Geometric approach to the Weil–Petersson symplectic form

Axelsson, Reynir; Schumacher, Georg

doi:10.4171/cmh/194

Cited by 7 publications

(8 citation statements)

References 15 publications

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“…We will briefly recall a description of the classical Weil-Petersson form, on the Teichmüller/moduli space of Riemann surfaces of genus larger that one, in terms of canonical ( [Si]) and horizontal lifts (cf. [Sch,AS1]). Let ρ s : T s S → H 1 (X s , T 0 Xs ) be the Kodaira-Spencer map associated to the deformation ν(ψ) (see (3)).…”

Section: Relative Tangent Cohomology For Hurwitz Spacesmentioning

confidence: 99%

Kähler structure on Hurwitz spaces

Axelsson¹,

Biswas

Schumacher

2015

manuscripta math.

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The classical Hurwitz spaces, that parameterize compact Riemann surfaces equipped with covering maps to P 1 of fixed numerical type with simple branch points, are extensively studied in the literature. We apply deformation theory, and present a study of the Kähler structure of the Hurwitz spaces, which reflects the variation of the complex structure of the Riemann surface as well as the variation of the meromorphic map. We introduce a generalized Weil-Petersson Kähler form on the Hurwitz space. This form turns out to be the curvature of a Quillen metric on a determinant line bundle. Alternatively, the generalized Weil-Petersson Kähler form can be characterized as the curvature form of the hermitian metric on the Deligne pairing of the relative canonical line bundle and the pull back of the anti-canonical line bundle on P 1 . Replacing the projective line by an arbitrary but fixed curve Y , we arrive at a generalized Hurwitz space with similar properties. The determinant line bundle extends to a compactification of the (generalized) Hurwitz space as a line bundle, and the Quillen metric yields a singular hermitian metric on the compactification so that a power of the determinant line bundle provides an embedding of the Hurwitz space in a projective space.2000 Mathematics Subject Classification. 32G15, 14H10, 53C55.

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Section: Relative Tangent Cohomology For Hurwitz Spacesmentioning

confidence: 99%

Kähler structure on Hurwitz spaces

Axelsson¹,

Biswas

Schumacher

2015

manuscripta math.

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“…Tr g (u * 0 Φ 0 )dµ g ≤ 6E d 2 E dt2 , where the second equality holds by (3.10), the last inequality follows from(3.15). Combining with(3.18) shows thatdE dt 6E d 2 E dt 2 .…”

mentioning

confidence: 94%

“…In [22], Wolf presented a precise formula for the second derivative of l(γ) along a Weil-Petersson geodesic. By using the methods of Kähler geometry, Axelsson and Schumacher [2,3] obtained the formulas for the first and the second variation of l(γ), and proved that its logarithm log l(γ) is strictly plurisubharmonic.…”

Section: Introductionmentioning

confidence: 99%

Plurisubharmonicity and geodesic convexity of energy function on Teichmüller space

Kim,

Wan,

Zhang

2018

Preprint

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Let π : X → T be Teichmüller curve over Teichmüller space T , such that the fiber Xz = π −1 (z) is exactly the Riemann surface given by the complex structure z ∈ T . For a fixed Riemannian manifold M and a continuous map u0 : M → Xz 0 , let E(z) denote the energy function of the harmonic map u(z) : M → Xz homotopic to u0, z ∈ T . We obtain the first and the second variations of the energy function E(z), and show that log E(z) is strictly plurisubharmonic on Teichmüller space, from which we give a new proof on the Steinness of Teichmüller space. We also obtain a precise formula on the second variation of E 1/2 if dim M = 1. In particular, we get the formula of Axelsson-Schumacher on the second variation of the geodesic length function. We give also a simple and corrected proof for the theorem of Yamada, the convexity of energy function E(t) along Weil-Petersson geodesics. As an application we show that E(t) c is also strictly convex for c > 5/6 and convex for c = 5/6 along Weil-Petersson geodesics. We also reprove a Kerckhoff's theorem which is a positive answer to the Nielsen realization problem. Contents 9 1.3. Harmonic maps 11 2. Variations of energy on Teichmüller space 13 2.1. The first variation 13 2.2. The second variation 14 2.3. Plurisubharmonicity 23 3. Convexity of energy along Weil-Petersson geodesic 26 References 30

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“…The curvature Ric (m) (µ, µ) has been known to admit an expansion of the form c 2 m 2 + c 1 m + c 0 + · · · for large m. 3 In [21] the first two coefficients have been found explicitly for general fiberation of Kähler manifolds. We determine c 0 and the remainder in Corollary 2.…”

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confidence: 99%

“…, m → ∞. 3 We began by computing the coefficients of m −k for k ∈ N using the expression of the curvature given in Proposition 1. These coefficients become increasingly complicated expressions at an exponential rate, and in the end, they vanished in each increasingly lengthy calculation.…”

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confidence: 99%

Second variation of Selberg zeta functions and curvature asymptotics

2019

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We give an explicit formula for the second variation of the logarithm of the Selberg zeta function, Z(s), on Teichmüller space. We then use this formula to determine the asymptotic behavior as s → ∞ of the second variation. As a consequence, we determine the signature of the Hessian of log Z(s) for sufficiently large s. As a further consequence, the asymptotic behavior of the second variation of log Z(s) shows that the Ricci curvature of the Hodge bundle H 0 (K m t ) → t over Teichmüller space agrees with the Quillen curvature up to a term of exponential decay, O(s 2 e −l 0 s ), where l 0 is the length of the shortest closed hyperbolic geodesic.

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Geometric approach to the Weil–Petersson symplectic form

Cited by 7 publications

References 15 publications

Kähler structure on Hurwitz spaces

Kähler structure on Hurwitz spaces

Plurisubharmonicity and geodesic convexity of energy function on Teichmüller space

Second variation of Selberg zeta functions and curvature asymptotics

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