Carlos Simpson

Moduli of representations of the fundamental group of a smooth projective variety I

© Publications mathématiques de l’I.H.É.S., 1994, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Higgs bundles and local systems

© Publications mathématiques de l’I.H.É.S., 1992, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » ( http://www. ihes.fr/IHES/Publications/Publications.html), implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization

The fundamental group is one of the most basic topological invariants of a space. The aim of this paper is to present a method of constructing representations of fundamental groups in complex geometry, using techniques of partial differential equations. A representation of the fundamental group of a manifold is the same thing as a vector bundle over the manifold with a connection whose curvature vanishes, and this condition amounts to a differential equation. On the other hand, the natural objects of geometry over a complex manifold are the holomorphic vector bundles and holomorphic maps between them. We will adopt a philosophy based on algebraic geometry, that these holomorphic objects are understandable, and this leads us to try to produce flat connections starting from holomorphic data. Briefly, the results are as follows. We solve the Yang-Mills equations on holomorphic vector bundles with interaction terms, over compact and some noncompact complex Kahler manifolds, yielding flat connections when certain Chern numbers vanish. An application in the compact case gives necessary and sufficient conditions for a variety to be uniformized by any particular bounded symmetric domain. The first such construction was the theorem of Narasimhan and Seshadri relating holomorphic vector bundles and unitary connections on a curve. It was later extended to higher dimensions by Donaldson, Uhlenbeck, and Yau. Their work serves as a paradigm for what we will prove, so it is worth describing first. Let X be a compact complex manifold. One can produce unitary connections using holomorphic vector bundles as follows. There is a natural operator 8 which reflects the holomorphic structure of a bundle E. Given a metric on E , there is an operator a defined by the condition that the sum D = a + 8 is a connection which preserves the metric. The curvature of D is a two-form F = D2 with coefficients in the endomorphisms of E. The equation F = 0 is usually overdetermined, but there is a natural intermediate equation, itself of

Harmonic bundles on noncompact curves

The purpose of this paper is to extend the correspondence between Higgs bundles and local systems [2,5,6,7, 13, 17, 19,20,21] to the case when X is a noncom pact algebraic curve. The basic result is that there is a class of analytic objects on X which will be put in one-to-one correspondences with two different classes of algebraic geometric objects on the completion X. The analytic objects are harmonic bundles on X satisfying a growth condition at the punctures which we call tameness. The two types of algebraic objects are Higgs bundles and g-x-modules, both with regular singularities at the punctures, together with additional data of filtrations of the fibers over the punctures. The filtered regular g-x-modules also correspond to topological objects, local systems (i.e. representations of the fundamental group), together with filtrations assigned to the punctures. For the algebraic or topological objects, there is an

Homotopy theory of higher categories

This is the first draft of a book about higher categories approached by iterating Segals method, as in Tamsamanis definition of

On the classification of rank two representations of quasiprojective fundamental groups

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Variations of mixed Hodge structure attached to the deformation theory of a complex variation of Hodge structures

-nerve and Pelissiers thesis. If

The Construction Problem in Kähler Geometry

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Computer Theorem Proving in Mathematics

is a tractable left proper cartesian model category, we construct a tractable left proper cartesian model structure on the category of

Constructing Buildings and Harmonic Maps

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