Hitchin Map for Higher Dimensional Varieties

program

November 11th talk by Juan Sebastian Numpaque Roa (Universidade do Porto)

Title:The spectral description of the Hitchin fibration

Abstract: Let X be a compact Riemann surface of genus g > 1. A Higgs bundle over X is a pair (E, φ) where E is a holomorphic vector bundle and φ : E → E ⊗ K a vector bundle morphism. These bundles emerged in the late 80’s in Nigel Hitchin’s study of the selfduality equations over a Riemann surface (1987) and in Carlos Simpon’s subsequent work on non-abelian Hodge theory (1992). In the celebrated paper “Stable bundles and integrable systems”, Nigel Hitchin showed that the moduli space of stable Higgs bundles of fixed rank r and degree d over X, M(r, d), is better understood through a map from this space

to sum(H^{0}(X,K^{n})) known, today, as the Hitchin fibration. He did this by introducing the so-called spectral curves and linking a very special abelian variety over these to the fiber over a point, through the Hitchin fibration.

In this talk we will focus on making a clear picture of Hitchin’s strategy to study M(r, d) discussed above, which has inspired tremendous breakthroughs in mathematics such as Ngo’s proof of the Fundamental lemma in Langlands program (2010).

November 22th talk by Cesare Goretti

Title: Cameral and Spectral cover

Abstract: The aim of the talk is to give a characterization of the category of Higgs bundle with a fixed cameral cover. The first minutes of the talk will be devoted to explain the difference between spectral and cameral covers and why we will focus on the latter. We will then proceed in defining the categories of higgs bundles and of cameral cover over a scheme X and a natural functor between them. From that point on we will focus on the study of the (sheaf of) category given by the fibers of such functor, the fiber over \tilde{X} will be denoted by Higgs_{\tilde{X}}. We will present two big results linked to Higgs_{\tilde{X}}. The first one is that Higgs_{\tilde{X}} is a gerbe over Tors_{T_{\tilde{X}}}. To do that we will need to recall the definitions of (sheaf of) Picard category, of gerbe and define T_{\tilde{X}}. The second and principal result is that Higgs_{\tilde{X}} is naturally equivalent to the sheafification of the category of R-twisted N-shifted W-equivariant T-bundle on \tilde{X}.

December 6th talk by Marwan Benyousef

Title:  The Hitchin morphism in the language of stacks

Abstract: The talk will be four-fold. We start with a split group scheme G on a smooth projective curve where G is étale-locally modeled on a constant reductive group. We state a result of Kostant-Veldkamp providing the Chevalley's restriction theorem together with the Chevalley morphism and the section of Kostant.

In the second part, these maps will be the building blocks for constructing the Hitchin functor in the language of stacks for any reductive group. We will prove that this functor is representable by an affine space, recovering the original Hitchin description of the affine base when G is the constant group scheme. 

In the third part, we lay the foundations of the construction of a Picard stack P and see in the last part which Hitchin fibers are a P-gerbe with open dense orbits.

December 13th talk by Marwan Benyousef

Title: The Hitchin map and the spectral data morphism  

Abstract: Intrinsically, the construction of the Hitchin map for higher dimensional varieties relies on a natural map between two quotients of the commuting scheme for the diagonal adjoint action of a split reductive group:  the stack quotient and the GIT quotient. We give two equivalent descriptions of the commuting scheme giving rise to the Hitchin map using the Weyl polarisation method. We then see that the Hitchin map factors through a thickening of a closed subscheme of the Hitchin base and we conjecture that the resulting spectral data morphism is surjective.

January 10th talk by me

title: Spectral covers

Abstract: We review the construction of the universal spectral cover for dimension one (section 6, [CN20]).

Then, using Weyl’s polarisation construction, we build a universal spectral covering of the Chow scheme classifying zero dimensional cycles of length n of A_d (Proposition 6.1, [CN20]). We then prove that any fibre over an open locus B^♡_X(k) is isomorphic to the stack of maximal Cohen-Macaulay sheaves of generic rank one on the spectral cover corresponding to the fibre (Proposition 6.3, [CN20]).

January 17th talk by Jorge Esquivel.