Tian's $\alpha_{m,k}^{\hat K}$-invariants on group compactifications

TL;DR

This paper computes Tian's $eta$-invariants on polarized group compactifications, confirming the conjecture for a specific case and providing counterexamples for the general case.

Contribution

It explicitly calculates Tian's $eta$-invariants on group compactifications and verifies the conjecture for $k=1$, while showing it fails for $k eq 1$.

Findings

Tian's conjecture holds for $k=1$ on these manifolds.

Counterexamples demonstrate the conjecture fails for $k eq 1$.

Explicit computation of invariants on group compactifications.

Abstract

In this paper, we compute Tian's $\alpha_{m,k}^{K\times K}$-invariant on a polarized $G$-group compactification, where $K$ denotes a maximal compact subgroup of a connected complex reductive group $G$. We prove that Tian's conjecture (see Conjecture 1.1 below) is true for $\alpha_{m,k}^{K\times K}$-invariant on such manifolds when $k=1$, but it fails in general by producing counter-examples when $k\ge 2$.