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Singular Hermitian Metrics / Analytic Theory of Moduli Spaces

Subject Area Mathematics
Term from 2007 to 2020
Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 48226760
 
Final Report Year 2023

Final Report Abstract

The construction of coarse moduli spaces is a possible answer to the question of classifying projective varieties/Kähler manifolds or holomorphic vector bundles. The complex structure of such a moduli space reflects the variation of the structures of the analyzed objects in algebraic/holomorphic families. More precise results can be expected from differential geometric studies (construction of a Weil-Petersson metric) and a determinant bundle which has a Quillen metric. As far as degenerations of the situation are concerned, the aim is to construct singular Hermitian metrics on certain line bundles whose curvature forms are positive, closed currents. In a series of articles with Nicholas Buchdahl, the moduli space of polystable vector bundles on Kähler manifolds was constructed. These objects occur "at the boundary" of the moduli space of stable vector bundles. Since the known analytical methods are not applicable (non-existence of universal deformations), the methods of Geometric Invariant Theory were developed for the analytical situation and complex spaces with local analytic GIT structures were introduced. The above mentioned program could be carried out for this moduli space. Furthermore, moduli spaces of instantons were investigated. In a project with Indranil Biswas, moduli spaces of quiver bundles were constructed and investigated, which are related to theoretical physics. For families of stable bundles that degenerate as coherent sheaves, it was shown (together with I. Biswas) that the Weil-Petersson form degenerates as a positive current. In a longer paper it was shown that for families of canonically polarized manifolds (the natural analogues of Riemann surfaces of genus greater than one) the relative canonical line bundle on the total space is positive, and a further generalized Weil-Petersson metric can be constructed which takes into account higher Kodaira-Spencer mappings. From negative curvature properties, a longer open question about the hyperbolicity of the moduli space could be answered in the affirmative way. In addition, the quasi-projectivity of the moduli space was proved by analytical methods. In work with Young-Jun Choi and Matthias Braun and Y.-J. Choi, respectively, families of polarized Calabi-Yau manifolds and their degeneracies were studied. After a natural Kähler form had been constructed by the author on the space of all submanifolds of a given Kähler manifold (Douady space), in a joint work with Reynir Axelsson ist degeneracies on the boundary of the moduli space were investigated, results which in turn allowed the existence of a singular Hermitian metric on a determinant bundle. The classical Weil-Petersson metric was considered in a joint work with Stefano Trapani, based on earlier joint results. Precise estimates of the mean Ricci curvature and applications to certain intersection numbers were found. During the reporting period, 42 papers were published in scientific journals.

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