arXiv subjects

Alexei Rosly

Publications and source records attributed to Alexei Rosly.

5 recordsLinked to original sources

arXiv2022

Coherent Sheaves, Chern Classes, and Superconnections on Compact Complex-Analytic Manifolds

We construct a twist-closed enhancement of the category ${\mathcal D}^b_{\rm coh}(X)$, the bounded derived category of complexes of ${\mathcal O}_X$-modules with coherent cohomology, by means of the DG-category of $\bar\partial$-superconnections. Then we apply the techniques of $\bar\partial$-superconnections to define Chern classes and Bott-Chern classes of objects in the category, in particular, of coherent sheaves.

math.AG↗

arXiv2018

Ultraviolet Properties of the Self-Dual Yang-Mills Theory

We compute the ultraviolet divergences in the self-dual Yang-Mills theory, both in the purely perturbative (zero instanton charge) and topologically non-trivial sectors. It is shown in particular that the instanton measure is precisely the same as the one-loop result in the standard Yang-Mills theory.

hep-th↗

arXiv2015

A polar complex for locally free sheaves

We construct the so-called polar complex for an arbitrary locally free sheaf on a smooth variety over a field of characteristic zero. This complex is built from logarithmic forms on all irreducible subvarieties with values in a locally free sheaf. We prove that cohomology groups of the polar complex are canonically isomorphic to the cohomology groups of the locally free sheaf. Relations of the polar complex with Rost's cycle modules, algebraic cycles, Cousin complex, and adelic complex are discussed. In particular, the polar complex is a subcomplex in the Cousin complex. One can say that the polar complex is a first order pole part of the Cousin complex, providing a much smaller, but, in fact, quasiisomorphic subcomplex.

math.AG↗

arXiv2000

Symplectic geometry on moduli spaces of holomorphic bundles over complex surfaces

We give a comparative description of the Poisson structures on the moduli spaces of flat connections on real surfaces and holomorphic Poisson structures on the moduli spaces of holomorphic bundles on complex surfaces. The symplectic leaves of the latter are classified by restrictions of the bundles to certain divisors. This can be regarded as fixing a "complex analogue of the holonomy" of a connection along a "complex analogue of the boundary" in analogy with the real case.

math.AG↗

arXiv2000

Polar Homology

For complex projective manifolds we introduce polar homology groups, which are holomorphic analogues of the homology groups in topology. The polar k-chains are subvarieties of complex dimension k with meromorphic forms on them, while the boundary operator is defined by taking the polar divisor and the Poincare residue on it.

math.AG↗