On the Futaki Invariants of Complete Intersections

On the modified Futaki invariant of complete intersections in projective spaces

It is known that a necessary condition for the existence of Kähler-Ricci solitons is the vanishing of the modified Futaki invariant introduced by Tian-Zhu. In a recent work of Berman-Nyström, it was generalized for (singular) Fano varieties and the notion of algebro-geometric stability of the pair $(M,V)$ of a Fano manifold $M$ and a holomorphic vector field $V$ was introduced. In this paper, we propose a method of computing the modified Futaki invariant for Fano complete intersections in projective spaces.

The Futaki invariant and the Mabuchi energy of a complete intersection

We study the Futaki invariant and the Mabuchi K-energy of a Kähler manifold $M$ using the Deligne pairing technique developed in earlier papers. We first prove a rather simple characterization of the Futaki character: The Futaki character on a Q-Fano variety is the eigenvalue of the action of $Aut(M)$ on $Chow(M)$, the Chow point of $M$. We use this to give a new proof of Lu's theorem on the Futaki invariant of a complete intersection. In the last theorem, we prove a non-linear generalization of Lu's formula which expresses the K-energy of a smooth complete intersection as a singular norm on t…

Generalized Futaki Invariant of Almost Fano Toric Varieties, Examples

The interpretation, due to T. Mabuchi, of the classical Futaki invariant of Fano toric manifolds is extended to the case of the Generalized Futaki invariant, introduced by W. Ding and G. Tian, of almost Fano toric varieties. As an application it is shown that the real part of the Generalized Futaki invariant is positive for all degenerations of the Fano manifold V_{38}, obtained by intersection of the Veronese embedding of ${\bf P}^3\times{\bf P}^2 \subset {\bf P}^{11}$ with codimension-two hyperplanes.

Cohomology formula for obstructions to asymptotic Chow semistability

Odaka and Wang proved the intersection formula for the Donaldson-Futaki invariant. In this paper, we generalize this result for the higher Futaki invariants which are obstructions to asymptotic Chow semistability.

On the K-stability of complete intersections in polarized manifolds

We consider the problem of existence of constant scalar curvature Kaehler metrics on complete intersections of sections of vector bundles. In particular we give general formulas relating the Futaki invariant of such a manifold to the weight of sections defining it and to the Futaki invariant of the ambient manifold. As applications we give a new Mukai-Umemura-Tian like example of Fano 5-fold admitting no Kaehler-Einstein metric and a strong evidence of K-stability of complete intersections on Grassmannians.

具有不同Hodge数的微分同胚完全交的例子

Examples of diffeomorphic complete intersections with different Hodge numbers

本文构造了多对微分同胚但Hodge数不同的复完全交，并指出复3维情形下Chern数也不同。