---
date: 2021-04-30
tags: [ web/quick-notes ]
---


# 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](Hodge%20diamonds.md).

- Inverse Hodge problem: when can a Hodge diamond be realized by a smooth [projective variety](projective%20variety)?
  Very hard problem.
  Want to use this to get information about $h_{\crys}$, i.e. [crystalline cohomology](crystalline%20cohomology.md).

- Easier question: look at linear/polynomial relations satisfied by all Hodge diamonds of $\Vark(\smooth, \proj)$?


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

  - Linear relations are spanned by [Serre duality](Serre%20duality.md) in positive characteristic.
  - In $\ch(k) > 0$, the only polynomial relations are $h^{0,0} = 1$ and Serre Duality.
  - In $\ch(k) = 0$, 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](blowup.md),

\[
h(\Bl_2 X) 
&= h(X) - h(z) + h(E) \\
&= h(X) - h(z) + h(z)(1 + \LL + \LL^2 + \cdots + \LL^{c-1} )\\
&= h(z) + (\LL + \LL^2 + \cdots + \LL^{c-1} )
,\]

  where $z$ is the point removed and $E$ is the [exceptional divisor](exceptional%20divisor).


## l-adic Representations

- Too many primes with [supersingular reduction](supersingular%20reduction) implies [CM](CM).
  Primes are supersingular about half of the time.

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