We will discuss various versions of the derived category of coherent sheaves on compact complex-analytic manifolds and dg-enhancement of the suitable version of the category via Dolbeault superconnections. This enhancement allows us to define Chern and Bott-Chern classes of objects of the category, in particular of coherent sheaves. Also it allows us to construct the complex-analytic moduli spaces of objects in the category. The theory is generalized to non-compact complex-analytic manifolds, where one should consider the category of restricted objects of the derived category. We will analyze when this notion is homological.

About the speaker: Alexey Bondal is a leading researcher at the Steklov Mathematical Institute in Moscow, the director of the Center for Pure Mathematics at the Moscow Institute of Physics and Technology, and a senior fellow at the Kavli Institute for the Physics and Mathematics of the Universe. He studied at the Department of Mechanics and Mathematics of Moscow State University from 1978 to 1983 and was a graduate student at the Steklov Mathematical Institute from 1985 to 1988. His research fields cover algebraic geometry, homological algebra, representation theory, noncommutative geometry, and mathematical physics. In terms of professional experience, from 2009 to 2022, he served as the principal investigator at the Kavli IPMU in Japan, and from 2022 to 2024, he worked as a visiting senior scientist at the same institute. He has also given some important presentations, such as 2002 Invited Speaker at the International Congress of Mathematicians in Beijing, 2017 Plenary speaker at 3-rd Pacific Rim Mathematical Association Congress in Mexico, 2023 Plenary speaker at the First International Congress of Basic Sciences in Beijing and 2024 Plenary speaker at the Second International Congress of Basic Sciences in Beijing.