Recent zbMATH articles in MSC 32G13https://zbmath.org/atom/cc/32G132024-03-13T18:33:02.981707ZWerkzeugWeil-Petersson forms for families of polystable bundles over compact Kähler manifoldshttps://zbmath.org/1528.530212024-03-13T18:33:02.981707Z"Buchdahl, Nicholas"https://zbmath.org/authors/?q=ai:buchdahl.nicholas-p"Schumacher, Georg"https://zbmath.org/authors/?q=ai:schumacher.georgThe authors show that the fiber integral formula of \textit{G. Schumacher} and \textit{M. Toma} [Math. Ann. 293, No. 1, 101--107 (1992; Zbl 0771.53037)] for the Kähler form associated to the \(L^2\) metric on the moduli space of stable bundles (the Weil-Petersson form) extends to the moduli space of polystable bundles at least as a closed \((1,1)\)-current, defined locally by a continuous plurisubharmonic function that is given by an explicit formula (explicit insofar as solutions of the Hermite-Einstein equations can be written down explicitly). In this way they obtain a clearer picture of the behavior of the metric as stable bundles degenerate into polystable bundles of the same topological type. The moduli space of polystable holomorphic vector bundles over a compact Kähler manifold previously constructed by the authors is equipped with a positive closed current that possesses local continuous potentials, which are given by explicit integral formulae. Its restriction to the moduli space of stable holomorphic vector bundles is the classical Weil-Petersson form. It agrees with a semiclassical smooth Weil-Petersson form on an open dense subspace of the polystable moduli space.
The paper is organized as follows. In Section 1 the authors provide some details on the construction of Weil-Petersson forms and some generalizations, and list some of the basic properties. Section 2 gives an explicit formula for a potential function for generalized Weil-Petersson forms. Section 3 deals with Weil-Petersson potentials. After a brief review of some of the main results of \textit{N. Buchdahl} and \textit{G. Schumacher} [Complex Manifolds 9, 78--113 (2022; Zbl 1484.14091)], the authors introduce a particular family of complex connections of the relevant kind, and a potential function for its associated Weil-Petersson form is calculated. Section 4 is devoted to the base connection and its Weil-Petersson form. The results of the previous section are used in this section to show that the extended Weil-Petersson potential descends from the polystable locus to a function on the analytic GIT quotient in the sense that it is invariant under the action of the relevant group. Sections 5 and 6 are devoted to proving that the potential function for the Weil-Petersson form obtained from the construction in Section 2 is continuous. Sections 7 and 8 deal with continuity of the potential function, and Weil-Petersson forms on moduli spaces of polystable bundles. It is shown that the Weil-Petersson current restricted to an open dense subspace of the polystable moduli space is a smooth semiclassical Weil-Petersson form, agreeing with the classical form when restricted to the stable locus.
Reviewer: Ahmed Lesfari (El Jadida)