The Curvature of the Petersson-Weil Metric on the Moduli Space of Kähler-Einstein Manifolds

Schumacher, Georg

doi:10.1007/978-1-4757-9771-8_14

Cited by 46 publications

(64 citation statements)

References 12 publications

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“…In this section we describe the harmonic lift of a vector field on the moduli space to the universal curve due to Royden, Siu [16] and Schumacher [15]. Details can also be found in [10].…”

Section: The Curvature Formulasmentioning

confidence: 99%

Geometric aspects of the moduli space of riemann surfaces

Liu

Sun

Yau

2005

Sci. China Ser. A-Math.

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Contents 1. Introduction 1 2. The Topological Aspects of the Moduli Space 3 3. The Background of the Teichmüller Theory 4 4. Metrics on the Teichmüller Space and the Moduli Space 5 5. The Curvature Formulas 8 6. The Asymptotics 11 7. The Equivalence of the Complete Metrics 16 8. Bounded Geometry of the Kähler-Einstein Metric 18 9. Application to Algebraic Geometry 19 10. Final Remarks 20 References 20

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“…In this section we describe the harmonic lift of a vector field on the moduli space to the universal curve due to Royden, Siu [16] and Schumacher [15]. Details can also be found in [10].…”

Section: The Curvature Formulasmentioning

confidence: 99%

Geometric aspects of the moduli space of riemann surfaces

Liu

Sun

Yau

2005

Sci. China Ser. A-Math.

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“…One is the recent survey paper of Todorov [17], which gives a complete summary of the recent progress in the subject. The other one is by Schumacher [13], in which the author computed the curvature of the Weil-Petersson metric of Kähler-Einstein manifolds, using the idea of Siu [14] of horizontal liftings.…”

Section: Introductionmentioning

confidence: 99%

On the Weil-Petersson Volume and the First Chern Class of the Moduli Space of Calabi-Yau Manifolds

Sun

2005

Commun. Math. Phys.

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“…(ii), (iii) and (iv) can be found in [Sch,Lemma 2.1], [Sch,Lemma 2.6] and [Sch,Lemma 2.2], respectively. We remark that among the conditions assumed in [Sch], the proofs of [Sch,Lemma 2.6] and [Sch,Lemma 2.2] work under the mere condition that ω X is d-closed.…”

Section: §0 Introductionmentioning

confidence: 95%

“…Thus (z, s) = (z α , s i ) 1≤α≤n,1≤i≤m gives local holomorphic coordinates for X and we will use capital letters I, J, K to index coordinates on X so that I can be i or α, etc. Also we write [Sch,§1,equation (1.2)]). Here gβ α denotes the components of the inverse of g αβ (and not that of g IJ , which may not be invertible).…”

Section: §0 Introductionmentioning

confidence: 99%

𝐿²-metrics, projective flatness and families of polarized abelian varieties

Weng

2003

Trans. Amer. Math. Soc.

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Abstract. We compute the curvature of the L 2 -metric on the direct image of a family of Hermitian holomorphic vector bundles over a family of compact Kähler manifolds. As an application, we show that the L 2 -metric on the direct image of a family of ample line bundles over a family of abelian varieties and equipped with a family of canonical Hermitian metrics is always projectively flat. When the parameter space is a compact Kähler manifold, this leads to the poly-stability of the direct image with respect to any Kähler form on the parameter space. §0. Introduction It is well known that for a family π : A → S of principally polarized abelian varieties, the theta bundles on the fibers of π form a line bundle Θ over A. From the fact that the theta characters of level n, n ∈ Z + , satisfy a certain heat equation, one knows that the direct image vector bundles π * Θ ⊗n admit projectively flat connections such that the theta characters are parallel sections. Moreover, from the fact that the theta characters form an orthonormal basis with respect to a certain natural L 2 -pairing, one knows that the above projectively flat connections are indeed Hermitian connections (cf. [APW, §5]).As a generalization of the above, one considers in conformal field theory a family of Riemann surfaces and an associated family π : M → S of moduli spaces of stable vector bundles over the Riemann surfaces. As before, the so-called generalized theta bundles over the fibers of π form a line bundle Θ over M. An important result in conformal field theory in [TUY], [H] and [APW] was to show that for n ∈ Z + , π * Θ ⊗n admits a projectively flat connection, whose construction depends on the fact that certain generalized heat equations, or KZ equations, hold for sections of generalized theta bundles.In another direction, instead of just considering the theta bundles, Welters [W], following [M1], showed that for a family of polarized abelian varieties π : A → S and any relatively ample line bundle L over A, π * L always admits a projectively flat connection. This was established from hypercohomological considerations in [W], which led to certain generalized heat equations for sections of L (cf. also [H]).However the projectively flat connections constructed in [H] and [W] are not Hermitian connections in general. From the viewpoint of representation theory, a vector bundle of rank r over S admits a projectively flat connection (resp. projectively flat Hermitian connection) if and only if E arises from a representation of

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The Curvature of the Petersson-Weil Metric on the Moduli Space of Kähler-Einstein Manifolds

Cited by 46 publications

References 12 publications

Geometric aspects of the moduli space of riemann surfaces

Geometric aspects of the moduli space of riemann surfaces

On the Weil-Petersson Volume and the First Chern Class of the Moduli Space of Calabi-Yau Manifolds

𝐿²-metrics, projective flatness and families of polarized abelian varieties

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