A Torelli Theorem for the Moduli Space of Higgs Bundles on a Curve (original) (raw)
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Given a smooth complex projective variety M and a smooth closed curve X ⊂ M such that the homomorphism of fundamental groups π 1 (X) −→ π 1 (M) is surjective, we study the restriction map of Higgs bundles, namely from the Higgs bundles on M to those on X. In particular, we investigate the interplay between this restriction map and various types of branes contained in the moduli spaces of Higgs bundles on M and X. We also consider the setup where a finite group is acting on M via holomorphic automorphisms or anti-holomorphic involutions, and the curve X is preserved by this action. Branes are studied in this context.
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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 Petersson-Weil Kähler form is computed. We also show that, under certain assumptions, a moduli space of Higgs bundles supports of natural hyper-Kähler structure.
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In this paper we study GGG-Higgs bundles over an elliptic curve when the structure group GGG is a classical complex reductive Lie group. Modifying the notion of family, we define a new moduli problem for the classification of semistable GGG-Higgs bundles of a given topological type over an elliptic curve and we give an explicit description of the associated moduli space as a finite quotient of a product of copies of the cotangent bundle of the elliptic curve. We construct a bijective morphism from this new moduli space to the usual moduli space of semistable GGG-Higgs bundles, proving that the former is the normalization of the latter. We also obtain an explicit description of the Hitchin fibration for our (new) moduli space of GGG-Higgs bundles and we study the generic and non-generic fibres.
ON MODULI SPACE OF HIGGS Gp(2n, C)-BUNDLES OVER A RIEMANN SURFACE
Let X be a compact connected Riemann surface; the holomorphic cotangent bundle of X will be denoted by K X. Let M H denote the moduli space of semistable Higgs Gp(2n, C)-bundles over X of fixed topological type. The complex variety M H has a natural holomorphic symplectic structure. On the other hand, for any ≥ 1, the Liouville symplectic from on the total space of K X defines a holomorphic symplectic structure on the Hilbert scheme Hilb (K X) parametrizing the zero-dimensional subschemes of K X. We relate the symplectic form on Hilb (K X) with the symplectic form on M H .
A compactification of the moduli space of principal Higgs bundles over singular curves
2011
A principal Higgs bundle (P,phi)(P,\phi)(P,phi) over a singular curve XXX is a pair consisting of a principal bundle PPP and a morphism phi:XtotextAdPotimesOmega1X\phi:X\to\text{Ad}P \otimes \Omega^1_Xphi:XtotextAdPotimesOmega1X. We construct the moduli space of principal Higgs G-bundles over an irreducible singular curve XXX using the theory of decorated vector bundles. More precisely, given a faithful representation rho:GtoSl(V)\rho:G\to Sl(V)rho:GtoSl(V) of GGG, we consider principal Higgs bundles as triples (E,q,phi)(E,q,\phi)(E,q,phi) where EEE is a vector bundle with rkE=dimV\rk{E}=\dim VrkE=dimV over the normalization xtilde\xtildextilde of XXX, qqq is a parabolic structure on EEE and phi:EabtoL\phi:E\ab{}\to Lphi:EabtoL is a morphism of bundles, being LLL a line bundle and Eabdoteqdot(Eotimesa)oplusbE\ab{}\doteqdot (E^{\otimes a})^{\oplus b}Eabdoteqdot(Eotimesa)oplusb a vector bundle depending on the Higgs field phi\phiphi and on the principal bundle structure. Moreover we show that this moduli space for suitable integers a,ba,ba,b is related to the space of framed modules.
Stratifications on the moduli space of Higgs bundles
Portugaliae Mathematica, 2017
The moduli space of Higgs bundles has two stratifications. The Bia lynicki-Birula stratification comes from the action of the non-zero complex numbers by multiplication on the Higgs field, and the Shatz stratification arises from the Harder-Narasimhan type of the vector bundle underlying a Higgs bundle. While these two stratifications coincide in the case of rank two Higgs bundles, this is not the case in higher rank. In this paper we analyze the relation between the two stratifications for the moduli space of rank three Higgs bundles.
Enumerative geometry on the moduli space of rank 2 Higgs bundles
2020
The P=W conjecture identifies the perverse filtration of the Hitchin system on the cohomology of the moduli space of Higgs bundles with the weight filtration of the corresponding character variety. In this paper, we introduce an enumerative approach to to this problem; our technique only uses the structure of the equivariant intersection numbers on the moduli space of Higgs bundles, and little information about the topology of the Hitchin map. In the rank 2 case, starting from the known intersection numbers of the moduli of stable bundles, we derive the equivariant intersection numbers on the Higgs moduli, and then verify the top perversity part of our enumerative P=W statement for even tautological classes. A key in this calculation is the existence of polynomial solutions to the Discrete Heat Equation satisfying particular vanishing properties. For odd classes, we derive a determinantal criterion for the enumerative P=W.
Illinois Journal of Mathematics, 2002
We prove a Hitchin-Kobayashi correspondence for extensions of Higgs bundles. The results generalize known results for extensions of holomorphic bundles. Using Simpson's methods, we construct moduli spaces of stable objects. In an appendix we construct Bott-Chern forms for Higgs bundles.
Communications in Mathematical Physics, 2001
The moduli space of triples of the form (E, θ, s) are considered, where (E, θ) is a Higgs bundle on a fixed Riemann surface X, and s is a nonzero holomorphic section of E. Such a moduli space admits a natural map to the moduli space of Higgs bundles simply by forgetting s. If (Y, L) is the spectral data for the Higgs bundle (E, θ), then s defines a section of the line bundle L over Y. The divisor of this section gives a point of a Hilbert scheme, parametrizing 0-dimensional subschemes of the total space of the canonical bundle K X , since Y is a curve on K X. The main result says that the pullback of the symplectic form on the moduli space of Higgs bundles to the moduli space of triples coincides with the pullback of the natural symplectic form on the Hilbert scheme using the map that sends any triple (E, θ, s) to the divisor of the corresponding section of the line bundle on the spectral curve.
A Torelli theorem for moduli spaces of principal bundles over a curve
Annales de l’institut Fourier, 2012
Let X and X ′ be compact Riemann surfaces of genus ≥ 3, and let G and G ′ be nonabelian reductive complex groups. If one component M d G (X) of the coarse moduli space for semistable principal G-bundles over X is isomorphic to another component M d ′ G ′ (X ′), then X is isomorphic to X ′ .