Some remarks on motivical and derived invariants

TL;DR

This paper explores conjectures about derived invariants of algebraic varieties, showing that invariance of Hasse-Weil Zeta functions over finite fields implies invariance of Hodge diamonds over complex fields.

Contribution

It establishes a link between the conjectural invariance of Zeta functions and Hodge diamonds across different characteristics.

Findings

Derived invariance of Zeta functions implies invariance of Hodge diamonds.

Discusses implications of conjectures on derived equivalences.

Addresses invariants over fields of arbitrary characteristics.

Abstract

We discuss several conjectures about derived equivalent varieties, defined over fields of arbitrary characteristics, and implications among them. In particular we show that the (conjectural) derived invariance of the Hasse-Weil Zeta functions of smooth projective varieties over finite fields, implies the derived invariance of the Hodge diamond of complex algebraic varieties.