The study of vector bundles on an algebraic manifold with the trivial canonical bundle

具有平凡正则丛的代数流形上向量丛的研究

• 批准号: 12640024
• 负责人: YOSHIOKA Kota
• 金额: $ 2.43万
• 依托单位: KOBE UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 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/grant/KAKENHI-PROJECT-12640024/
• 关键词: vector bundle; K3 surface; abelian surface; Fourier-Mukai transform; Painleve equation; Gromov-Witten不変量

Yoshioka find the condition for the existence of stable sheaves on K3 and abelian surfaces and showed the connectedness of the moduli spaces. If the moduli space is compact, then he determined the albanese map and showed that the fiber is a hyperkaehler manifold. Moreover he showed that the deformation type of the manifold is determined by the dimension. By this result, the study of the topological type of the moduli space is reduced to the rank 1 case. In order to get these results, he used the Fourier-Mukai transform and the deformation of the underlying K3 surfaces. Hence he also studied the relation between the stability and the Fourier-Mukai transform. Moreover he studied the moduli of stable sheaves on an elliptic surface, surface components of the moduli of stable sheaves on a K3 surface and the Gromov-Witten invariants and got some results.Saito studied Gopakumar-Vafa conjecture on BPS states. He proposed a mathematical definition of the BPS invariants by using the moduli of purely 1-dimensional sheaves and checked the consistency for some cases.Yamada studied D-brane on a rational elliptic surface. Mathematical beautiful structures such as monodromy group, Mordell-Weil group, affine Weyl group are studied from the Physical point of view.Saito and yamada studied the Painleve equation in terms of the symmetry and the geometry. In particular, Saito showed that the Backlund transform is the flop in the birational geometry and the Painleve equation is derived from the deformation theory of a rational surface, which has an application of the classification of the Riccati solution.

Yoshioka找到了在K3和Abelian表面上存在稳定的滑轮的条件，并显示了模量空间的连接性。如果模量空间紧凑，则确定了阿尔巴尼斯图，并表明纤维是hyperkaehler歧管。此外，他表明歧管的变形类型取决于维度。通过此结果，对模量空间的拓扑类型的研究减少到等级1情况。为了获得这些结果，他使用了傅立叶 - 木叶变换和基础K3表面的变形。因此，他还研究了稳定性与傅立叶木粉转换之间的关系。此外，他研究了在椭圆表面上的稳定滑轮的模量，在K3表面和Gromov-witten不变性板上的模量模量的表面成分，并得到了一些结果。Saito研究了BPS状态的Gopakumar-Vafa猜想。他提出了通过使用纯1维或骨的模量对BPS不变的数学定义，并检查了某些情况的一致性。Yamada在有理的椭圆表面研究了D-Brane。从物理的角度研究了数学美丽的结构，例如monodromy群，mordell-weil群，息叶群。Saito和Yamada都根据对称性和几何形状研究了Painleve方程。特别是，saito表明反弹变换是植物几何形状中的插槽，而painleve方程源自有理表面的变形理论，该理论具有riccati溶液分类的应用。

项目成果

K.Yoshioka: "ATwisted stability and Fourier-Mukai transform II"Manuscripta Math.. (出版予定). (2003)

K. Yoshioka：“AT 扭曲稳定性和 Fourier-Mukai 变换 II”Manuscripta Math..（即将出版）。

Fukae, M., Yamada, Y., Yang, S.: "Mordell-Weyl lattice via string junctions"Nuclear Phys. B. 572. 71-94 (2000)

Fukae, M.、Yamada, Y.、Yang, S.：“通过弦连接的 Mordell-Weyl 晶格”核物理。

Hosoho, S., Saito M-H., Takahashi, A.: "Relative Lefschetz action and BPS state counting"Internat. Math. Res. Notices. No. 15. 783-816 (2001)

Hosoho, S.、Saito M-H.、Takahashi, A.：“相对 Lefschetz 作用和 BPS 状态计数”Internat。

K. Yoshioka: "Twisted stability and Fourier-Mukai transform I"Compositio Math.. to appear.

K. Yoshioka：“扭曲稳定性和 Fourier-Mukai 变换 I”Compositio Math.. 出现。

K. Yoshioka: "Twisted stability and Fourier-Mukai transform II"Manuscripta Math.. to appear.

K. Yoshioka：“扭曲稳定性和 Fourier-Mukai 变换 II”Manuscripta Math.. 出现。