ON DEGENERATIONS OF MODULI OF HITCHIN PAIRS

. The purpose of this note is to announce certain basic results on the construction of a degeneration of M HX k ( n, d ) as the smooth curve X k degenerates to an irreducible nodal curve with a single node.

Let X k be a smooth projective curve of genus g ≥ 2 over an algebraically closed field k of characteristic zero and let L be a line bundle on X k . A Hitchin pair (E, θ) is comprised of a torsion-free O X k -module E together with a O X k -morphism θ : E → E ⊗L called the Higgs structure. Let M H X k (n, d) denote the moduli space of semistable Hitchin pairs on X k with Higgs structure given by the line bundle L. The geometry of Hitchin pairs or Higgs bundles has been extensively studied for over twenty-five years beginning with Hitchin ([4], [5]), Nitsure ([8]), and Simpson ([11], [12], [13]).
More precisely, let R be a discrete valuation ring with quotient field K and residue field an algebraically closed field k, for instance R = k[[t]]. Let S = Spec R, and Spec K the generic point and let s be the closed point of S. Let X → S be a proper, flat family with generic fibre X K a smooth projective curve of genus g ≥ 2 and with closed fibre X s a irreducible nodal curve C with a single node p ∈ C. Assume that X is regular as a scheme over k. Let L be a relative line bundle on X and assume that deg(L| C ) > deg(ω C ), where ω C is the dualizing sheaf on C. Let (n, d) be a pair of integers such that gcd(n, d) = 1. We now make the key definitions (motivated by the constructions in Gieseker [3] and Nagaraj-Seshadri [7]) before we state our principal results. LetC be its normalization and let ν :C → C be the normalization map and let ν −1 (p) = {p 1 , p 2 }.
Definition 2. Let E be a vector bundle of rank n on a chain R (m) . Let Say that E is strictly standard if moreover, for every i there is an index j such that a ij = 1.
Definition 3.Let C (m) denote the semi-stable curve which is semistably equivalent to C, which is obtained as follows: the normalizationC is a component of C (m) and further, if ν : Let p : X → S be as before a family of smooth curves degenerating to the singular curve C. For an S-scheme T , let X T := X × S T .
for some m and ν restricts to the morphism which contracts the P 1 's on C (m) .
Definition 5. (see [7] and [10]) A vector bundle V on C (m) of rank n is called a Gieseker vector bundle if it satisfies the following conditions: (1) for m ≥ 1, the restriction V | R (m) is strictly standard, (2) the direct image ν * (V ) to be a torsion-free O C -module.
A Gieseker vector bundle on a modification X (mod) T is a vector bundle such that its restriction to each C (m) in it is a Gieseker vector bundle.
Let L mod be the line bundle on X i.e., a morphism φ T : V T → V T ⊗ L mod satisfying the following: is a family of stable Hitchin pairs on X T over T (for the notion of (semi)stability of torsion-free Hitchin pairs, see [12], [13] and [1]).
and a line bundle D T on the parameter space T such that Equivalently, for each closed point t ∈ T over s ∈ S, there exists an automorphism g of C (m) which is the identity automorphism on the normalizationC, with the property that g * (V t , φ t ) (V t , φ t ).
Let M H S (n, d) be the functor which associates to every S-scheme T , the set M H S (n, d)(T ) of the equivalence classes of families of psemistable torsion-free Hitchin pairs (E, θ) on X T := X × S T with Hilbert polynomial P given by n and d, where , i.e., equivalence classes such that (V T , φ T ) is a stable Gieseker-Hitchin pair on X (mod) T and ν * (V T , φ T ) ∈ M H S (n, d)(T ). Our principal results are the following: (1) There is a quasi-projective S-scheme G H S (n, d) of Gieseker-Hitchin pairs which coarsely represents the functor G H S (n, d); the Sscheme G H S (n, d) is flat over S and regular over k, with the closed fibre a divisor with (analytic) normal crossing singularities.
(2) The generic fibre is isomorphic to the classical Hitchin space M H X K (n, d).
Theorem 2. We have a Hitchin morphism of S-schemes to an affine space A S over S which extends the classical Hitchin map on M H X K (n, d). Furthermore, g S is proper and has the following properties: (1) To a general section ξ : S → A S we can associate a spectral fibered surface Y ξ over S with smooth projective generic fibre Y ξ,K and whose closed fibre Y ξ,s is an irreducible vine curve with n-nodes (cf. [2]). (2) Let δ = d + deg(L) n(n−1) 2 and let P δ,Y ξ denote the compactified relative Picard S-scheme of the spectral fibered surface Y ξ over S (see [2]). Then we have a proper birational morphism which is an isomorphism over the generic fibre and this map coincides with the classical Hitchin isomorphism of the Hitchin fibre with the Jacobian of Y ξ,K .
gives a new compactification of the Picard variety, whose fibre over s is a divisor with analytic normal crossing singularities.
The compactified Picard variety P δ,Y ξ,s of the irreducible vine curve Y ξ,s with n-nodes, has a stratification in terms of the complexity of the torsion-freeness of the sheaves. This can be given as follows: where P δ,Y ξ,s (j) := {η | η is non-free at exactly j nodes}.
In this description the stratum P δ,Y ξ,s (0) corresponds to the open subset of line bundles on Y ξ,s of degree δ. The fibres of the morphism ν * to the compactified Picard variety of the vine curve Y ξ,s gets the following description: Theorem 3. The morphism ν * is an isomorphism over the subscheme of locally free sheaves of rank 1 and for each j, over the stratum P δ,Y ξ,s (j) the fibres are canonical toric subvarieties of the wonderful compactification P GL(j) obtained from the closures of the maximal tori of P GL(j).
These are toric varieties associated to the Weyl chamber of P GL(j) (see [9]).
For the details of this announcement see [1].