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Minimal Legendrian Surfaces in the Five-Dimensional Heisenberg Group Read chapter Read first chapter

2016 | OriginalPaper | Chapter

Futaki Invariant and CM Polarization

Author : Gang Tian

Published in: Geometry and Topology of Manifolds

Publisher: Springer Japan

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Abstract

This is an expository paper. We will discuss various formulations of Futaki invariant and its relation to the CM line bundle. We will discuss their connections to the K-energy. We will also include proof for certain known results which may not have been well presented or less accessible in the literature. We always assume that M is a compact Kähler manifold. By a polarization, we mean a positive line bundle L over M, then we call (M, L) a polarized manifold.

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previous chapter The Space of Left-Invariant Riemannian Metrics
Footnotes
1
In [21], the integrand is given in a different but equivalent form.
 
2
In [6], we assume that \(M_0\) is a \({\mathbb Q}\)-Fano variety with \(L_0\) being the anti-canonical bundle \(K_{M_0}^{-1}\). However, as one can see in the subsequent discussion, the general cases can be done by following similar arguments.
 
3
We refer the readers to [12] for definition of the twisted K-energy which extends the usual K-energy to conic cases.
 
4
Our presentation here differs slightly from that in [7], but they are equivalent.
 
5
One can even have an \({\mathbf G}_0\)-equivaraint resolution, but it is not needed here.
 
6
This is a pointwise version of the adjunction formula.
 
7
Here the \({\mathbf G}_0\)-action on \(M_0'\) is given by \(v'\) which covers the m-multiple of the action generated by v on \(M_0\), so we need to add m when using \(\hat{\theta }\) et al. in the subsequent computations.
 
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Metadata
Title
Futaki Invariant and CM Polarization
Author
Gang Tian
Copyright Year
2016
Publisher
Springer Japan
Book
Geometry and Topology of Manifolds

Print ISBN: 978-4-431-56019-7

Electronic ISBN: 978-4-431-56021-0

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-4-431-56021-0_18

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