Derived invariants of irregular varieties and Hochschild homology

TL;DR

This paper investigates how derived equivalences preserve various invariants of irregular varieties, including Hochschild homology, Albanese dimension, and Hodge numbers, using techniques from birational geometry.

Contribution

It introduces a generalized Hochschild homology framework and proves the derived invariance of several geometric and cohomological invariants for irregular varieties.

Findings

Derived invariance of Hochschild homology

Invariance of Albanese dimension for certain varieties

Stability of Hodge numbers under derived equivalence

Abstract

We study the behavior of cohomological support loci of the canonical bundle under derived equivalence of smooth projective varieties. This is achieved by investigating the derived invariance of a generalized version of Hochschild homology. Furthermore, using techniques coming from birational geometry, we establish the derived invariance of the Albanese dimension for varieties having non-negative Kodaira dimension. We apply our machinery to study the derived invariance of the holomorphic Euler characteristic and of certain Hodge numbers for special classes of varieties. Further applications concern the behavior of particular types of fibrations under derived equivalence.