A generalization of Higgs bundles to higher dimensional varieties

Abstract

Let X be a smooth n-dimensional projective variety defined over \(\mathbb{C}\) and let L be a line bundle on X. In this paper we shall construct a moduli space \(\mathcal{P}(L)\) parametrizing \((n-1)\)-cohomology L-twisted Higgs pairs, i.e., pairs \((E,\bar \phi)\) where E is a vector bundle on X and \(\bar \phi \in H^{n-1}(X,\mathcal{E}\kern-0.6em\hbox{\it nd}(E) \otimes L)\). If we take \(L =\omega_X\), the canonical line bundle on X, the variety \(\mathcal{P}(L)\) 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 \(s \in H^0(X,\omega_X^{-1} \otimes L)\), one can define, in a natural way, a Poisson structure \(\theta_s\) on \(\mathcal{P}(L)\). We also analyze the relations between this Poisson structure on \(\mathcal{P}(L)\) 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.