Hilbert series and obstructions to asymptotic semistability

TL;DR

This paper investigates obstructions to asymptotic Chow semistability on polarized manifolds, showing that certain integral invariants are independent and can be derived from derivatives of the Hilbert series, extending volume minimization concepts.

Contribution

It demonstrates that multiple obstructions to asymptotic semistability are linearly independent and can be expressed as derivatives of the Hilbert series on toric Fano manifolds.

Findings

Obstructions span at least two dimensions on certain toric Fano threefolds.

Obstructions are derivatives of the Hilbert series.

Extension of volume minimization principles to these obstructions.

Abstract

Given a polarized manifold there are obstructions for asymptotic Chow semistability described as integral invariants. One of them is an obstruction to the existence for the first Chern class of the polarization to admit a constant scalar curvature K\"ahler (cscK) metric. A natural question is whether or not the other obstructions are linearly dependent on the obstruction to the existence of a cscK metric. The purpose of this paper is to see that this is not the case by exhibiting toric Fano threefolds in which these obstructions span at least two dimension. To see this we show that on toric Fano manifolds these obstructions are obtained as derivatives of the Hilbert series. This last observation should be regarded as an extension of the volume minimization of Martelli, Sparks and Yau.