y, \Phi ^{-1} {\mathscr {O}}_y) = \chi ({\mathscr {O}}_Y, {\mathscr {O}}_y) = 1$. If $\dim X = 4$ then either $v = 0$, which again is impossible, or $\chi (v,v) > 0$, which contradicts the fact that $\chi (\Phi ^{-1} {\mathscr {O}}_y, \Phi ^{-1} {\mathscr {O}}_y) = \chi ({\mathscr {O}}_y, {\mathscr {O}}_y) = 0$. $\square$

Proof of Theorem A.1

Because ${{\,\mathrm{rank}\,}}(\Phi L) \ne 0$, the natural map of algebra objects

$$\begin{aligned} {\mathscr {O}}_Y \rightarrow \mathrm{R}{{\,\mathrm{{\mathscr {H}}\!{ om}}\,}}_Y(\Phi L,\Phi L) \end{aligned}$$

is split by the trace map

$$\begin{aligned} \mathrm{R}{{\,\mathrm{{\mathscr {H}}\!{ om}}\,}}_Y(\Phi L,\Phi L) \rightarrow {\mathscr {O}}_Y, \end{aligned}$$

so it induces an injection

$$\begin{aligned} \mathrm{H}^{*}({\mathscr {O}}_Y) \hookrightarrow {{\,\mathrm{Ext}\,}}^{*}_Y(\Phi L, \Phi L) ={{\,\mathrm{Ext}\,}}^{*}_X(L,L) = \mathrm{H}^{*}({\mathscr {O}}_X). \end{aligned}$$

Symmetrically we get an injection $\mathrm{H}^{*}({\mathscr {O}}_X) \hookrightarrow \mathrm{H}^{*}({\mathscr {O}}_Y)$. $\square$

We remark that this proof fails in characteristic p because an equivalence might take all line bundles to objects whose rank is a multiple of p.

Appendix B. A higher-dimensional example in any characteristic (by Alexander Petrov)

Let p be an arbitrary prime number. Denote $\overline{{\mathbb {F}}}_p$ by k.

Theorem A.1

There exist smooth projective derived equivalent varieties $X_1, X_2$ over k such that

$$h^{0,3}(X_1)\ne h^{0, 3}(X_2)$$

Moreover, for both $i=1,2$ the variety $X_i$ satisfies the following properties:

• (a) $X_i$ can be lifted to a smooth formal scheme ${\mathfrak {X}}_i$ over W(k) such that Hodge cohomology groups $H^r({\mathfrak {X}}_i,\Omega ^s_{{\mathfrak {X}}_i/W(k)})$ are torsion-free for all r, s.

• (b) The Hodge-to-de Rham spectral sequence for $X_i$ degenerates at the first page.

• (c) The crystalline cohomology groups $H^n_{{{\,\mathrm{cris}\,}}}(X_i/W(k))$ are torsion-free for all n.

• (d) The Hochschild-Kostant-Rosenberg spectral sequence for $X_i$ degenerates at the second page. That is, there exists an isomorphism ${{\,\mathrm{HH}\,}}_n(X_i/k)\simeq \bigoplus \limits _{s}H^s(X_i,\Omega ^{n+s}_{X_i/k})$ for every n.

• (e) $X_i$ cannot be lifted to a smooth algebraic scheme over W(k).

The varieties $X_1,X_2$ are both obtained as approximations of the quotient stack associated to a finite group acting on an abelian variety. The key to the construction is the appropriate choice of such finite group action that relies on complex multiplication and Honda-Tate theory.

Let $G= \mathbf {Z}/l\mathbf {Z}$ be the cyclic group of order l where l is an arbitrary odd prime divisor of a number of the form $p^{2r}+1$, for an arbitrary $r\ge 1$.

Proposition A.2

There exists an abelian variety A over k equipped with an action of G by endomorphisms of A such that

$$\begin{aligned} \dim _k H^3(A,{\mathscr {O}}_A)^{G}\ne \dim _k H^3(\widehat{A},{\mathscr {O}}_{\widehat{A}})^G \end{aligned}$$

(B.1)

Here $\widehat{A}$ denotes the dual abelian variety. Moreover, A can be lifted to a formal abelian scheme ${\mathfrak {A}}$ over W(k) together with an action of G.

Proof

Take $A={\mathfrak {Z}}\times _{W(k')}k$ with ${\mathfrak {Z}}$, $k'$ provided by [44], Proposition 3.1. The inequality (B.1) follows because there are G-equivariant isomorphisms $H^3(\widehat{A},{\mathscr {O}})\simeq \Lambda ^3H^1(\widehat{A},{\mathscr {O}}_{\widehat{A}})\simeq \Lambda ^3 (H^0(A,\Omega ^1_{A/k})^{\vee })\simeq H^0(A,\Omega ^3_{A/k})^{\vee }$ (the last isomorphism exists even if $p=3$) and $\dim _k H^0(A,\Omega ^3_{A/k})^G=\dim _k (H^0(A,\Omega ^3_{A/k})^{\vee })^G$ as the order of G is prime to p. $\square$

This proposition is specific to positive characteristic. For an abelian variety B equipped with an action of a finite group $\Gamma$ over a field F of characteristic zero there must exist $\Gamma$-equivariant isomorphisms $H^i(B,\Omega ^{j}_{B/F})\simeq H^i(\widehat{B},\Omega ^j_{\widehat{B}/F})^{\vee }$ for all i, j as follows either from Hodge theory or thanks to the existence of a separable $\Gamma$-invariant polarization on B.

A more subtle feature of this construction is that it is impossible to find an abelian variety B with an action of a finite group $\Gamma$ with $p\not \mid |\Gamma |$ that would have $\dim _k H^i(B,{\mathscr {O}}_B)^{\Gamma }\ne \dim _k H^i(\widehat{B},{\mathscr {O}}_{\widehat{B}})^{\Gamma }$ for $i=1$ or $i=2$. This can be deduced from Corollary 2.2 of [44] applied to an approximation of the stack $[{\mathfrak {B}}/G]$ where ${\mathfrak {B}}$ is a formal $\Gamma$-equivariant lift of B that exists by Grothendieck-Messing theory combined with the fact that the order of $\Gamma$ is prime to p.

Proof of Theorem A.1

Let A be the abelian variety provided by Proposition A.2. By Proposition 15 of [48] there exists a smooth complete intersection Y of dimension 4 over k equipped with a free action of G. The diagonal action of G on the product of $A\times Y$ is free as well.

Define $X_1=(A\times Y)/G$ and $X_2=(\widehat{A}\times Y)/G$ where $\widehat{A}$ is the dual abelian variety of A equipped with the induced action of G. In both cases the quotient is taken with respect to the free diagonal action. The equivalence of $D^b(X_1)$ and $D^b(X_2)$ will follow from the Mukai equivalence between derived categories of an abelian scheme and its dual. Indeed, consider $X_1$ and $X_2$ as abelian schemes over Y/G. The base changes of both ${{\,\mathrm{Pic}\,}}^0_{Y/G}(X_1)$ and $X_2$ along $Y\rightarrow Y/G$ are isomorphic to $\widehat{A}\times Y$ compatibly with the G-action. By étale descent, ${{\,\mathrm{Pic}\,}}^0_{Y/G}(X_1)\simeq X_2$ as abelian schemes over Y/G. Proposition 6.7 of [9] implies that $D^b(X_1)\simeq D^b(X_2)$.

Next, we compare the Hodge numbers of $X_1$ and $X_2$. By Théorème 1.1 of Exposé XI [18] we have $H^i(Y,{\mathscr {O}}_Y)=0$ for $1\le i\le 3$. Hence, there are G-equivariant identifications $H^3(A\times Y,{\mathscr {O}}_{A\times Y})\simeq H^3(A,{\mathscr {O}}_A)$ and $H^3(\widehat{A}\times Y,{\mathscr {O}}_{\widehat{A}\times Y})\simeq H^3(\widehat{A},{\mathscr {O}}_{\widehat{A}})$. Since G acts freely on both $A\times Y$ and $\widehat{A}\times Y$, the projections $A\times Y\rightarrow X_1$ and $\widehat{A}\times Y\rightarrow X_2$ are étale G-torsors and, since the order of G is prime to p, we have $H^3(X_1,{\mathscr {O}}_{X_1})\simeq H^3(A\times Y,{\mathscr {O}}_{A\times Y})^G$ and $H^3(X_2,{\mathscr {O}}_{X_2})\simeq H^3(\widehat{A}\times Y,{\mathscr {O}}_{\widehat{A}\times Y})^G$.

The inequality (B.1) therefore says that $h^{0,3}(X_1)\ne h^{0,3}(X_2)$. Condition (a) can be fulfilled as it is possible to choose Y that lifts to a smooth projective scheme over W(k) together with an action of G, by Proposition 4.2.3 of [46]. Denote by ${\mathfrak {X}}_1$ and ${\mathfrak {X}}_2$ the resulting formal schemes over W(k) lifting $X_1$ and $X_2$. Since ${\mathfrak {X}}_i$ for $i=1,2$ can be presented as a quotient by a free action of G of a product of an abelian scheme with a complete intersection, the Hodge cohomology modules $H^r({\mathfrak {X}}_i,\Omega ^s_{{\mathfrak {X}}_i/W(k)})$ are free for all r, s.

Both properties (b) and (d) would be immediate if we had $\dim _k X_i\le p$ but this is not always possible to achieve. Instead, we can argue using the lifts ${\mathfrak {X}}_i$. For (b), consider the Hodge-Tate complex . By Proposition 4.15 of [10] there is a morphism in the derived category of ${\mathfrak {X}}_i$ that induces an isomorphism on first cohomology. Taking n-th tensor power of s and precomposing it with the antisymmetrization map $\Omega ^n_{{\mathfrak {X}}_i/W(k)}\rightarrow (\Omega ^{1}_{{\mathfrak {X}}_i/W(k)})^{\otimes n}$ we obtain maps that induce a quasi-isomorphism . In particular, the differentials in the Hodge-Tate spectral sequence vanish modulo torsion. But, as we established above, the Hodge cohomology of ${\mathfrak {X}}_i$ has no torsion, so the Hodge-Tate spectral sequence degenerates at the second page. Therefore the conjugate spectral sequence for $X_i$ degenerates at the second page as well and, equivalently, the Hodge-to-de Rham spectral sequence degenerates at the first page.

Similarly, for (d) consider the Hochschild-Kostant-Rosenberg spectral sequence $E_2^{r,s}=H^r({\mathfrak {X}}_i,\Omega ^{-s}_{{\mathfrak {X}}_i/W(k)})$ converging to ${{\,\mathrm{HH}\,}}_{-r-s}({\mathfrak {X}}_i/W(k))$. There exist maps $\varepsilon _n:\Omega ^n_{{\mathfrak {X}}_i/W(k)}[n]\rightarrow {{\,\mathrm{HH}\,}}({\mathfrak {X}}_i/W(k))$ into the Hochschild complex inducing multiplication by n! on the n-th cohomology: $\varepsilon _n=n!:\Omega ^n_{{\mathfrak {X}}_i/W(k)}\rightarrow {\mathscr {H}}^{-n}({{\,\mathrm{HH}\,}}({\mathfrak {X}}_i/W(k)))\simeq \Omega ^n_{{\mathfrak {X}}_i/W(k)}$. Therefore, the HKR spectral sequence always degenerates modulo torsion, hence degenerates at the second page in our case. Passing to the mod p reduction gives (d).

The property (c) follows from (a) and (b) as $H^n_{{{\,\mathrm{cris}\,}}}(X_i/W(k))\simeq H^n_{{{\,\mathrm{dR}\,}}}({\mathfrak {X}}_i/W(k))$.

Finally, to prove (e), note that by the same computation as above one sees that $h^{0,3}(X_1)=h^{3,0}(X_2)\ne h^{0,3}(X_2)=h^{3,0}(X_1)$ so both $X_1$ and $X_2$ violate Hodge symmetry. Denote by K the fraction field of W(k). If ${\mathscr {X}}_i$ is a smooth scheme over W(k) lifting $X_i$ then we have

$$\begin{aligned} \dim _{K}H^r({\mathscr {X}}_{i,K},\Omega ^s_{{\mathscr {X}}_{i,K}/K})\le \dim _k H^r(X_i,\Omega ^s_{X_i/k}) \end{aligned}$$

(B.2)

for all r, s by semi-continuity while $\dim _K H^n_{{{\,\mathrm{dR}\,}}}({\mathscr {X}}_{i,K}/K)=\dim _k H^n_{{{\,\mathrm{dR}\,}}}(X_i/k)$ because $H^n_{{{\,\mathrm{dR}\,}}}({\mathscr {X}}_i/W(k))\simeq H^n_{{{\,\mathrm{cris}\,}}}(X_i/W(k))$ is torsion-free for all n. Since Hodge-to-de Rham spectral sequences for $X_i$ and ${\mathscr {X}}_{i,K}$ degenerate at the first page, we deduce that $\sum \limits _{r,s}\dim _{K}H^r({\mathscr {X}}_{i,K},\Omega ^s_{{\mathscr {X}}_{i,K}/K}) =\sum \limits _{r,s}\dim _k H^r(X_i,\Omega ^s_{X_i/k})$ so (B.2) is in fact equality for all r, s. But this means that the smooth proper algebraic variety ${\mathscr {X}}_{i,K}$ over a field of characteristic zero violates Hodge symmetry which is impossible. $\square$