Frobenius splitting and M\"obius inversion

TL;DR

This paper connects Frobenius splitting, M"obius inversion, and K-homology to provide a method for computing fundamental classes and K_0-classes of certain subvarieties in flag manifolds, enhancing algebraic geometry tools.

Contribution

It introduces a novel approach to compute K-homology classes using M"obius inversion and relates Chow classes to K_0-classes for multiplicity-free unions of Schubert varieties.

Findings

Fundamental class in K-homology can be expressed as an alternating sum over irreducible varieties.

K_0-classes of certain subvarieties can be derived from Chow classes using M"obius inversion.

The method applies to subvarieties homologous to multiplicity-free unions of Schubert varieties.

Abstract

We show that the fundamental class in K-homology of a Frobenius split scheme can be computed as a certain alternating sum over irreducible varieties, with the coefficients computed using M\"obius inversion on a certain poset. If G/P is a generalized flag manifold and X is an irreducible subvariety homologous to a multiplicity-free union of Schubert varieties, then using a result of Brion we show how to compute the K_0-class [X] in K_0(G/P) from the Chow class in A_*(G/P).