Derived invariance of the number of holomorphic 1-forms and vector fields

TL;DR

This paper proves that derived equivalent smooth projective varieties share invariants like the number of holomorphic 1-forms, vector fields, and Hodge numbers, highlighting deep geometric invariances.

Contribution

It establishes the derived invariance of the number of holomorphic 1-forms and vector fields, linking derived categories to classical geometric invariants.

Findings

Derived equivalent varieties have isogenous Picard varieties.

Irregularity and vector field counts are preserved under derived equivalence.

Hodge numbers are invariant for derived equivalent threefolds.

Abstract

We prove that smooth projective varieties with equivalent derived categories have isogenous (and sometimes isomorphic) Picard varieties. In particular their irregularity and number of independent vector fields are the same. This is turn implies, in combination with the invariance of Hochschild homology, that all Hodge numbers are the same for arbitrary derived equivalent threefolds.