K-theory of stratified vector bundles

Stratified Fiber Bundles, Quinn Homology and Brane Stability of Hyperbolic Orbifolds

We revisit the problem of stability of string vacua involving hyperbolic orbifolds using methods from homotopy theory and K-homology. We propose a definition of Type II string theory on such backgrounds that further carry stratified systems of fiber bundles, which generalise the more conventional orbifold and symmetric string backgrounds, together with a classification of wrapped branes by a suitable generalized homology theory. For spaces stratified fibered over hyperbolic orbifolds we use the algebraic K-theory of their fundamental groups and Quinn homology to derive criteria for brane stabi…

光滑图册分层空间、K同调定向与Gysin映射

Smooth atlas stratified spaces, K-Homology Orientations, and Gysin maps

该研究引入光滑图册分层空间并证明其与Thom-Mather分层空间等价，在双变KK-理论中构造定向Witt分层空间纤维丛的Gysin映射并证明其保持解析符号差类，还建立了解析符号差类与拓扑Siegel-Sullivan定向的联系。

向量丛自同构的全纯因子分解

Holomorphic Factorization of Vector Bundle Automorphisms

该研究在复几何领域证明了Stein空间上秩2向量丛的零伦特殊全纯自同构可分解为有限个幂幺自同构或指数映射的乘积，推进了全纯向量丛自同构的结构研究。

Existence of vector bundles and global resolutions for singular surfaces

We prove two results about vector bundles on singular algebraic surfaces. First, on proper surfaces there are vector bundles of rank two with arbitrarily large second Chern number and fixed determinant. Second, on separated normal surfaces any coherent sheaf is the quotient of a vector bundle. As a consequence, for such surfaces the Quillen K-theory of vector bundles coincides with the Waldhausen K-theory of perfect complexes. Examples show that, on nonseparated schemes, usually many coherent sheaves are not quotients of vector bundles.

Unipotent Factorization of Vector Bundle Automorphisms

We provide unipotent factorizations of vector bundle automorphisms of real and complex vector bundles over smooth manifolds. This generalises work of Thurston-Wasserstein and Wasserstein for trivial vector bundles. We also address two symplectic cases and propose a complex geometric analog of the problem in the setting of holomorphic vector bundles over Stein manifolds.

Standard Vector Bundles

We construct the categories of standard vector bundles over schemes and define direct sum and tensor product. These categories are equivalent to the usual categories of vector bundles with additional properties. The tensor product is strictly associative, strictly commutative with line bundles, and strictly functorial on base change.