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Some Topics in Several Complex Variables




报告人:Prof. Xiaokui Yang (Institute of Mathematics, AMSS) 

  目:Curvature relations in differential geometry and complex algebraic geometry 

  要:In this talk, I will present a variety of curvature relations in differential geometry and complex algebraic geometry including a recent work confirming Yau's conjecture on the relationship between the negativity of holomorphic sectional curvature and ampleness of the canonical line bundle over compact Kähler manifolds.  




报告人:Prof. Valentino Tosatti (Northwestern University) 

  目:Degenerations of Calabi-Yau metrics  

  要:In this talk, a Calabi-Yau manifold will be a compact Kähler manifold with vanishing real first Chern class. By a fundamental theorem of Yau, these manifolds admit Ricci-flat Kähler metric, a unique one in each Kähler class. I will discuss the problem of understanding how Ricci-flat Calabi-Yau manifolds degenerate when we vary either the complex structure or the Kähler class. The hardest (and most interesting) case is when the manifolds collapse to lower-dimensional spaces, and understanding this case is relevant to the Strominger-Yau-Zaslow picture of mirror symmetry. I will present several results about these questions.  




报告人:Prof. Valentino Tosatti (Northwestern University) 

  目:Kähler currents in complex geometry 

  要:A Kähler current on a compact complex manifold is a closed positive real (1,1) current which is strictly positive definite, in the weak sense. These are fundamental objects for the study of complex manifolds, and have been used in recent years to prove several striking results. I will discuss a theorem, joint with Tristan Collins, which gives a geometric characterization of the smallest singular set among all Kähler currents in a given nef cohomology class. This generalizes (and reproves) results by Nakamaye and Ein-Lazarsfeld-Mustata-Nakamaye-Popa in algebraic geometry. I will give some applications of this result, and some recent further developments.