Stringy Hodge numbers of strictly canonical nondegenerate singularities

TL;DR

This paper introduces a class of strictly canonical nondegenerate hypersurface singularities that contribute polynomially to Batyrev's stringy E-function, confirming nonnegativity of stringy Hodge numbers in certain cases.

Contribution

It characterizes a new class of singularities based on Newton polyhedron facets and proves Batyrev's conjecture for these cases under specific conditions.

Findings

Singularities are strictly canonical with polynomial stringy E-function contributions.

Batyrev's conjecture on nonnegativity of stringy Hodge numbers holds for these singularities.

Results extend previous work on Brieskorn singularities.

Abstract

We describe a class of isolated nondegenerate hypersurface singularities that give a polynomial contribution to Batyrev's stringy E-function. These singularities are obtained by imposing a natural condition on the facets of the Newton polyhedron, and they are strictly canonical. We prove that Batyrev's conjecture concerning the nonnegativity of stringy Hodge numbers is true for complete varieties with such singularities, under some additional hypotheses on the defining polynomials (e.g. convenient or weighted homogeneous). The proof uses combinatorics on lattice polytopes. The results form a strong generalisation of previously obtained results for Brieskorn singularities.