CM Stability and the Generalized Futaki Invariant I

By S. Paul, G. Tian. 2006

103 citations recorded by the source corpus.

Based on the Cayley, Grothendieck, Knudsen Mumford theory of determinants we extend the CM polarization to the Hilbert scheme. We identify the weight of this refined line bundle with the generalized Futaki invariant of Donaldson. We are able to conclude that CM stability implies K-Stability. An application of the Grothendieck Riemann Roch Theorem shows that this refined sheaf is isomorphic to the CM polarization introduced by Tian in 1994 on any closed, simply connected base .

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/19acb11047d99938f4927678a72458397d8d25b1

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Citations in the corpus

Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics (2008, background): “A complete discussion of these notions is given in [30] .”
DELIGNE PAIRINGS AND THE KNUDSEN-MUMFORD EXPANSION (2006, unknown): “Then h0e −φ is a metric on L0 and 〈h0e −φ, ..., hn〉 = 〈h0, ..., hn〉 · e −ψ (2.5) where ψ : B → C is the function ψ = ∫ X/B φ · ω1 ∧ · · · ∧ ωn (2.6) and ωj = − √ −1 2π ∂∂̄ log hj is the curvature of hj .”
Compact moduli space of Kahler-Einstein Fano varieties (2014, background): “ the completely same picture for general Q-Fano varieties. Question 4. What about the projectivity of our moduli space? The expectation is that “descended” Q-line bundle from the CM line bundle [FS], [PT] explained (with the proof of descending phenomenon) at the end of [OSS], will be ample on the c”
Lectures on Stability and Constant Scalar Curvature (2008, background): “ is the restriction of ρto the central ﬁber. In [46], Donaldson deﬁned L→X to be K-stable if F(T) >0 for any non-trivial test conﬁguration T. As we had mentioned earlier, it was noted by Paul-Tian [99] that F(T) coincided with the numerical invariant F of the bundle ηK. Thus the deﬁnition of K-st”
Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs (2015, result): “In the case of the Mabuchi energy, this result is closely related to [69, 70, 71].”
Test configurations for K-stability and geodesic rays (2006, methodology): “We then use [ 22 ] which relates the line bundle λCM on the Hilbert scheme to LCM.”
Extremal K\"ahler metrics (2014, result): “Note that when L = K−1M , then related results were shown by Tian [54] and Paul-Tian [40].”
K‐Stability and Kähler‐Einstein Metrics (2012, background): “X/ in terms of asymptotic expansion of the K-energy: In his thesis [30] (also see [39]), Chi Li observed that for any algebraic subgroup G0 D f .”
Positivity of the CM line bundle for families of K-stable klt Fano varieties (2020, background): “In the second case, we are not able to present such connections, because the lack of literature would force us to work out many details about the Paul-Tian type definition [101,102], and then prove the equivalence with that: this would be beyond the scope of the present article.”
Greatest lower bounds on the Ricci curvature of Fano manifolds (2009, background): “Futaki invariant of the test-conﬁguration is negative, ie. Mis not K-semistable. We want to show that R(M) <1. Since the total space of the test-conﬁguration is smooth, according to [17] (see also [16]) we have limsup t→∞ d dt M(ωt) <0. We want to show that for suitably small ǫ>0 we have (4”