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The semiregularity theorem for equivariant noncommutative varieties

Alexander Perry

Abstract

We generalize the classical semiregularity theorem of Buchweitz and Flenner to the setting of noncommutative algebraic geometry, with group actions. This applies in particular to twisted derived categories, in which case it answers a question of Markman and streamlines part of his proof of the Hodge conjecture for abelian fourfolds. Along the way, we prove that for many finite group actions on derived categories of varieties, the invariant category is of geometric origin.

The semiregularity theorem for equivariant noncommutative varieties

Abstract

We generalize the classical semiregularity theorem of Buchweitz and Flenner to the setting of noncommutative algebraic geometry, with group actions. This applies in particular to twisted derived categories, in which case it answers a question of Markman and streamlines part of his proof of the Hodge conjecture for abelian fourfolds. Along the way, we prove that for many finite group actions on derived categories of varieties, the invariant category is of geometric origin.

Paper Structure

This paper contains 23 sections, 16 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Relation to Markman's work
  3. Organization of the paper
  4. Conventions
  5. Acknowledgements
  6. Semiregularity
  7. Semiregularity morphisms
  8. Semiregularity of objects
  9. Hodge theory of categories
  10. General formalism
  11. Twisted derived categories
  12. Invariant categories
  13. Smoothness of moduli spaces
  14. Moduli spaces
  15. Smoothness
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $f \colon X \to S$ be a smooth proper family of complex varieties. Let $0 \in S(\mathbf{C})$ be a point and let $E_0 \in \mathrm{D}_{\mathrm{perf}}(X_0)$ be a semiregular perfect complex with $\mathop{\mathrm{Ext}}\nolimits^{<0}(E_0,E_0)=0$. Assume that $B_0 \in \mathrm{H}^2(X_0, \mathbf{Q}(1))$ remains Hodge along $S$, i.e. lifts to a global section $w$ of the local system $\bigoplus_{k \geq

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • ...and 41 more