Moduli of Higgs Bundles
نویسنده
چکیده
2 Local symplectic, complex and Kähler geometry: a quick review 10 2.1 Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Symplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Symplectic quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Complex manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5 Type decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.6 Holomorphic vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.7 Hermitian and Kähler metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.8 Hodge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.9 Kähler quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.10 Holomorphic symplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . . 25
منابع مشابه
Geometry of Moduli Spaces of Higgs Bundles
We construct a Petersson-Weil type Kähler form on the moduli spaces of Higgs bundles over a compact Kähler manifold. A fiber integral formula for this form is proved, from which it follows that the Petersson-Weil form is the curvature of a certain determinant line bundle, equipped with a Quillen metric, on the moduli space of Higgs bundles over a projective manifold. The curvature of the Peters...
متن کاملModuli Spaces of Parabolic Higgs Bundles and Parabolic K(d) Pairs over Smooth Curves: I
This paper concerns the moduli spaces of rank two parabolic Higgs bundles and parabolic K(D) pairs over a smooth curve. Precisely which parabolic bundles occur in stable K(D) pairs and stable Higgs bundles is determined. Using Morse theory, the moduli space of parabolic Higgs bundles is shown to be a noncompact, connected, simply connected manifold, and a computation of its Poincaré polynomial ...
متن کاملHyperkähler Geometry and the Moduli Space of Higgs Bundles
Today will be mostly preliminaries, including some complex and symplectic geometry, such as the symplectic quotient, and an introduction to Kähler and hyperkähler geometry. Over the rest of the week, we’ll discuss some examples (which are usually left implicit) such as quiver varieties, introduce the moduli space of Higgs bundles, and more. A good reference for this is Andy Neitzke’s lecture no...
متن کاملBetti Numbers of the Moduli Space of Rank 3 Parabolic Higgs Bundles
Parabolic Higgs bundles on a Riemann surface are of interest for many reasons, one of them being their importance in the study of representations of the fundamental group of the punctured surface in the complex general linear group. In this paper we calculate the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles, using Morse theory. A key point is that certain critical submani...
متن کاملNathan Clement December 30 , 2016 Research Statement Parabolic Higgs Bundles
My main project to date ([3]) is concerned with certain moduli spaces of Higgs bundles and isomorphisms between them, which arise from natural operations on the bundles. Conventionally1, a Higgs bundle is a pair (V, φ) on a Riemann surface X. Here V is a vector bundle (algebraic) and φ is a linear map V → ΩX⊗V called the Higgs field. Why consider such pairs? If nothing else, maybe a Higgs bundl...
متن کاملHiggs Bundles and Geometric Structures on Surfaces
Introduction 1 1. Representations of the fundamental group 3 2. Abelian groups and rank one Higgs bundles 5 3. Stable vector bundles and Higgs bundles 6 4. Hyperbolic geometry: G = PSL(2,R) 8 5. Moduli of hyperbolic structures and representations 13 6. Rank two Higgs bundles 19 7. Split R-forms and Hitchin’s Teichmüller component 21 8. Hermitian symmetric spaces: Maximal representations 24 Refe...
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