Modified Futaki invariant and equivariant Riemann–Roch formula

Abstract

Abstract. In this paper, we give a new version of the modified Futaki invariant for a test configuration associated to the soliton action on a Fano manifold. Our version will naturally come from toric test configurations defined by Donaldson for toric manifolds. As an application, we show that the modified K-energy is proper for toric invariant Kähler potentials on a toric Fano manifold. IntroductionLet (M, g) be a Fano manifold with a Kähler form ω g ∈ 2πc 1 (M) of g. Denote η(M) to be the linear space of holomorphic vector fields on M. Then by Hodge Theorem, for any X ∈ η(M), there exists a unique smooth complex-valued functionwhere h g is the Ricci potential of g such thatIt was shown there that F X (v) is a holomorphic invariant independent of the choice of g with ω g ∈ 2πc 1 (M), and so it defines an obstruction to the existence of Kähler-Ricci solitons with respect to an element X ∈ η r (M), where η r (M) is the reductive part of η(M). In particular, when. It was also proved by Tian and Zhu that there exists a unique X such thatRecently, by using Ding-Tian's idea of generalizing Futaki invariant [DT], Xiong and Berman, gave a generalization of the modified Futaki invariant F X (·) for any special degeneration associated to X, independently [Xi,Be2]. As a consequence, they both proved that F X (·) is nonnegative if M admits a Kähler-Ricci soliton. Berman also gaves an algebraic formula for F X (·), which depends on weights of the automorphisms group on holomorphic sections of multi-line bundles on the center fibre induced by the test configuration. The purpose of present paper is to define the modified Futaki invariant F X (·) for any test configuration associated to X. Our motivation is inspired by Berman's algebraic formula for special degenerations and is to modify his formula for general test configurations. Then by applying the We note that the above energy argument was used by other people, such as in [JMR, LS, T3, LZ] to study the conical Kähler-Einstein metrics on general Fano manifolds. Theorem 0.1 and Theorem 0.2 will be proved in Section 2-3 and Section 4, respectively.Acknowledgements. The third author would like to thank professor Gang Tian for his interest to the paper and sharing his insight in Kähler geometry.

K-energy on polarized compactifications of Lie groups

“…Propostion 5.2 implies Theorem 5.1 by the following lemma, which can be derived in a same way as for Lemma 4.14 (also see [30,Lemma 3.4]).…”

“…Reduction of Modified K-energy. The following is a generalization of Proposition 3.1 in[30]. Let φ ∈ H K×K (ω 0 ) and u be the Legendre function of ψ = ψ 0 +φ.Then µ X ω0 (φ) = 1 V K X (u) + const.,where K X (u) = N X (u) + L X (u), and L X (u) = 2P+ y − 4ρ, ∇u e θ X (y) π dy, .…”

“…A natural question is how to verify the K-stability by reducing it to a finite dimensional progress. The answer is known for Fano surfaces by Tian [22] and for toric Fano manifolds by Wang and Zhu [29] (see also [30]). In fact, in both cases the existence is equivalent to the vanishing of Futaki invariant.…”

G-Sasaki Manifolds and K-Energy

“…Similar to Kähler geometry, one can introduce a modified K-energy on H 1 2 dη as in [43,12,44,33], etc.. We note that an analogy of Theorem 7.1 for Kähler-Einstein G-manifolds has been recently estibalished in [20] and [33], respectively. By following the argument in [33], one can extend the proof of Theorem 0.1 to Theorem 7.1 by taking f a (v) = f X (v) = e X i vi in Theorem 5.3.…”

Twisted and conical Kähler–Ricci solitons on Fano manifolds

“…where (z 1 , · · · , z n ) are local holomorphic coordinates such that D = {z n = 0}. As in [8], we give the definition of conical Kähler-Ricci soliton with respect to the holomorphic vector field X, which has been studied on the toric manifold by [8] and [29].…”