Vector Bundles Near Negative Curves: Moduli and Local Euler Characteristic

Ballico, Edoardo; Gasparim, Elizabeth; Köppe, Thomas

doi:10.1080/00927870802562351

Cited by 18 publications

(44 citation statements)

References 14 publications

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“…Our third result (Theorem 6.14) shows that any holomorphic vector bundle on Z k (τ ) splits as a direct sum of algebraic line bundles. This is somewhat surprising, given the existence of nontrivial moduli of vector bundles on the original Z k surfaces proved in [8]. Our fourth result (Theorem 6.18) shows that any nontrivial deformation Z k (τ ) is affine.…”

Section: Statement Of Resultsmentioning

confidence: 87%

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Classical deformations of noncompact surfaces and their moduli of instantons

Barmeier

Gasparim

2019

Journal of Pure and Applied Algebra

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We describe semiuniversal deformation spaces for the noncompact surfaces Z k := Tot(O P 1 (−k)) and prove that any nontrivial deformation Z k (τ ) of Z k is affine.It is known that the moduli spaces of instantons of charge j on Z k are quasi-projective varieties of dimension 2j − k − 2. In contrast, our results imply that the moduli spaces of instantons on any nontrivial deformation Z k (τ ) are empty.

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Section: Statement Of Resultsmentioning

confidence: 87%

“…In contrast with Z k (τ ), the surfaces Z k have nontrivial moduli of vector bundles. For example, [8,Thm. 4.11] shows that the moduli of rank 2 bundles on Z k with splitting type j has dimension 2j − k − 2.…”

Section: Vector Bundles On Z K (τ )mentioning

confidence: 99%

Classical deformations of noncompact surfaces and their moduli of instantons

Barmeier

Gasparim

2019

Journal of Pure and Applied Algebra

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“…Note that this result is in strong contrast with the case of surfaces (n = 1), for which h 0 W ; (π * F ) ∨∨ /π * F attains a wide variety of values (compare with[5, Theorem 2.16]).…”

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confidence: 88%

Local moduli of holomorphic bundles

Ballico

Gasparim

Köppe

2009

Journal of Pure and Applied Algebra

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“…The case when Z is the total space of a negative line bundle on P 1 was studied in [BGK1] and [GKM]. Unfortunately, the width vanishes in higher dimensions.…”

Section: Numerical Invariantsmentioning

confidence: 99%

“…The lower bound is attained by a class of generic bundles, and the upper bound by the split bundle O(−j) ⊕ O(j). This can be seen by direct computation as explained in [BGK1] and [Kö].…”

Section: Threefoldsmentioning

confidence: 99%

Sheaves on singular varieties

Gasparim¹,

Köppe²

2010

J. Sing.

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Abstract. We prove existence of reflexive sheaves on singular surfaces and threefolds with prescribed numerical invariants and study their moduli. MotivationSheaves on singular varieties have become very popular recently because of their appearance in Physics, String Theory and Mirror Symmetry. In particular, many open questions about sheaves on singular varieties have come to light. The corresponding mathematical tools, however, are waiting to be developed. Our aim in this paper is to entice singularists to develop some basic techniques needed to approach such questions. It is extremely common for a physicist or string theorist to start up a lecture by giving a partition function for a theory, and now even algebraic geometers are quite often doing the same. It is not just a fashion, but the fact is that this is an extremely efficient way to present results. The general format of such partition functions is of an infinite sum whose terms contain integrals over moduli spaces. Here are some examples. We will not need details from these expressions, just the observation that they all contain integrals over moduli spaces.Example 1.1. (String Theory) The Nekrasov partition function for N = 2 supersymmetric SU (r) pure gauge theory on a complex surface X is given by an expression of the formwhere M r,d,n (X) is the moduli space of framed torsion-free sheaves or rank r, and Chern classes c 1 = d and c 2 = n. For the case of gauge theories with matter, one writes a similar expression but with more interesting integrands, see [GL]. Example 1.2. (Donaldson-Thomas Theory) For a Calabi-Yau threefold X, the partition function for Donaldson-Thomas theory is given by: where M g (X, β) is the moduli space of genus-g curves representing the class β ∈ H 2 (X, Z).There is a precise sense in which this partition function is equivalent to the one in Example 1.2, see [MNOP].These examples illustrate the appearance of integrals over moduli spaces of sheaves. Even in the case of moduli spaces of maps of Example 1.3 the theory is still related to a theory given by integration over moduli of sheaves. Observe that the definition of moduli spaces itself requires a choice of numerical invariants: in Example 1.1 the Chern classes and in Example 1.2 the Euler characteristic. So, we now agree that we are interested in moduli spaces of sheaves on surfaces and threefolds. Of course, the physics motivation is just a bonus, and we could have been interested in such moduli spaces for purely geometric reasons, as they are part of classical algebraic geometry. Now physics dictates that we should consider theories defined over singular varieties. In fact, some of the most popular categories considered currently by physicists and string theorists turn out empty in the absence of singularities; such is the case of the FukayaSeidel category and the Orlov category of singularities. Thus we arrive at the conclusion that we need to understand moduli of sheaves on singular varieties. Both the case of global moduli of sheaves on projective varieties and the ...

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Vector Bundles Near Negative Curves: Moduli and Local Euler Characteristic

Cited by 18 publications

References 14 publications

Classical deformations of noncompact surfaces and their moduli of instantons

Classical deformations of noncompact surfaces and their moduli of instantons

Local moduli of holomorphic bundles

Sheaves on singular varieties

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