A: Arithmetik und Geometrie von Calabi-Yau Räumen; B: Lagrangesche Zyklen und Lagrangesche Singularitäten; C: Geometrie von Hilbertschemata; D: Irreduzible holomorph sympletkrischen Mannigfaltigkeiten; E: Flächensingularitäten mit rationalen Homologiesphä

点击翻译按钮获取中文摘要

Calabi-Yau spaces are objects of key interest in algebraic geometry and complex analysis. The phenomenon of mirror symmetry gives an unexpected relation between physics and geometry. A particular interesting subclass of Calabi-Yaus is that provided by rigid Calabi-Yau spaces, which are of arithmetical significance. Lagrangian singularities play a significant role in several parts of mathematics. Their systematic study was initiated by Arnold and Givental. The Lagrangian nature of the vanishing cycles is at the basis of categorical mirror symmetry for Fano varieties. Many instances of these relations are still to be studied in detail. Hilbert schemes provide natural resolutions of symplectic singularities. They can be studied as archetypical examples for general conjectures on the relation between orbifold cohomology and the cohomology of resolutions. Irreducible holomorphic manifolds are hard to construct. Only two series and two exceptional examples are known up to deformation. It is major challenge to study symplectic singularities as a tool to construct new examples. The topologically simplest class of surface singularities are those whose links are rational homology spheres. The surprising conjecture that the universal abelian cover should be a complete intersection is one of the outstanding open problems in the field.