Bott-Chern Characteristic Forms And Index Theorems For Coherent Sheaves On Complex Manifolds

In the paper \cite{Block2010}, Block constructed a dg-category $\mc{P}{\mc{A}^{0, \bullet}}$ using cohesive modules which is a dg-enhancement of $D^b{\mr{Coh}}(X)$, 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 \cite{Quillen1985} 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 \cite{Bismut1988} \cite{Botta} \cite{Donaldson1987} 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 \cite{Bismut2013}. 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 \cite{Berline1991}. Applying the heat kernel method and a theorem of Getzler in \cite{Getzler1991}, 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.