On the algebraic stringy Euler number

TL;DR

This paper investigates the algebraic stringy Euler number as a stringy invariant of singular algebraic varieties, proving its monotonicity under certain birational modifications for varieties with group actions, especially spherical varieties.

Contribution

It proves the conjecture that the algebraic stringy Euler number is a monotone invariant for varieties with group actions and equivariant desingularizations, including spherical varieties.

Findings

Proves the conjecture for varieties with a connected algebraic group action.

Establishes the invariance for varieties with finitely many G-orbits.

Applies results specifically to projective spherical varieties.

Abstract

We are interested in stringy invariants of singular projective algebraic varieties satisfying a strict monotonicity with respect to elementary birational modifications in the Mori program. We conjecture that the algebraic stringy Euler number is one of such invariants. In the present paper, we prove this conjecture for varieties having an action of a connected algebraic group G and admitting equivariant desingularizations with only finitely many G-orbits. In particular, we prove our conjecture for arbitrary projective spherical varieties.