Since April 2011, I am working in Kumamoto University. My research interest is Kaehler Geometry, especially, canonical Kaehler metrics such as Kaehler-Einstein metrics.
We give a purely algebro-geometric proof that if the alpha-invariant of a
Q-Fano variety X is greater than dim X/(dim X+1), then (X,O(-K_X)) is K-stable.
The key of our proof is a relation among the Seshadri constants, the
alpha-invariant and K-stability. It also gives applications concerning the
automorphism group.
It is shown that the diameter of a compact shrinking Ricci soliton has a
universal lower bound. This is proved by extending universal estimates for the
first non-zero eigenvalue of Laplacian on compact Riemannian manifolds with
lower Ricci curvature bound to a twisted Laplacian on compact shrinking Ricci
solitons.
In this expository article we first give an overview on multiplier ideal
sheaves and geometric problems in Kählerian and Sasakian geometries. Then we
review our recent results on the relationship between the support of the
subschemes cut out by multiplier ideal sheaves and the invariant whose
non-vanishing obstructs the existence of Kähler-Einstein metrics on Fano
manifolds.
Donaldson proved that if a polarized manifold has constant scalar
curvature Kähler metrics in and its automorphism group Aut is
discrete, is asymptotically Chow stable. In this paper, we shall show
an example which implies that the above result does not hold in the case when
Aut is not discrete.
The purpose of this paper is to calculate the support of the multiplier ideal
sheaves derived from the Kähler-Ricci flow on certain toric Fano manifolds
with large symmetry. The early idea of this paper has already been in Appendix
of \cite{futaki-sano0711}.
