Bott-Chern Characteristic Forms and Index Theorems for Coherent Sheaves on Complex Manifolds / Hua Qiang.

Format:

Book

Thesis/Dissertation

Author/Creator:

Qiang, Hua, author.

Contributor:

Block, Jonathan, degree supervisor.

Zhu, Yuecheng, degree committee member.

Pantev, Tony, 1963- degree committee member.

Kadision, Richard Vincent, degree committee member.

Block, Jonathan, degree committee member.

University of Pennsylvania. Mathematics, degree granting institution.

Language:

English

Subjects (All):

Mathematics.

Mathematics--Penn dissertations.

Penn dissertations--Mathematics.

Local Subjects:

Mathematics.

Mathematics--Penn dissertations.

Penn dissertations--Mathematics.

Genre:

Academic theses.

Physical Description:

1 online resource (89 pages)

Contained In:

Dissertation Abstracts International 79-01B(E).

Place of Publication:

[Philadelphia, Pennsylvania]: University of Pennsylvania ; Ann Arbor : ProQuest Dissertations & Theses, 2017.

Language Note:

English

System Details:

Mode of access: World Wide Web.

text file

Summary:

In the paper [Blo10], Block constructed a dg-category P A0,• using cohesive modules which is a dg-enhancement of Db Coh(X), the bounded derived category of complexes of analytic sheaves with coherent cohomology. This enables us to study coherent sheaves using global differential geometric constructions.

In the first part of my thesis, we construct natural superconnections in the sense of Quillen [Qui85] on cohesive modules and use them to define characteristic classes with values in Bott-Chern cohomology. In addition, we generalize the double transgression formulas in [BGS88a] [BC65] [Don87] and prove the invariance of these characteristic classes under derived equivalences. This provides an extension of Bott-Chern characteristic classes to coherent sheaves on complex manifolds and answers the question raised in [Bis13].

In the second part of my thesis, we define the generalized Dolbeault-Dirac operator on the generalized Dolbeault complex for a cohesive module. We identify it with a generalized Dirac operator in the sense of Clifford modules and Clifford superconnections as in [BGV91]. Applying the heat kernel method and a theorem of Getzler in [Get91], we first derive a generalization of the Hirzebruch-Riemann-Roch formula to compute the Euler characteristic. Then, generalizing Bismut's proof of the family index theorem, we derive a generalization of the classical Grothendieck-Riemann-Roch formula with values in Bott-Chern cohomology in special cases.

Notes:

Source: Dissertation Abstracts International, Volume: 79-01(E), Section: B.

Advisors: Jonathan Block; Committee members: Jonathan Block; Richard Vincent Kadision; Tony Pantev; Yuecheng Zhu.

Department: Mathematics.

Ph.D. University of Pennsylvania 2017.

Local Notes:

School code: 0175

ISBN:

9780355129427

Access Restriction:

Restricted for use by site license.