Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics

By S. Paul. 2008

58 citations recorded by the source corpus.

Let X be a smooth, linearly normal algebraic variety. It is shown that the Mabuchi energy of X restricted to the Bergman metrics is completely determined by the X-hyperdiscriminant of format (n-1) and the Chow form of X. As a corollary it is shown that the Mabuchi energy is bounded from below for all degenerations in G if and only if the hyperdiscriminant polytope dominates the Chow polytope for all maximal algebraic tori H of G .

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c42efa35cf853b529a19772260d8a06b1aa7eb0c

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Citations in the corpus

K‐Stability and Kähler‐Einstein Metrics (2012, background): “It follows from [36] that for all 2 G, we have (6.”
Regularity of Kähler–Ricci flows on Fano manifolds (2013, unknown): “own that K(ω(0),ω(t)) is non-increasing in t(cf. [51]). So it suﬃces to show a uniform lower bound of K(ω(0),ωi) where ωi = ω(ti). We will establish this by using a result of S. Paul. It is proved in [28, 29] that if Mis K-stable, then the K-energy is bounded from below on the space of Bergman-type ”
Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature K\ (2010, background): “au-Tian-Donaldson conjecture(see [Th]). The notion of “stability” in this case is the so-called “K-stability”, see [Ti1], [D5]. There are also other related notion of stability, see for example [RT], [Pa], etc. Note that the Kempf-Ness theorem consists of both the existence and uniqueness part. It i”
Space of Ricci Flows I (2009, background): “inally pointed out by G. Sze¨kelyhidi(c.f. [91]) along the classical continuity path. Corollary 4. Suppose (M,J) is a Fano manifold with Aut(M,J) discrete. If it is stable in the sense of S.Paul(c.f. [64]), then it admits a Ka¨hler Einstein metric. An important application of our structure theory is”
Extremal K\"ahler metrics (2014, background): “f stability: K-stability, and a slight reﬁnement of it which we call Kb-stability. As a consequence of work of Tian [54], Donaldson [18, 20], Mabuchi [34], Stoppa [44], Stoppa-Sz´ekelyhidi [45], Paul [39], Berman [7], and others, there are now many satisfactory results that show that the existence o”
Uniform K-stability and asymptotics of energy functionals in Kähler geometry (2016, methodology): “metric ˚ ˙ on L. Note that M Jis bounded below on H 1 ’GL(N;C)=U(N) i M(˚ ˙) J(˚ ˙) bounded below for ˙2G, by translation invariance of Mand J. The key ingredient is the following result of S. Paul [Pau12]. Theorem 5.5. The functionals E, Jand Mall have log norm singularities on G. Granted this resu”