A GENERALIZATION OF HIGGS BUNDLES TO HIGHER DIMENSIONAL VARIETIES

Abstract. 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, φ̄) where E is a vector bundle on X and φ ̄ ∈ Hn−1(X, End(E)⊗L). If we take L = ωX, the canonical line bundle on X, the variety P(L) is canonically identified with the cotangent bundle of the smooth locus of the moduli space of stable vector bundles on X and, 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 ∈ H0(X,ω−1X ⊗ L), one can define, in a natural way, a Poisson structure θs onP(L). We also analyze the relations between this Poisson structure on P(L) and the canonical symplectic structure of the cotangent 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 [Bo1] in the case of curves. Introduction. Let C be a smooth projective curve of genus ≥ 2 defined over the complex fiel