Derived invariants and motives, Part II integral derived invariants and some applications

TL;DR

This paper develops new integral derived invariants using motif theory, demonstrating their invariance for complex surfaces and applications to Hodge-Witt cohomology, fundamental groups, and cubic fourfolds.

Contribution

It introduces novel derived invariants with integral coefficients and applies them to complex surfaces, cohomology, and algebraic geometry conjectures.

Findings

Torsion in abelianized fundamental group is a derived invariant.

Hodge-Witt cohomology groups are preserved under derived equivalence.

Serre's ordinary density conjecture holds for certain cubic fourfolds.

Abstract

In this paper we construct new derived invariants with integral coefficients using the theory of motifs, and give several applications. Specifically, we obtain the following results: For complex algebraic surfaces, we prove that certain torsion in the abelianized fundamental group is a derived invariant. We prove that the collection of Hodge-Witt cohomology groups is a derived invariant. In particular, Hodge-Witt reduction and ordinary reduction are preserved by derived equivalence when the characteristic is sufficiently large. Finally, using the techniques of non-commutative algebraic geometry, we prove that Serre's ordinary density conjecture is true for cubic $4$-folds which contain a $\mathbb{P}^2$.