A construction of stable vector bundles on Calabi–Yau manifolds

Nakashima, Tohru

doi:10.1016/s0393-0440(03)00089-5

Cited by 7 publications

(6 citation statements)

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“…Elementary transformations are a classical method for constructing bundles introduced by Maruyama [29] [30], and recently for example Nakashima constructs stable vector bundles on CY threefolds using this technique [34]. The following proposition says that E is obtained by an elementary transformation along C, and determines the structure of φ.…”

Section: The Case Of Reducible Spectral Surfacementioning

confidence: 99%

Obstructed Bundles of Rank Two on a Quintic Surface

Mestrano

Simpson

2011

Int. J. Math.

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To the memory of Masaki MaruyamaIn this note we consider the moduli space of stable bundles of rank two on a very general quintic surface. We study the potentially obstructed points of the moduli space via the spectral covering of a twisted endomorphism. This analysis leads to generically nonreduced components of the moduli space, and components which are generically smooth of more than the expected dimension. We obtain a sharp bound asked for by O'Grady on when the moduli space is good. must be a singular point. The moduli space might on the other hand be smooth but overdetermined, i.e. having dimension bigger than the expected dimension. And of course it could also be overdetermined and singular too.As is well known (see [9;46;39, Sec. 1;24]), the dual of the space of obstructions is H 0 (X, End 0 (E) ⊗ K X ) by Serre duality. An element φ in this dual space is a trace-free morphism φ : E → E ⊗ K X . Such a φ corresponds, by Kuranishi theory, to an equation of the moduli space locally at E, and we call it a co-obstruction. A pair (E, φ) consisting of a bundle together with a nonzero co-obstruction, may be thought of as a K X -valued Hitchin pair on X [19]. These pairs are different from those considered in [41] for the surface X: the Higgs bundles corresponding to representations of π 1 are endomorphisms taking values in Ω 1 X . Over a curve these two notions coincide and indeed Hitchin used the notation K X in his original paper [19]. Generalizing his notation as written leads to the notion of a Higgs field E → E ⊗ K X which is exactly a co-obstruction, often called a "twisted endomorphism."A basic tool in the analysis of Hitchin pairs is the notion of spectral cover [19,8,3,46]. A twisted endomorphism φ : E → E ⊗ K X gives E the structure of coherent sheaf on the total space of the vector bundle K X , and the support of the coherent sheaf is the spectral covering associated to φ. It consists of the set of pairs (x, u) where x ∈ X and u ∈ K X,x such that u is an eigenvalue of φ x .In our rank-two case the spectral cover is particularly simple to describe: it is the divisor Z ⊂ K X determined by the equationWe investigate in a very basic way the possible classification of such spectral covers, and the implications for the locus of singularities of the moduli space. This follows Donaldson's original proof of generic smoothness [9], as it has been developed by Zuo in [46], and more recently by Langer [24].Many authors have shown that the moduli spaces of bundles of odd degree on abelian and K3 surfaces are smooth, going back to [11] and Mukai [32] (see the discussions and references in [43,44]). O'Grady has observed an important example of symplectic singularities in the moduli of rank-two bundles on a K3 surface [40], along the locus of reducible bundles. In view of these properties and examples, for understanding bundles on surfaces of general type it seems like a good idea to look at surfaces of general type which are as close as possible to K3 surfaces. This motivates our consideration of the example of a ver...

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Section: The Case Of Reducible Spectral Surfacementioning

confidence: 99%

Obstructed Bundles of Rank Two on a Quintic Surface

Mestrano

Simpson

2011

Int. J. Math.

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“…The above conjecture is false as proved in [4], [10] and [11]. In [10] and [11] the counterexamples were constructed using the method of elementary transformation on arbitrary surfaces as in the conjecture.…”

Section: Introductionmentioning

confidence: 99%

“…In [10] the author of that paper posed the following problem: Problem 1.1 Construct a sequence of µ-stable vector bundles E m with effectively computable discriminants δ m such that δ m go to infinity with m.…”

Section: Introductionmentioning

confidence: 99%

“…In [10] and [11] the counterexamples were constructed using the method of elementary transformation on arbitrary surfaces as in the conjecture. In [4] there are two families of counterexamples: on K3 surfaces using the specific methods for such surfaces from [7] and on surfaces in P 3 of degree d ≥ 7 using Horrocks's theory of monads and methods from [5].…”

Section: Introductionmentioning

confidence: 99%

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On large families of bundles over algebraic surfaces

Anghel

Buruiana

2016

Journal of Geometry and Physics

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The aim of this note is to construct sequences of vector bundles with unbounded rank and discriminant on an arbitrary algebraic surface. This problem, on principally polarized abelian varieties with cyclic Neron-Severi group generated by the polarization, was considered by Nakashima in connection with the Douglas-Reinbacher-Yau conjecture on the Strong Bogomolov Inequality. In particular we show that on any surface, the Strong Bogomolov Inequality SBI l is false for all l > 4.

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“…Such an asymptotic existence problem may be considered as a preliminary step to the solution of the original, much harder existence problem and is closely related to the strong Bogomolov inequality which has been discussed in the recent study of string theory ( [1], [8]). In [7], we gave examples of Calabi-Yau manifolds (X, H) such that SBI does not hold by constructing certain large families of μ-stable bundles on them.…”

Section: Introductionmentioning

confidence: 99%

Untitled

2013

Proc. Amer. Math. Soc.

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A construction of stable vector bundles on Calabi–Yau manifolds

Cited by 7 publications

References 9 publications

Obstructed Bundles of Rank Two on a Quintic Surface

Obstructed Bundles of Rank Two on a Quintic Surface

On large families of bundles over algebraic surfaces

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