CM Stability and the Generalized Futaki Invariant I

TL;DR

This paper extends the CM polarization to the Hilbert scheme using determinant theory, linking it to the generalized Futaki invariant, and shows that CM stability implies K-stability.

Contribution

It introduces a refined CM polarization on the Hilbert scheme and connects it to the generalized Futaki invariant, establishing a new stability implication.

Findings

CM stability implies K-stability.

The refined sheaf is isomorphic to Tian's CM polarization.

Application of Grothendieck Riemann-Roch theorem confirms the isomorphism.

Abstract

Based on the Cayley, Grothendieck, Knudsen Mumford theory of determinants we extend the CM polarization to the Hilbert scheme. We identify the weight of this refined line bundle with the generalized Futaki invariant of Donaldson. We are able to conclude that CM stability implies K-Stability. An application of the Grothendieck Riemann Roch Theorem shows that this refined sheaf is isomorphic to the CM polarization introduced by Tian in 1994 on any closed, simply connected base .