Hodge diamonds that cannot be realised

Posted on 30 September 2020 by Remy

In Paulsen–Schreieder [PS19] and vDdB–Paulsen [DBP20], the authors/we show that any block of numbers

$\left(\begin{array}{ccccc} & & h^{n,n} & & \\ & \iddots & & \ddots & \\ h^{n,0} & & & & h^{0,n} \\ & \ddots & & \iddots & \\ & & h^{0,0} & & \end{array}\right) \in \big(\mathbf Z/m\big)^{(n+1)^2}$

satisfying $h^{0,0} = 1$ , $h^{p,q} = h^{n-p,n-q}$ , and $h^{p,q} = h^{q,p}$ (characteristic $0$ only) can be realised as the modulo $m$ reduction of a Hodge diamond of a smooth projective variety.

While preparing for a talk on [DBP20], I came up with the following easy example of a Hodge diamond that cannot be realised integrally, while not obviously violating any of the conditions (symmetry, nonnegativity, hard Lefschetz, …).

Lemma. There is no smooth projective variety (in any characteristic) whose Hodge diamond is

$\begin{array}{ccccc} & & 1 & & \\ & 1 & & 1 & \\ 0 & & 1 & & 0 \\ & 1 & & 1 & \\ & & 1.\!\! & & \end{array}$

Proof. If $\operatorname{char} k > 0$ , we have $\sum_{p+q = i}h^{p,q} \geq h_{\operatorname{dR}}^i \geq h_{\operatorname{cris}}^i$ , with equality for all $i$ if and only if the Hodge–de Rham spectral sequence degenerates and $H_{\operatorname{cris}}^i$ is torsion-free for all $i$ . Because $H^2_{\operatorname{cris}}$ contains an ample class, we must have equality on $h^2$ , hence everywhere because of how spectral sequences and universal coefficients work.

Thus, in any characteristic, we conclude that $h^1_{\operatorname{Weil}}(X) = 2$ , so $\dim \mathbf{Pic}_X^0 = 1$ and the same for $\dim \mathbf{Alb}_X$ . Thus, $X \to \mathbf{Alb}_X$ is a fibration, so a fibre and a relatively ample divisor are linearly independent in the Néron–Severi group, contradicting the assumption $h^{1,1}(X) = 1$ . $\qedsymbol$

Remark. In characteristic zero, the Hodge diamonds

$\begin{array}{ccccc} & & 1 & & \\ & a & & a & \\ 0 & & 1 & & 0 \\ & a & & a & \\ & & 1 & & \end{array}$

cannot occur for any $a \geq 1$ , by essentially the same argument. Indeed, the only thing left to prove is that the image $X \to \mathbf{Alb}_X$ cannot be a surface. If it were, then $X$ would have a global 2-form; see e.g. [Beau96, Lemma V.18].

This argument does not work in positive characteristic due to the possibility of an inseparable Albanese map. It seems to follow from Bombieri–Mumford’s classification of surfaces in positive characteristic that the above Hodge diamond does not occur in positive characteristic either, but the analysis is a little intricate.

Remark. On the other hand, the nearly identical Hodge diamond

$\begin{array}{ccccc} & & 1 & & \\ & a & & a & \\ 0 & & 2 & & 0 \\ & a & & a & \\ & & 1 & & \end{array}$

is realised by $C \times \mathbf P^1$ , where $C$ is a curve of genus $a$ . This is some evidence that the full inverse Hodge problem is very difficult, and I do not expect a full classification of which Hodge diamonds are possible (even for surfaces this might be out of reach).

References.

[Beau96] A. Beauville, Complex algebraic surfaces. London Mathematical Society Student Texts 34 (1996).

[DBP20] R. van Dobben de Bruyn and M. Paulsen, The construction problem for Hodge numbers modulo an integer in positive characteristic. Forum Math. Sigma (to appear).

[PS19] M. Paulsen and S. Schreieder, The construction problem for Hodge numbers modulo an integer. Algebra Number Theory 13.10, p. 2427–2434 (2019).

Higher pushforwards along finite morphisms

Posted on 12 January 2018 by Remy

This post is about one of my favourite answers I have given on MathOverflow, although it seems to have gone by mostly unnoticed. In the post, Qixiao asks (essentially) the following:

Question. If $f \colon X \to Y$ is a finite morphism of schemes, is the pushforward $f_* \colon \Sh(X) \to \Sh(Y)$ exact?

Note that this is true on the subcategory of quasicoherent sheaves because affine morphisms have no quasicoherent higher pushforwards. Also, in the étale topology the pushforward along a finite morphism is exact on the category of all abelian sheaves; see e.g. Tag 03QP.

However, we show that the answer to the question above is negative.

Example. Let $Y$ be the spectrum of a DVR $(R,\mathfrak m)$ , let $R \to S$ be a finite extension of domains such that $S$ has exactly two primes $\mathfrak p, \mathfrak q$ above $\mathfrak m$ , and let $X = \Spec S$ . For example, $R = \Z_{(5)}$ and $S = \Z_{(5)}[i]$ , or $R = k[x]_{(x)}$ and $S = k[x]_{(x)}[\sqrt{x+1}]$ if you prefer a more geometric example.

By my previous post, the global sections functor $\Gamma \colon \Sh(Y) \to \Ab$ is exact. If the same were true for $f_* \colon \Sh(X) \to \Sh(Y)$ , then the global sections functor on $X$ would be exact as well. Thus, it suffices to prove that this is not the case, i.e. to produce a surjection $\mathscr F \to \mathscr G$ of sheaves on $X$ such that the map on global sections is not surjective.

The topological space of $X$ consists of closed points $x,y$ and a generic point $\eta$ . Let $U = \{\eta\}$ and $Z = U^{\operatorname{c}} = \{x,y\}$ ; then $U$ is open and $Z$ is closed. Hence, for any sheaf $\mathscr F$ on $X$ , we have a short exact sequence (see e.g. Tag 02UT)

$0 \to j_! (\mathscr F|_U) \to \mathscr F \to i_* (\mathscr F|_Z) \to 0,$

where $j \colon U \to X$ and $i \colon Z \to X$ are the inclusions. Let $\mathscr F$ be the constant sheaf $\Z$ ; then the same goes for $\mathscr F|_U$ and $\mathscr F|_Z$ . Then the map

$H^0(X,\mathscr F) \to H^0(X,i_*(\mathscr F|_Z)) = H^0(Z,\mathscr F|_Z)$

is given by the diagonal map $\Z \to \Z \oplus \Z$ , since $X$ is connected by $Z$ has two connected components. This is visibly not surjective. $\qedsymbol$

Cohomology of a local scheme

Posted on 6 January 2018 by Remy

This is a continuation of my previous post on local schemes. Here is a ridiculous lemma.

Lemma. Let $(X,x)$ be a local scheme, and let $\mathscr F$ be any abelian sheaf on $X$ . Then $H^i(X,\mathscr F) = 0$ for all $i > 0$ .

Proof. It suffices to show that the global sections functor $\Gamma \colon \Sh(X) \to \Ab$ is exact. Let $\mathscr F \to \mathscr G$ be a surjection of abelian sheaves on $X$ , and let $s \in \mathscr G(X)$ be a global section. Then $s$ can be lifted to a section of $\mathscr F$ in an open neighbourhood $U$ of $x$ . But the only open neighbourhood of $x$ is $X$ . Thus, $s$ can be lifted to a section of $\mathscr F(X)$ . $\qedsymbol$

What’s going on is that the functors $\mathscr F \mapsto \Gamma(X,\mathscr F)$ and $\mathscr F \mapsto \mathscr F_x$ are naturally isomorphic, due to the absence of open neighbourhoods of $x$ .

Remark. It seems believable that there are suitable site-theoretic versions of this lemma as well. For example, a strictly Henselian local ring has no higher cohomology in the étale topology. The argument is essentially the same: every open neighbourhood of the closed point has a section; see e.g. the proof of Tag 03QO.

Lovely little lemmas

Or maybe ‘amusing observations’

Category Archives: Cohomology

Hodge diamonds that cannot be realised

Higher pushforwards along finite morphisms

Cohomology of a local scheme