On stringy Euler characteristics of Clifford non-commutative varieties

TL;DR

This paper introduces a modified version of stringy Hodge numbers for Clifford non-commutative varieties and proves that their Euler characteristics match those of complete intersections of quadrics, aligning with mirror symmetry predictions.

Contribution

It defines a new modification of stringy Hodge numbers for Clifford varieties and establishes the equality of Euler characteristics with complete intersections of quadrics.

Findings

Modified stringy Hodge numbers for Clifford varieties are consistent with mirror symmetry.

Euler characteristics of Clifford varieties and complete intersections of quadrics are equal under the new definition.

The approach works in arbitrary dimensions.

Abstract

It was shown by Kuznetsov that complete intersections of $n$ generic quadrics in ${\mathbb P}^{2n-1}$ are related by Homological Projective Duality to certain non-commutative (Clifford) varieties which are in some sense birational to double covers of ${\mathbb P}^{n-1}$ ramified over symmetric determinantal hypersurfaces. Mirror symmetry predicts that the Hodge numbers of the complete intersections of quadrics must coincide with the appropriately defined Hodge numbers of these double covers. We observe that these numbers must be different from the well-known Batyrev's stringy Hodge numbers, else the equality fails already at the level of Euler characteristics. We define a natural modification of stringy Hodge numbers for the particular class of Clifford varieties, and prove the corresponding equality of Euler characteristics in arbitrary dimension.