Constructing Variations of Hodge Structure Using Yang-Mills Theory and Applications to Uniformization

Simpson, Carlos

doi:10.2307/1990994

Cited by 371 publications

(874 citation statements)

References 22 publications

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“…A 2-dimensional oriented polyhedral surface is automatically Kähler (since SO.2/ D U.1/) and complete classification of such structures is given by Troyanov [20] (we recall this classification in Section 2). In the rest of this work we deal mostly with 4-dimensional polyhedral Kähler manifolds.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…In this section we recall the classification of PK structures on complex curves in Troyanov [20] and give several examples of polyhedral Kähler manifolds of higher dimension.…”

Section: Examples Of Polyhedral Kähler Manifoldsmentioning

confidence: 99%

“…Theorem 2.1 (Troyanov [20]) Consider a complex curve of genus g with pairwise distinct marked points x 1 ; : : : ; x n . Let˛1; : : : ;˛n be real positive numbers such that P .˛i 1/ D 2g 2.…”

Section: Flat Metrics On Surfacesmentioning

confidence: 99%

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Polyhedral Kähler manifolds

Panov

2009

Geom. Topol.

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In this article we introduce the notion of polyhedral Kähler manifolds, even dimensional polyhedral manifolds with unitary holonomy. We concentrate on the 4-dimensional case, prove that such manifolds are smooth complex surfaces and classify the singularities of the metric. The singularities form a divisor and the residues of the flat connection on the complement of the divisor give us a system of cohomological equations. A parabolic version of the Kobayshi-Hitchin correspondence of T Mochizuki permits us to characterize polyhedral Kähler metrics of nonnegative curvature on CP 2 with singularities at complex line arrangements.53C56; 32Q15, 53C55

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Section: Introduction and Resultsmentioning

confidence: 99%

“…In this section we recall the classification of PK structures on complex curves in Troyanov [20] and give several examples of polyhedral Kähler manifolds of higher dimension.…”

Section: Examples Of Polyhedral Kähler Manifoldsmentioning

confidence: 99%

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Polyhedral Kähler manifolds

Panov

2009

Geom. Topol.

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“…These correspond to flat irreducible Sl(N,C) bundles over the Riemann surface Σ [13,16]. One might wonder which property characterizes the flat Sl(N,C) connections that correspond to points in the W moduli space.…”

Section: The Moduli Space For W Gravitymentioning

confidence: 99%

“…The relevant component is specified by the topological type of the real vector bundle on which the flat Sl(N, IR) connection lives. To really construct this flat connection explicitly, one needs to know the so-called Hermitian-Yang-Mills metric on the Higgs bundle, see [16]. This metric, and the associated flat connection are very easy to describe if W = 0.…”

Section: The Moduli Space For W Gravitymentioning

confidence: 99%

Covariant W gravity and its moduli space from gauge theory

Goeree

1993

Nuclear Physics B

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In this paper we study arbitrary W algebras related to embeddings of sl 2 in a Lie algebra g. We will give a simple general formula for all W transformations, which will enable us to construct the covariant action for general W gravity. It turns out that this covariant action is nothing but a Fourier transform of the WZW action. The same general formula provides a 'geometrical' interpretation of W transformations: they are just a homotopy contraction of ordinary gauge transformations. This is used to argue that the moduli space relevant to W gravity is part of the moduli space of G-bundles over a Riemann surface. ‡

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Generating series for the E‐polynomials of GL(n,C)$GL(n,{\mathbb {C}})$‐character varieties

Florentino

Nozad

Zamora

2022

Mathematische Nachrichten

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With 𝐺 = 𝐺𝐿(𝑛, ℂ), let  Γ 𝐺 be the 𝐺-character variety of a given finitely presented group Γ, and let  𝑖𝑟𝑟 Γ 𝐺 ⊂  Γ 𝐺 be the locus of irreducible representation conjugacy classes. We provide a concrete relation, in terms of plethystic functions, between the generating series for 𝐸-polynomials of  Γ 𝐺 and the one for  𝑖𝑟𝑟 Γ 𝐺, generalizing a formula of Mozgovoy-Reineke. The proof uses a natural stratification of  Γ 𝐺 coming from affine GIT, the combinatorics of partitions, and the formula of MacDonald-Cheah for symmetric products; we also adapt it to the so-called Cartan brane in the moduli space of Higgs bundles. Combining our methods with arithmetic ones yields explicit expressions for the 𝐸-polynomials, and Euler characteristics, of the irreducible stratum of 𝐺𝐿(𝑛, ℂ)-character varieties of some groups Γ, including surface groups, free groups, and torus knot groups, for low values of 𝑛. K E Y W O R D Scharacter varieties, E-polynomials, Hodge theory, representations of finitely presented groups INTRODUCTIONLet 𝐺 be a complex reductive algebraic group, Γ be a finitely presented group, such as the fundamental group of a compact manifold or a finite 𝐶𝑊-complex, and letThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Constructing Variations of Hodge Structure Using Yang-Mills Theory and Applications to Uniformization

Cited by 371 publications

References 22 publications

Polyhedral Kähler manifolds

Polyhedral Kähler manifolds

Covariant W gravity and its moduli space from gauge theory

Generating series for the E‐polynomials of GL(n,C)$GL(n,{\mathbb {C}})$‐character varieties

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