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1
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Rank-two vector bundles on Hirzebruch surfaces

100%
Aprodu M., Brînzănescu V., Marchitan M.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1321-1330
EN
We survey some parts of the vast literature on vector bundles on Hirzebruch surfaces, focusing on the rank-two case.
2
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The Picard group of a coarse moduli space of vector bundles in positive characteristic

100%
Hoffmann N.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1306-1313
EN
Let C be a smooth projective curve over an algebraically closed field of arbitrary characteristic. Let M r,Lss denote the projective coarse moduli scheme of semistable rank r vector bundles over C with fixed determinant L. We prove Pic(M r,Lss) = ℤ, identify the ample generator, and deduce that M r,Lss is locally factorial. In characteristic zero, this has already been proved by Drézet and Narasimhan. The main point of the present note is to circumvent the usual problems with Geometric Invariant Theory in positive characteristic.
3
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On localization in holomorphic equivariant cohomology

88%
Bruzzo U., Rubtsov V.
Open Mathematics
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2012
|
tom 10
|
nr 4
1442-1454
EN
We study a holomorphic equivariant cohomology built out of the Atiyah algebroid of an equivariant holomorphic vector bundle and prove a related localization formula. This encompasses various residue formulas in complex geometry, in particular we shall show that it contains as special cases Carrell-Liebermann’s and Feng-Ma’s residue formulas, and Baum-Bott’s formula for the zeroes of a meromorphic vector field.
4
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Elliptic operators on refined Sobolev scales on vector bundles

63%
Zinchenko T.
Open Mathematics
|
2017
|
tom 15
|
nr 1
907-925
EN
We introduce a refined Sobolev scale on a vector bundle over a closed infinitely smooth manifold. This scale consists of inner product Hörmander spaces parametrized with a real number and a function varying slowly at infinity in the sense of Karamata. We prove that these spaces are obtained by the interpolation with a function parameter between inner product Sobolev spaces. An arbitrary classical elliptic pseudodifferential operator acting between vector bundles of the same rank is investigated on this scale. We prove that this operator is bounded and Fredholm on pairs of appropriate Hörmander spaces. We also prove that the solutions to the corresponding elliptic equation satisfy a certain a priori estimate on these spaces. The local regularity of these solutions is investigated on the refined Sobolev scale. We find new sufficient conditions for the solutions to have continuous derivatives of a given order.
5
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K-theory from the point of view of C*-algebras and Fredholm representations

63%
Mishchenko A.
Open Mathematics
|
2005
|
tom 3
|
nr 4
766-793
EN
These notes represent the subject of five lectures which were delivered as a minicourse during the VI conference in Krynica, Poland, “Geometry and Topology of Manifolds”, May, 2–8, 2004.
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