DELIGNE PAIRINGS AND THE KNUDSEN-MUMFORD EXPANSION

By Duong H. Phong, J. Ross, Jacob Sturm. 2006

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Let X → B be a proper flat morphism between smooth quasiprojective varieties of relative dimension n, and L → X a line bundle which is ample on the fibers. We establish formulas for the first two terms in the Knudsen-Mumford expansion for det(π�L k ) in terms of Deligne pairings of L and the relative canonical bundle K. This generalizes the theorem of Deligne [1], which holds for families of relative dimension one. As a corollary, we show that when X is smooth, the line bundle η associated to X → B, which was introduced in Phong-Sturm [12], coincides with the CM bundle defined by Paul-Tian [10, 11]. In a second and third corollaries, we establish asymptotics for the K-energy along Bergman rays generalizing the formulas obtained in [11].

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/605379e160851375aa27ccdece936aa539b8ee8c

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Citations in the corpus

Compact Moduli Spaces of Del Pezzo Surfaces and K\"ahler-Einstein metrics (2012, unknown): “We refere to [52], [53] for more details.”
Uniform K-stability and asymptotics of energy functionals in Kähler geometry (2016, background): “:;˚ refi X = s Klog X=P1 Ln + o(s): (3.2) The collection of metrics 1 2 logMA( j X˝ ) with ˝6= 0 denes a smooth metric on K log X=C over , but the diculty here (as opposed to the situation in [PRS08]) is that will not a priori extend to a smooth (or even locally bounded) metric on Klog X=C over . ”
The partial $C^0$-estimate along the continuity method (2013, background): “ Donaldson [17] in K-stability. Indeed, in general the latter is only related to Mabuchi energy asymptotics in a ﬁxed projective space up to a correction term (see Paul-Tian [28] and Phong-Ross-Sturm [29]). We will now sketch the general idea in obtaining a lower bound on the Bergman kernel for a ﬁx”
A variational approach to the Yau–Tian–Donaldson conjecture (2015, result): “When U has algebraic singularities, U NA belongs to H NA , and we then have the following result, which is a reformulation of [BHJ19, Theorem 3.6] (see also [SD18, Theorem 4.9], and [PRS08] for a previous result in the same direction).”
Lectures on Stability and Constant Scalar Curvature (2008, background): “The equivalence with the above Deligne intersection pairing formulation is in [101].”
Smooth and singular Kahler-Einstein metrics (2014, background): “4 since we are not aware of a single easy-to-read reference for this (we refer to [198, 227] for the related Deligne pairing formalism).”
K-stability of constant scalar curvature Kähler manifolds (2008, unknown): “[11], Phong–Sturm–Ross [12] and others.”
Greatest lower bounds on the Ricci curvature of Fano manifolds (2009, background): “ppose that the 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&”