We here study the Brill-Noether theory for rank two vector bundles generated by their sections. We generalize the vanishing theorem, the Clifford theorem and the existence theorem to such bundles.
We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invari...We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invariant differential forms. When the action is cocompact, we develop a generalized complex Hodge theory for the Dolbeault cohomology of invariant differential forms. We prove that every cyclic cohomology class of these two Hopf algebroids can be represented by a generalized harmonic form. This implies that the space of cyclic cohomology of each Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we generalize the Serre duality and prove a Kodaira type vanishing theorem.展开更多
We prove the following vanishing theorem. Let M be an irreducible symmetric space of noncompact type whose dimension exceeds 2 and M ≠SO0（2, 2）/SO（2） × SO（2）. Let π ： E →＊ M be any vector bundle. Then ...We prove the following vanishing theorem. Let M be an irreducible symmetric space of noncompact type whose dimension exceeds 2 and M ≠SO0（2, 2）/SO（2） × SO（2）. Let π ： E →＊ M be any vector bundle. Then any E-valued L2 harmonic 1-form over M vanishes. In particular we get the vanishing theorem for harmonic maps from irreducible symmetric spaces of noncompact type.展开更多
Using non-linear connection of Finsler manifold M, the existence of local coordinates which is normalized at a point x is proved, and the Laplace operator A on 1-form of M is defined by non-linear connection and its c...Using non-linear connection of Finsler manifold M, the existence of local coordinates which is normalized at a point x is proved, and the Laplace operator A on 1-form of M is defined by non-linear connection and its curvature tensor. After proving the maximum principle theorem of Hopf-Bochner on M, the Bochner type vanishing theorem of Killing vectors and harmonic 1-form are obtained.展开更多
In this paper, we prove a local odd dimensional equivariant family index theorem which generalizes Freed＇s odd dimensional index formula. Then we extend this theorem to the noncommuta- tive geometry framework. As a c...In this paper, we prove a local odd dimensional equivariant family index theorem which generalizes Freed＇s odd dimensional index formula. Then we extend this theorem to the noncommuta- tive geometry framework. As a corollary, we get the odd family Lichnerowicz vanishing theorem and the odd family Atiyah-Hirzebruch vanishing theorem.展开更多
基金Supported by the National Key R and D Program of China(2020YFA0713100)the Natural Science Foundation of Jiangsu Province(BK20230900)National Natural Science Foundation of China(12141104)。
文摘We here study the Brill-Noether theory for rank two vector bundles generated by their sections. We generalize the vanishing theorem, the Clifford theorem and the existence theorem to such bundles.
文摘We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invariant differential forms. When the action is cocompact, we develop a generalized complex Hodge theory for the Dolbeault cohomology of invariant differential forms. We prove that every cyclic cohomology class of these two Hopf algebroids can be represented by a generalized harmonic form. This implies that the space of cyclic cohomology of each Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we generalize the Serre duality and prove a Kodaira type vanishing theorem.
文摘We prove the following vanishing theorem. Let M be an irreducible symmetric space of noncompact type whose dimension exceeds 2 and M ≠SO0（2, 2）/SO（2） × SO（2）. Let π ： E →＊ M be any vector bundle. Then any E-valued L2 harmonic 1-form over M vanishes. In particular we get the vanishing theorem for harmonic maps from irreducible symmetric spaces of noncompact type.
基金Project supported by the Natural Science Foundation of China(10271097)
文摘Using non-linear connection of Finsler manifold M, the existence of local coordinates which is normalized at a point x is proved, and the Laplace operator A on 1-form of M is defined by non-linear connection and its curvature tensor. After proving the maximum principle theorem of Hopf-Bochner on M, the Bochner type vanishing theorem of Killing vectors and harmonic 1-form are obtained.
基金Supported by National Natural Science Foundation of China(Grant No.11271062)Program for New Century Excellent Talents in University(Grant No.13-0721)
文摘In this paper, we prove a local odd dimensional equivariant family index theorem which generalizes Freed＇s odd dimensional index formula. Then we extend this theorem to the noncommuta- tive geometry framework. As a corollary, we get the odd family Lichnerowicz vanishing theorem and the odd family Atiyah-Hirzebruch vanishing theorem.
