Algebraic vaieties with Kodaira dimansion O and A generalization of the Bogomolor-decomposition

具有 Kodaira 维 O 的代数簇和 Bogomolor 分解的 A 推广

• 批准号: 12440007
• 负责人: NAMIKAWA Yoshinori
• 金额: $ 3.46万
• 依托单位: Osaka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440007/
• 关键词: Complex symplectic variety; derived category; deformation theory; singularity; Period; Torelli problem; higher dimmsional algebraic varieries; カラビ・ヤウ多様体; トレリ問題; モジュライ空間; カラビーヤウ多様体; 双有理幾何; 基本群; 倉西空間

Does the period of the second cohomology determine an isomorphism class of a complex irreducible symplectic manifold? This is the Torelli problem. It is true for a K3 surface, but there is an counter-example of Debarre for dim 【greater than or equal】 4. Mukai, Huybrechts and others have posed the birational Torelli problem modifying the original one. We construeted a counter-example even for this problem.On the other hand, for a complex irreducible symplectic manifold, which kind of information on the original variety can be recoverd from the derived category of coherent sheaves ? When do we have an equivalence of derived categories of two complex irreducible manifolds ? Such questions are quite interesting. We proved that the derived categories are equivalent if two smooth projective varieties are connected by a Mukai flop. As an application, one knows that birationally equivalent, complex, projective symplectic 4-folds have equivalent derived categories.We also studied the deformation theory of complex symplectic varieties and generalized several important facts on a complex symplectic manifold to a singular symplectic variety.

复不可约辛流形的二阶上同调的周期决定同构类吗？这就是Torelli的问题。这对于K3曲面是正确的，但对于dim [大于或等于] 4，存在Debarre的反例。Mukai，Huybrechts和其他人提出了修改原始问题的双理性Torelli问题。另一方面，对于复的不可约辛流形，从凝聚层的导出范畴中可以得到关于原簇的哪类信息？什么时候两个复不可约流形的导范畴是等价的？这类问题很有意思。我们证明了如果两个光滑投射簇由一个Mukai翻转连接，则导出范畴是等价的。作为应用，我们知道双有理等价的复射影辛4-folds有等价的导范畴，我们还研究了复辛簇的变形理论，并将复辛流形上的几个重要事实推广到奇异辛簇上。

项目成果

宮西正宜(共著): "Open algebraic surfaces with finite group actions"Transform.Groups. 7. 185-207 (2002)

Masanori Miyanishi（合著者）：“具有有限群作用的开放代数曲面”Transform.Groups。 7. 185-207 (2002)

並河良典: "Calabi-Yaus and Deformation Theory"Sugaku exposition, American Math.Soc.. (刊行予定). (2001)

Yoshinori Namikawa：“Calabi-Yaus 和变形理论”Sugaku 阐述，American Math.Soc.（即将出版）（2001 年）。

藤木明: "Compact self-dual manifolds with torus actions"J.Diff Geometry. (刊行予定).

Akira Fujiki：“具有环面作用的紧凑自对偶流形”J.Diff Geometry（即将出版）。

"Stratified local moduli of Calabi-Yau threefolds"Topology. 41. 1219-1237 (2002)

“卡拉比-丘三重分层局部模”拓扑。

藤木明: "Topology of compact self-dual manifolds whose twbfor space is of positive algebraic dimension"J.Math.Soc.Japan. 54. 587-608 (2002)

Akira Fujiki：“twb 空间具有正代数维数的紧致自对偶流形的拓扑”J.Math.Soc.Japan 54. 587-608 (2002)。