On the algebraic stringy Euler number

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.

The stringy Euler number of Calabi–Yau hypersurfaces in toric varieties and the Mavlyutov duality

“…More generally, the stringy Euler number e str (X) of any minimal projective algebraic variety X does not depend on the choice of this model and coincides with the stringy Euler number of its canonical model, because all these birational models are K-equivalent to each other. There exist some versions of the stringy Euler number that are conjectured to have minimum exactly on minimal models in a given birational class [BG18]. We remark that in general the stringy Euler number may not be an integer, and so far no example of mirror symmetry is known if the stringy Euler number e str (X) of a Calabi-Yau variety X is not an integer.…”

Canonical models of toric hypersurfaces

“…Unfortunately, we can not expect that minimal models of toric hypersurfaces can be also characterized by the minimality of their stringy Euler numbers. However, one may consider the minimality of another stringy invariant, an algebraic stringy Euler number e str alg (X) [BG18], which is expected to attain minimum for minimal models in a given birational class. Non-degenerate toric hypersurfaces is a good class of algebraic varieties for testing this expectation.…”