Homological units

TL;DR

This paper introduces homological units and explores their invariance properties, applying these concepts to demonstrate derived invariance of Hodge numbers for certain algebraic varieties and providing geometric interpretations.

Contribution

It defines homological units, studies their invariance, and proves derived invariance of Hodge numbers for specific four-dimensional varieties with equal Picard numbers.

Findings

Derived equivalent smooth projective fourfolds with same Picard number have identical Hodge numbers.

Homological units are invariant under derived equivalences.

Provides geometric interpretation via derived Jacobians.

Abstract

We define and study the invariance properties of homological units. Some applications are given to the derived invariance of Hodge numbers. In particular, we prove that if X and Y are derived equivalent smooth projective varieties of dimension 4 having the same Picard number, then they have the same Hodge numbers. We also give a geometric interpretation of the conjectural invariance of homological units in terms of derived Jacobians.