# 2021-04-30

## Remy van Dobben de Bruyn, “Constructing varieties with prescribed Hodge numbers modulo m in positive characteristic”.

Reference: Remy van Dobben de Bruyn (Princeton and IAS), “Constructing varieties with prescribed Hodge numbers modulo m in positive characteristic”. Stanford AG Seminar.

• One of Serre’s tricks: use group quotients cleverly.

• There is a way to directly add/subtract/multiply Hodge diamonds.

• Inverse Hodge problem: when can a Hodge diamond be realized by a smooth projective variety? Very hard problem. Want to use this to get information about $$h_{{\mathrm{crys}}}$$, i.e. crystalline cohomology.

• Easier question: look at linear/polynomial relations satisfied by all Hodge diamonds of $${ {\mathsf{Var}}_{/k} }({\mathsf{sm}}, \mathop{\mathrm{proj}})$$?

• Main theorems: for a fixed dimension $$n$$,

• Linear relations are spanned by Serre duality in positive characteristic.
• In $$\operatorname{ch}(k) > 0$$, the only polynomial relations are $$h^{0,0} = 1$$ and Serre Duality.
• In $$\operatorname{ch}(k) = 0$$, one has to add in Hodge symmetry.
• Important tools:

• Kunneth formula for Hodge diamonds: there’s a graphical way to do this by summing over several different ways to place blocks in the diamonds.
• See Blowups,

\begin{align*} h(\operatorname{Bl}_2 X) &= h(X) - h(z) + h(E) \\ &= h(X) - h(z) + h(z)(1 + {\mathbb{L}}+ {\mathbb{L}}^2 + \cdots + {\mathbb{L}}^{c-1} )\\ &= h(z) + ({\mathbb{L}}+ {\mathbb{L}}^2 + \cdots + {\mathbb{L}}^{c-1} ) ,\end{align*}

where $$z$$ is the point removed and $$E$$ is the exceptional divisor.

## l-adic Representations

• Too many primes with supersingular reduction implies CM. Primes are supersingular about half of the time.

• Open image theorems: not known for abelian varieties in general.

#web/quick-notes