THE NORM MAP ON THE COMPACTIFIED JACOBAIN AND SPECTRAL DATA FOR G-HIGGS PAIRS

Abstract

The thesis is concerned with spectral datas for G-Higgs pairs. A Higgs pair is the datum (E, Φ) of a vector bundle E on a fixed smooth curve C over the complex numbers and an algebraic morphism Φ : E → E ⊗L where L is a line bundle on C. For G varying among classical groups as GL, SL, PGL, Sp, GSp, PSp, SO, a G-Higgs pair is essentially a (class of) Higgs pair(s), with additional structure on E (orthogonal, symplectic, etc...) and additional conditions on Φ; Higgs pairs correspond to the case G = GL. The moduli space MG of G-Higgs pairs with fixed rank and degree is endowed with the morphism H : MG → AG (called Hitchin morphism) to the affine space of characteristic polynomials, depending on G. To any characteristic a ∈ AG one can associate a spectral curve Xa π−→ C that encodes the data of the Hitchin fiber H−1 (a) in terms of torsion-free rank-1 sheaves on the spectral curve; by torsion-free rank-1 sheaf, we mean a pure coherent sheaf of dimension 1, with the same length of the structure sheaf at each generic point. When Xa is smooth, the description of the spectral data is well-known for all classical groups. In the case G = SL, the description of the spectral data for smooth spectral curves involves the Norm map associated to π between the Picard varieties of Xa and C. Lot of our research has been focused on the problem concerning whether the Norm map can be defined also for the compactified Jacobian parametrizing torsion-free rank-1 sheaves. In order to answer this question, we took in consideration also the geometric side of the problem, concerning the direct image for generalized divisors on curves. Here is a list of results appearing in the thesis: (1) For any finite, flat morphism of curves X π−→ Y , the direct image from generalized divisors on X to generalized divisors on Y can be defined, assuming that the curves are noetherian, embeddable, with pure dimension. The direct image is equivariant with respect to the 1 action of Cartier divisors by sum and is equal to the multiplication map by deg(π) when composed with the pullback. (2) When the codomain curve Y is not smooth (even with node singular ities!), the direct image fails to work properly for families of effective generalized divisors (i.e., it is not geometric). (3) Under the condition that Y is smooth, the direct image for families of effective generalized divisors is geometric, yielding a morphism between the Hilbert spaces parametrizing them. (4) Under the condition that Y is smooth, the Norm map can be defined from the compactified Jacobian of X to the Jacobian of Y . The Norm is equivariant with respect to the action of line bundles by tensor product and is equal to the deg(π)-th tensor power when composed with the pullback. (5) The Norm and the direct image are compatible via the Abel map. The generalization of the Norm allows us to define the Prym stack associated to π as the locus of torsion-free, rank-1 sheaves whose Norm with respect to π is isomorphic to OC. Using the Prym stack, we give a nice stacky description of the spectral data for any characteristic. The same intuition works also for G = PGL. When Xa is reduced, we show that the Prym variety inside the the Prym stack is a dense, open subset. In the case G = Sp, the spectral curve Xa admits an involution σ. When Xa has mild singularities, the quotient Xa/σ is smooth and the spectral data can be described with the aim of the Norm map associated to the quotient morphism Xa q −→ Xa/σ. Unfortunately, Xa/σ is not smooth in general and the Norm map associated to q cannot be defined. However, we give a nice stacky description of the spectral data for any characteristic involving the equalizes of some map of stacks; the same idea works also for G = GSp and G = PSp. This thesis is divided in two parts, introduced by a preliminary Chapter about torsion-free rank-1 sheaves and Higgs pairs (Chapter 1). In the first part (Chapter 2 and 3), motivated by the spectral correspondence for G = SL, we study the Norm map Nmπ on the compactified Jacobian associated to a finite, flat morphism X π−→ Y between projective curves; the Norm map happens to be well-defined only if Y is smooth. In such case, we define the Prym stack of X over Y as the (stacky) fiber Nm−1 (OY ). In the case that X is reduced with locally planar singularities, we show that the usual Prym scheme is contained in the Prym stack as an open and dense subset. In the second part (Chapter 4) we study the spectral correspondence for G Higgs pairs, in the case of G = SL(r, C), PGL(r, C), Sp(2r, C), GSp(2r, C), PSp(2r, C), over any fiber.