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It is well-known that on a non-projective complex manifold, a

coherent sheaf may not have a resolution by a complex of holomorphic vector

bundles. Nevertheless, J. Block showed that such resolution always exists if

we allow anti-holomorphic flat superconnections which generalize complexes

of holomorphic vector bundles. Block's result makes it possible to study

coherent sheaves with differential geometric and analytic tools. For

example, in a joint work with J.M Bismut, S, Shen, and I, we give an

analytic proof of the Grothendieck-Riemann-Roch theorem for coherent sheaves

on complex manifolds. In this talk I will present the ideas and applications

of anti-holomorphic flat superconnections. I will also talk about analogous

constructions of superconnections in other areas of geometry.