Relative Chow stability and extremal metrics

Extremal Metrics and Geometric Stability

This paper grew out of my lectures at Nankai Institute as well as a few other conferences in the last few years. The purpose of this paper is to describe some of my works on extremal Kähler metrics in the last fifteen years in a more unified way. In [Ti4], [Ti2], the author developed a method of relating certain stability of underlying manifolds to Kähler-Einstein metrics. A necessary and new c...

Extremal Kähler metrics and K-stability

Relative cycles and Chow sheaves

3 Relative cycles. 15 3.1 Relative cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Cycles associated with flat subschemes. . . . . . . . . . . . . . 20 3.3 Chow presheaves. . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.4 Relative cycles over geometrically unibranch schemes. . . . . . 32 3.5 Multiplicities of components of inverse images of equidimensional cycles over...

Conformal Capacities and Extremal Metrics

For any non-compact Riemannian manifold M of dimension n ≥ 2 we previously defined a function λM : M×M → R+ = R+∪{+∞] only dependent on the conformal structure of M , and proved that for a class of manifolds containing all the proper subdomains of R, λ 1 n M was a distance on M [F1, F2]. The case of a domain G of R has been the object of several investigations leading to estimations of λG[V1, ....

Extremal Almost-kähler Metrics

We generalize the notions of the Futaki invariant and extremal vector field of a compact Kähler manifold to the general almost-Kähler case and show the periodicity of the extremal vector field when the symplectic form represents an integral cohomology class modulo torsion. We also give an explicit formula of the hermitian scalar curvature in Darboux coordinates which allows us to obtain example...