2015
DOI: 10.1016/j.geomphys.2015.02.018
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Numerical Weil–Petersson metrics on moduli spaces of Calabi–Yau manifolds

Abstract: We introduce a simple and very fast algorithm to compute Weil-Petersson metrics on moduli spaces of Calabi-Yau varieties. Additionally, we introduce a second algorithm to approximate the same metric using Donaldson's quantization link between infinite and finite dimensional Geometric Invariant Theoretical (GIT) quotients that describe moduli spaces of varieties. Although this second algorithm is slower and more sophisticated, it can also be used to compute similar metrics on other moduli spaces (e.g. moduli sp… Show more

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Cited by 15 publications
(20 citation statements)
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“…The convergence in the previous corollary should be compared with the approximation results for the Weil-Petterson metric for moduli spaces of higher dimensional manifolds recently obtained by Keller-Lukic [29]. The approximating Kähler metrics ω ′ k in [29] are related to different balanced metrics, namely those defined wrt Donaldson'a original setting in [17] (where µ(φ) = M A(φ)).…”
Section: 3mentioning
confidence: 94%
“…The convergence in the previous corollary should be compared with the approximation results for the Weil-Petterson metric for moduli spaces of higher dimensional manifolds recently obtained by Keller-Lukic [29]. The approximating Kähler metrics ω ′ k in [29] are related to different balanced metrics, namely those defined wrt Donaldson'a original setting in [17] (where µ(φ) = M A(φ)).…”
Section: 3mentioning
confidence: 94%
“…A second direction would be to evaluate interesting geometric quantities given the metrics described here, such as curvature invariants and eigenvalues of various operators, 8 and to 6 These functionals share some of the properties of the Calabi energy [13], the Mabuchi K-energy [14], and its generalizations [9,15], although they are somewhat simpler. 7 The recent paper [15] uses a variational approach based on the Mabuchi K-energy [14] to prove a weak version of Yau's theorem. 8 The papers [1,6] discuss the calculation of scalar Laplacian eigenvalues on numerical CY metrics, with the second reference specializing in particular to algebraic metrics.…”
Section: Future Directionsmentioning
confidence: 99%
“…The argument has loopholes, as we discuss, but may nonetheless be useful as a quick way to understand why one should expect Yau's theorem to hold. 7 In Section 4 we combine the algebraic metrics with one of our energy functionals to define "optimal" metrics. We derive the Galerkin condition for the optimal metrics, and compare it to the one defining Donaldson's refined metrics.…”
Section: Introductionmentioning
confidence: 99%
“…where || · || H(k) is the norm on H 0 (X, L k ) defined by H(k). Since X is not contained in any proper linear subspace of P N −1 , this means that there exists a constant δ > 0 such that for all large enough k there exists a point q k ∈ X with dist H(k) (e θ * (v) • ι(q k ), ι(q k )) > δ. Recalling the equation (26), this contradicts (25).…”
Section: General Lemmas and Their Consequencesmentioning
confidence: 93%