A generalization of Higgs bundles to higher dimensional varieties
In the paper under review, the author studies a generalization of Higgs bundles to higher dimensional varieties. Properties of the moduli space parametrizing these generalized Higgs bundles are investigated intensively. In particular, a Poisson structure on this moduli space and its relation with parabolic structure over divisors are found. The main tool employed in this paper is deformation theory. In section one, for a fixed line bundle \(L\) on an \(n\)-dimensional smooth variety \(X\), the author introduces the notion of \((n-1)\)-cohomology \(L\)-twisted \textit{Higgs pairs} \((E, \overline{\phi})\) where \(E\) is a locally free sheaf on \(X\) and \(\overline{\phi} \in H^{n-1}(X, {\mathcal E}nd(E) \otimes L)\), and the notion of \textit{stability} for these pairs \((E, \overline{\phi})\). The moduli space of these pairs is denoted by \({\mathcal P}(L)\). The existence of a univeral Higgs pair on \({\mathcal P}(L)\) is discussed. In section two, the author uses infinitesimal deformation theory to study the tangent and cotangent bundles of \({\mathcal P}(L)\), and proves that they are related to the cohomology groups of certain complexes. A homomorphism \(B_s: T^*{\mathcal P}(L) \to T{\mathcal P}(L)\) is constructed, where \(s\) is a section of \(\omega_X^{-1} \otimes L\) and \(\omega_X\) denotes the canonical line bundle of \(X\). The map \(B_s\) is further investigated in section three. Section four is devoted to tangent vector fields on \({\mathcal P}(L)\). After reviewing some general results about symplectic and Poisson structures in section five, the author proves in section six that the antisymmetric contravariant \(2\)-tensor \(\theta_s\) coming from \(B_s\) defines a Poisson structure on \({\mathcal P}(L)\). This Poisson structure coincides with the canonical symplectic structure when \(L = \omega_X\) and \(s\) is the identity section. In section seven, relations between the common Higgs bundles and the generalized ones are analyzed. In the final section eight, the construction of the moduli space \(\mathcal M_{par}\) of parabolic vector bundles on \(X\) with parabolic structure over an effective divisor is recalled. Then the author describes the relations between the Poisson structure on \({\mathcal P}(L)\) and the canonical symplectic structure of the cotangent bundle to the smooth locus of the moduli space \(\mathcal M_{par}\).
scientific article; zbMATH DE number 1419356