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1
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Vector bundles on blown-up Hopf surfaces

100%
Toma M.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1356-1360
EN
We show that certain moduli spaces of vector bundles over blown-up primary Hopf surfaces admit no compact components. These are the moduli spaces used by Andrei Teleman in his work on the classification of class VII surfaces.
2
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Vector bundles of finite rank on complete intersections of finite codimension in ind-Grassmannians

100%
Ermakova S.
Complex Manifolds
|
2015
|
tom 2
|
nr 1
EN
In this article we establish an analogue of the Barth-Van de Ven-Tyurin-Sato theorem.We prove that a finite rank vector bundle on a complete intersection of finite codimension in a linear ind-Grassmannian is isomorphic to a direct sum of line bundles.
3
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On Cohen-Macaulay modules over non-commutative surface singularities

100%
Drozd Y., Gavran V.
Open Mathematics
|
2014
|
tom 12
|
nr 5
675-687
EN
We generalize the results of Kahn about a correspondence between Cohen-Macaulay modules and vector bundles to non-commutative surface singularities. As an application, we give examples of non-commutative surface singularities which are not Cohen-Macaulay finite, but are Cohen-Macaulay tame.
4
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Moduli of symplectic instanton vector bundles of higher rank on projective space ℙ3

100%
Bruzzo U., Markushevich D., Tikhomirov A.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1232-1245
EN
Symplectic instanton vector bundles on the projective space ℙ3 constitute a natural generalization of mathematical instantons of rank-2. We study the moduli space I n;r of rank-2r symplectic instanton vector bundles on ℙ3 with r ≥ 2 and second Chern class n ≥ r, n ≡ r (mod 2). We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus I n;r* of tame symplectic instantons is irreducible and has the expected dimension, equal to 4n(r + 1) −r(2r + 1).
5
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On the Brill-Noether theory for K3 surfaces

100%
Leyenson M.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1486-1540
EN
Let (S, H) be a polarized K3 surface. We define Brill-Noether filtration on moduli spaces of vector bundles on S. Assume that (c 1(E), H) > 0 for a sheaf E in the moduli space. We give a formula for the expected dimension of the Brill-Noether subschemes. Following the classical theory for curves, we give a notion of Brill-Noether generic K3 surfaces. Studying correspondences between moduli spaces of coherent sheaves of different ranks on S, we prove our main theorem: polarized K3 surface which is generic in the sense of moduli is also generic in the sense of Brill-Noether theory (here H is the positive generator of the Picard group of S). The harder part of the proof is proving the non-emptiness of the Brill-Noether loci. In the case of algebraic curves such a theorem, proved by Griffiths and Harris and, independently, by Lazarsfeld, is sometimes called the strong theorem of the Brill-Noether theory. We finish by considering a number of projective examples. In particular, we construct explicitly Brill-Noether special K3 surfaces of genus 5 and 6 and show the relation with the theory of Brill-Noether special curves.
6
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ACM bundles, quintic threefolds and counting problems

100%
Mohan Kumar N., Rao A.
Open Mathematics
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2012
|
tom 10
|
nr 4
1380-1392
EN
We review some facts about rank two arithmetically Cohen-Macaulay bundles on quintic threefolds. In particular, we separate them into seventeen natural classes, only fourteen of which can appear on a general quintic. We discuss some enumerative problems arising from these.
7
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Bubble tree compactification of moduli spaces of vector bundles on surfaces

63%
Markushevich D., Tikhomirov A., Trautmann G.
Open Mathematics
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2012
|
tom 10
|
nr 4
1331-1355
EN
We announce some results on compactifying moduli spaces of rank 2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so-called bubbling of vector bundles and connections in differential geometry. The new moduli spaces are algebraic spaces arising as quotients by group actions according to a result of Kollár. As an example, the compactification of the space of stable rank 2 vector bundles with Chern classes c 1 = 0, c 1 = 2 on the projective plane is studied in more detail. Proofs are only indicated and will appear in separate papers.
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