Asymptotic Curvature of Moduli Spaces for Calabi–Yau Threefolds

Trenner, Thomas; Wilson, P. Μ. H.

doi:10.1007/s12220-010-9152-1

Cited by 12 publications

(21 citation statements)

References 27 publications

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“…We now discuss the inequalities imposed on C by the Riemann-Hodge bilinear relations. Some related results, from the point of view of the mirror manifold, have been given by Trenner-Wilson and Trenner [TW11], [Tre10].…”

Section: The Hodge-riemann Bilinear Relationsmentioning

confidence: 99%

Semialgebraic horizontal subvarieties of Calabi–Yau type

Friedman¹,

Laza²

2013

Duke Math. J.

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We study horizontal subvarieties Z of a Griffiths period domain D. If Z is defined by algebraic equations, and if Z is also invariant under a large discrete subgroup in an appropriate sense, we prove that Z is a Hermitian symmetric domain D, embedded via a totally geodesic embedding in D. Next we discuss the case when Z is in addition of Calabi-Yau type. We classify the possible VHS of Calabi-Yau type parametrized by Hermitian symmetric domains D and show that they are essentially those found by Gross and Sheng-Zuo, up to taking factors of symmetric powers and certain shift operations. In the weight three case, we explicitly describe the embedding Z ֒→ D from the perspective of Griffiths transversality and relate this description to the Harish-Chandra realization of D and to the Korányi-Wolf tube domain description. There are further connections to homogeneous Legendrian varieties and the four Severi varieties of Zak. Convention:We abbreviate variation of Hodge structure by VHS. All VHS are polarizable/polarized, defined over Q unless otherwise specified, and satisfy the Griffiths transversality condition. A Hodge structure or VHS (of weight n) will be assumed to be effective (h p,q = 0 only for p, q ≥ 0, h n,0 = 0) unless otherwise noted. By a Tate twist, we can always arrange that a Hodge structure is effective. We denote by D a Griffiths period domain. Thus, D = G(R)/K, where G is an orthogonal or symplectic group defined over Q (the group preserving a pair (V, Q), where V is a Q-vector space and Q is a non-degenerate symmetric or alternating form defined over Q) and K is a compact subgroup of G(R), not in general maximal. Semi-algebraic implies Hermitian symmetricOur goal in this section is to prove Theorem 1.Definition 1.1. Let D = G(R)/K be a classifying space for Hodge structures with compact dualĎ = G(C)/P(C), where P(C) is an appropriate parabolic subgroup of G(C). A closed horizontal subvariety Z of D will be called semi-algebraic in D if Z is an open subset of its Zariski closureẐ ⊆Ď. Equivalently, there exists a closed subvarietyẐ of the projective varietyĎ such that Z is a connected component of Z ∩ D. Note that, if Z is semi-algebraic in D, then Z is a semi-algebraic set.Definition 1.2. Let D = G(R)/K be a classifying space for Hodge structures as above, and let Z be a closed horizontal subvariety of D. Let Γ = Γ Z be the stabilizer of Z in G(Z), i.e. Γ = {γ ∈ G(Z) : γ(Z) = Z}. Thus Γ acts properly discontinuously on Z. We call Γ\Z strongly quasi-projective if, for every subgroup Γ ′ of Γ of finite index, the analytic space Γ ′ \Z is quasi-projective, and thus the morphism Γ ′ \Z → Γ\Z is a morphism of quasi-projective varieties. In particular, if Γ\Z is strongly quasi-projective, then Γ\Z is quasi-projective.Remark 1.3.(i) If Γ acts on Z without fixed points and Γ\Z is quasi-projective, then by Riemann's existence theorem Γ\Z is automatically strongly quasiprojective.(ii) If D is Hermitian symmetric, so that the quotient of D by every arithmetic subgroup admits a Baily-Borel compactification, then ...

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Section: The Hodge-riemann Bilinear Relationsmentioning

confidence: 99%

Semialgebraic horizontal subvarieties of Calabi–Yau type

Friedman¹,

Laza²

2013

Duke Math. J.

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“…Clearly, the structure of the Riemann tensor on a Calabi-Yau Kähler moduli space is more intricate than that of these reference models. The empirical results just described are related to the conjectured bounds on the sectional curvature proposed in [27] (see also [28] for a related proposal and [29] for counterexamples). However, not all examples that we examined have a negative Ricci scalar.…”

Section: Explaining the Tails In The Spectramentioning

confidence: 65%

Heavy tails in Calabi-Yau moduli spaces

Long

McAllister

McGuirk

2014

J. High Energ. Phys.

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Abstract:We study the statistics of the metric on Kähler moduli space in compactifications of string theory on Calabi-Yau hypersurfaces in toric varieties. We find striking hierarchies in the eigenvalues of the metric and of the Riemann curvature contribution to the Hessian matrix: both spectra display heavy tails. The curvature contribution to the Hessian is non-positive, suggesting a reduced probability of metastability compared to cases in which the derivatives of the Kähler potential are uncorrelated. To facilitate our analysis, we have developed a novel triangulation algorithm that allows efficient study of hypersurfaces with h 1,1 as large as 25, which is difficult using algorithms internal to packages such as Sage. Our results serve as input for statistical studies of the vacuum structure in flux compactifications, and of the distribution of axion decay constants in string theory.

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“…The level sets

{scriptK}_{l} = f^{- 1} false(l false) \subset K

, where

l > 0

, with the induced Riemannian metric

g_{l}

was studied in [17, 32, 33]. Wilson explicitly computed the curvature tensor and the geodesics of

g_{l}

.…”

Section: Examplesmentioning

confidence: 99%

“…He conjectured that when

M

is a Calabi–Yau manifold,

K_{l}

should have nonpositive sectional curvatures, at least in the large Kähler structure limit, considering the correspondence to the Weil–Petersson metric on the complex moduli space under mirror symmetry. Now, there are some counterexamples in [31, 32].…”

Section: Examplesmentioning

confidence: 99%

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Conformal transformations of the pseudo‐Riemannian metric of a homogeneous pair

Kawai

2020

J. London Math. Soc.

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We introduce a new notion of a homogeneous pair for a pseudo‐Riemannian metric g and a positive function f on a manifold M admitting a free R>0‐action. There are many examples admitting this structure. For example, (a) a class of pseudo‐Hessian manifolds admitting a free R>0‐action and a homogeneous potential function such as the moduli space of torsion‐free G2‐structures, (b) the space of Riemannian metrics on a compact manifold and (c) many moduli spaces of geometric structures such as torsion‐free Spin(7)‐structures admit this structure. Hence we provide the unified method for the study of these geometric structures. We consider conformal transformations of the pseudo‐Riemannian metric g of a homogeneous pair (g,f). Showing that the pseudo‐Riemannian manifold (M,(v∘f)g), where v:double-struckR>0→double-struckR>0 is a smooth function, has the structure of a warped product, we study the geometric structures such as the sectional curvature, geodesics and the metric completion (if g is positive definite) with respect to (v∘f)g in terms of those on the level set of f. In particular, (1) we can generalize the result of Clarke and Rubinstein [Ann. Inst. H. Poincaré Anal. NonLinéaire 30 (2013) 251‐274] about the metric completion of the space of Riemannian metrics with respect to the conformal transformations of the Ebin metric, and (2) two canonical Riemannian metrics on the G2 moduli space have different metric completions.

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Asymptotic Curvature of Moduli Spaces for Calabi–Yau Threefolds

Cited by 12 publications

References 27 publications

Semialgebraic horizontal subvarieties of Calabi–Yau type

Semialgebraic horizontal subvarieties of Calabi–Yau type

Heavy tails in Calabi-Yau moduli spaces

Conformal transformations of the pseudo‐Riemannian metric of a homogeneous pair

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