A generalization of Higgs bundles to higher dimensional varieties

Let X be a smooth n-dimensional projective variety defined over C and let L be a line bundle on X. In this paper we shall construct a moduli space P(L ) parametrizing (n - 1)-cohomology L-twisted Higgs pairs, i.e., pairs (E, < (phi)over bar>) where E is a vector bundle on X and <(phi)over bar> is an e lement of Hn-1 (X, End(E) x L). If we take L = w(X), the canonical line bun dle on X, the variety P(L) is canonically identified with the cotangent bun dle of the smooth locus of the moduli space of stable vector bundles on X a nd, as such, it has a canonical symplectic structure. We prove that, in the general case, in correspondence to the choice of a non-zero section s is a n element of H-0(X, w(X)(-1) x L), one can define, in a natural way, a Pois son structure theta(s) on P(L). We also analyze the relations between this Poisson structure on P(L) and the canonical symplectic structure of the cot angent bundle to the smooth locus of the moduli space of parabolic bundles over X, with parabolic structure over the divisor D defined by the section s. These results generalize to the higher dimensional case similar results proved in [Bol] in the case of curves. Mathematics Subject Classification ( 1991): 14D20, 14J60, 14B10; 58F05.