Motivic integral of K3 surfaces over a non-archimedean field

TL;DR

This paper establishes a formula linking the motivic integral of K3 surfaces over non-archimedean fields with their limit Hodge structures, introduces a new monodromy pairing invariant, and proves a conjecture for Kummer K3 surfaces.

Contribution

It provides a new formula for motivic integrals of K3 surfaces, defines a birational invariant monodromy pairing, and verifies a conjecture for Kummer K3 surfaces.

Findings

Derived a formula relating motivic integral and limit Hodge structure for semi-stable K3 surfaces.

Defined a birational invariant monodromy pairing for varieties over non-archimedean fields.

Proved the conjectural formula for maximally degenerated K3 surfaces in the case of Kummer K3 surfaces.

Abstract

We prove a formula expressing the motivic integral (\cite{ls}) of a K3 surface over $\bC((t))$ with semi-stable reduction in terms of the associated limit Hodge structure. Secondly, for every smooth variety over a non-archimedean field we define an analogue of the monodromy pairing, constructed by Grothendieck in the case of Abelian varieties, and prove that our monodromy pairing is a birational invariant of the variety. Finally, we propose a conjectural formula for the motivic integral of maximally degenerated K3 surfaces over an arbitrary non-archimedean field and prove this conjecture for Kummer K3 surfaces.