Complex symplectic manifolds and related topics

首席调查员研究了从复杂对称性歧管中研究的全息图，在他获得了一系列关于预测空间和平滑超平衡的数字特征的基本结果。他的结果之一的断言是光滑的Fano n-fold X是一个同构的n空间，如果只有“X”的长度是n + 1或n，那么长度被定义为最小的（C，-Kx）， C在X上运行一组曲线。我们的新特征足够强大，足以应用于复杂的说明书，使我们能够证明从复杂manifolds中遵循的结构定理：·让Y成为维度2 n和π ： Y → Y^^ a birational morphism onto a normal variety。Let E denote an arbitrary irreducible component and put B = π（E）。然后B是2米[小于或等于] 2n的复杂对称性变异，对于一般点b μ B，反向图像π^<-1>（b）μ E是n + m的投影空间。让f ： Y → X是原始上的非线性光纤空间结构;维度2n的复杂对称性矩阵。如果我们接受全息部分，那么X是投影的n空间，并且f是拉格朗日子的纤维，他的研究的另一个产品是与J. Kollar， S.的联合工作。Mori and H。Takagi，它提供了Fano 3-文件的边界，只有规范性和单一性。

The head investigator studied holomorphic maps from complex symplectic manifolds, in which he obtained a series of fundamental results on numerical characterisations of projective space and smooth hyperquadrics. One of his results asserts that a smooth Fano n-fold X is isomorphic to projective n-space of a hyperquadric if and only if the"length of X"is n + 1 or n, where the length is defined to be the minimum of (C, -Kx), C running through the set of curves on X.Our new characterisations are strong enough to be applied to complex manifolds, enabling us to prove the following structure theorem on morphisms from complex manifolds :・ Let Y be a projective complex symplectic manifod of dimension 2n and π : Y → Y^^^ a birational morphism onto a normal variety. Let E denote an arbitrary irreducible component and put B = π(E). Then B is a complex symplectic variety of dimension 2m 【less than or equal】 2n and, for a general point b ∈ B, the inverse image π^<-1>(b) ∩ E is projective space of dimension n + m.・ Let f : Y → X be a nontrivial fiber space structure on a primitive ; complex symplectic manifold of dimension 2n. If f admits a holomorphic section, then X is projective n-space and the fibers of f are Lagrangian subvarieties.Another product of his research is a joint work with J. Kollar, S. Mori and H. Takagi, which proved the boundedness of Fano 3-folds with only canonical, singularities.