Parabolic vortex equations and instantons of infinite energy
For a compact Kähler manifold \(M\) there is a question about the existence of a Kähler-Einstein metric. Given a holomorphic vector bundle \(E\) on \(M\) an appropriate generalization of a Kähler-Einstein metric on \(M\) appears to be a Hermitian-Einstein metric on \(E\). This question has been studied by the famous works of Donaldson, Narasimhan-Seshadri, and Uhlenbeck-Yau, culminating in the Hitchin-Kobayashi correspondence that the existence of such a metric is equivalent to a notion of the stability of the bundle. This type of theorem can be generalized to bundles with a parabolic structure. On Riemann surfaces such a structure consists of a finite set of points and filtrations by vector spaces of the fibers at those points, together with some real numbers called weights. The desired metric correspondingly is the one adapted to the parabolic structure; in particular, its connection is singular along the parabolic points. In a related direction given two parabolic vector bundles on a compact Riemann surface \(X\) and a morphism or a meromorphic map \(f\) between them there is defined a notion of stability for this triple. The resulting metrics on two bundles satisfy a system of equations called parabolic vortex equations which also involves the section and a real parameter. The present paper studies the Hitchin-Kobayashi correspondence for solutions of parabolic vortex equations and the stable parabolic triples. In the case where there is no parabolic structures this question was studied in [\textit{S. B. Bradlow} and \textit{O. García-Prada}, Math. Ann. 304, No. 2, 225-252 (1996; Zbl 0852.32016) and \textit{O. García-Prada}, Int. J. Math. 5, No. 1, 1-52 (1994; Zbl 0799.32022)]. The present proof is based on a modification of the dimensional reduction arguments used in the ordinary case, which in turns is based on the relation between the usual Hitchin-Kobayashi correspondence on \(X\times \mathbb{P}^1\) and that for coupled vortex equations on \(X\). This step of arguments can be extended to the parabolic case when \(f\) is a morphism. In case \(f\) is meromorphic the proof uses a theorem of Simpson [\textit{C. T. Simpson}, J. Am. Math. Soc. 1, No. 4, 867-918 (1988; Zbl 0669.58008)] due to the extra analysis on non-compact manifolds resulting from removing parabolic points. The paper mentions that as a corollary there are examples of anti-selfdual connections on the complex surface \(X\times \mathbb{P}^1\) minus a divisor, with nontrivial monodromy around the divisor and infinite energy. It is however not known whether this corresponds to some algebraic object.
scientific article; zbMATH DE number 997711
scientific article