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.
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One of Serre’s tricks: use group quotients cleverly.
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There is a way to directly add/subtract/multiply Hodge diamonds.
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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.
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Easier question: look at linear/polynomial relations satisfied by all Hodge diamonds of \({ {\mathsf{Var}}_{/k} }({\mathsf{sm}}, \mathop{\mathrm{proj}})\)?
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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.
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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
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Too many primes with supersingular reduction implies CM. Primes are supersingular about half of the time.
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Open image theorems: not known for abelian varieties in general.