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.

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