Stable Vector Bundles Research Articles

Brauer obstruction for a universal vector bundle

Let X be a smooth complex projective curve with genus ( X ) > 2 , and let M be the moduli space parametrizing isomorphism classes of stable vector bundles E over X of rank r with ⋀ r E = ξ , where ξ is a fixed line bundle. We prove that the Brauer group Br ( M ) is Z / n Z , where n = g.c.d. ( r , degree ( ξ ) ) . Moreover, Br ( M ) is generated by the class of the projective bundle over M of relative dimension r − 1 obtained by restricting the universal projective bundle over X × M to a point of X. To cite this article: V. Balaji et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).

A counterexample to a conjecture due to Douglas, Reinbacher and Yau

In “Branes, Bundles and Attractors: Bogomolov and Beyond”, by Douglas, Reinbacher and Yau, the authors state the following conjecture: Consider a simply connected surface X with ample or trivial canonical line bundle. Then, the Chern classes of any stable vector bundle with rank r ≥ 2 satisfy 2 r c 2 − ( r − 1 ) c 1 2 − r 2 12 c 2 ( X ) ≥ 0 . The goal of this short note is to provide two sources of counterexamples to this strong version of the Bogomolov inequality.

Extension bundles and the standard model

We construct a new class of stable vector bundles suitable for heterotic string compactifications. Using these we describe a novel way to derive the fermionic matter content of the Standard Model from the heterotic string. More precisely, we can get either the Standard Model gauge group GSM times an additional U(1), or just GSM but with additional exotic matter. For this we compactify on an elliptically fibered Calabi-Yau threefold X with two sections, the B-fibration, a variant of the ordinary Weierstrass fibration, which allows X to carry a free involution. We construct rank five vector bundles, invariant under this involution, such that turning on a Wilson line we obtain the Standard Model gauge group and find various three generation models. This rank five bundle is derived from a stable rank four bundle that arises as an extension of bundles pulled-back from the base and twisted by suitable line bundles. We also give an account of various previous results and put the present construction into perspective.

Frobenius-unstable bundles and $p$-curvature

We use the theory of p-curvature of connections to analyze stable vector bundles of rank 2 on curves of genus 2 which pull back to unstable bundles under the Frobenius morphism. We take two approaches, first using explicit formulas for p-curvature to analyze low-characteristic cases, and then using degeneration techniques to obtain an answer for a general curve by degenerating to an irreducible rational nodal curve, and applying the results of additional works by the author. We also apply our explicit formulas to give a new description of the strata of curves of genus 2 of different p-rank.

Vanishing of the top Chern classes of the moduli of vector bundles

Let $Y$ be a smooth projective curve of genus $g\ge 2$ and let $M_{r,d}(Y)$ be the moduli space of stable vector bundles of rank $r$ and degree $d$ on $Y$. A classical conjecture of Newstead and Ramanan states that $ c_i(M_{2,1}(Y))=0$ for $i>2(g-1)$ i.e. the top $2g-1$ Chern classes vanish. The purpose of this paper is to generalize this vanishing result to the rank 3 case by generalizing Gieseker's degeneration method. More precisely, we prove that $c_i(M_{3,1}(Y))=0$ for $i>6g-5$. In other words, the top $3g-3$ Chern classes vanish. Notice that we also have $c_i(M_{3,2}(Y))=0$ for $i>6g-5$.

Strong Bogomolov inequality for stable vector bundles

Inspired by the recent conjectures concerning the existence of stable bundles on Calabi–Yau threefolds arising from string theory, we consider the possibility of strengthening the classical Bogomolov inequality. We show the existence of stable bundles violating such inequality on many complete intersections.

Injectivity of the determinantal map for space of sections of stable vector bundles on curves

Injectivity of the determinantal map for space of sections of stable vector bundles on curves

MAXIMAL SUBSHEAVES OF TORSION-FREE SHEAVES ON NODAL CURVES

Let Y be a reduced irreducible projective curve of arithmetic genus g ⩾ 2 with at most ordinary nodes as singularities. For a subsheaf F of rank r′, degree d′ of a torsion-free sheaf E of rank r, degree d, let s(E,F) = r′d−rd′. Define sr′(E) = min s(E,F), where the minimum is taken over all subsheaves of E of rank r′. For a fixed r′, sr′ defines a stratification of the moduli space U(r,d) of stable torsion-free sheaves of rank r, degree d by locally closed subsets Ur′,s. We study the nonemptiness and dimensions of the strata. We show that the general element in each nonempty stratum is a vector bundle and it has only finitely many (respectively unique) maximal subsheaves of rank r′ for s ⩽ r′(r-r′)(g − 1) (respectively s < r′(r-r′)(g − 1)). We prove that the tensor product of two general stable vector bundles on an irreducible nodal curve Y is nonspecial.

STABLE VECTOR BUNDLES OF RANK TWO ON ENRIQUES SURFACES

We describe the moduli spaces of stable vector bundles of rank 2 on Enriques surfaces. They all have the structure of the fibrations reflecting those of Enriques surfaces.

Infinitesimal deformations of the tangent bundle of a moduli space of vector bundles over a curve

Fix a line bundle $\xi$ on a connected smooth complex projective curve $X$ of genus at least three. Let $\mathcal{N}$ denote the moduli space of all stable vector bundles over $X$ of rank $n$ and determinant $\xi$. We assume that $n\geq 3$ and coprime to $\operatorname{degree}(\xi)$; If $\operatorname{genus}(X)\leq 4$, then we also assume that $n \geq 4$. We prove that $H^i(\mathcal{N}, \End(T\mathcal{N}\mkern2mu)) = H^i(X, \mathcal{O}_X)$ for $i= 0,1$.

Stable Vector Bundles as Generators of the Chow Ring

In this paper we show that the family of stable vector bundles gives a set of generators for the Chow ring, the K-theory and the derived category of any smooth projective variety.

A note on the Picard bundle over a moduli space of vector bundles

Let ℳ︁(n , d ) be a coprime moduli space of stable vector bundles of rank n ≥ 2 and degree d over a complex irreducible smooth projective curve X of genus g ≥ 2 and ℳ︁ξ ⊂ ℳ︁(n , d ) a fixed determinant moduli space. Assuming that the degree d is sufficiently large, denote by 𝒦 the vector bundle over X ×ℳ︁(n , d ) defined by the kernel of the evaluation map H 0(X , E ) → Ex , where E ∈ℳ︁(n , d ) and x ∈ X . We prove that 𝒦 and its restriction 𝒦ξ to X × ℳ︁ξ are stable. The space of all infinitesimal deformations of 𝒦 over X ×ℳ︁(n , d ) is proved to be of dimension 3g and that of 𝒦ξ over X × ℳ︁ξ of dimension 2g , assuming that g ≥ 3 and if g = 3 then n ≥ 4 and if g = 4 then n ≥ 3. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Deformations of the generalised Picard bundle

Let X be a nonsingular algebraic curve of genus g ⩾ 3 , and let M ξ denote the moduli space of stable vector bundles of rank n ⩾ 2 and degree d with fixed determinant ξ over X such that n and d are coprime. We assume that if g = 3 then n ⩾ 4 and if g = 4 then n ⩾ 3 , and suppose further that n 0 , d 0 are integers such that n 0 ⩾ 1 and nd 0 + n 0 d > nn 0 ( 2 g - 2 ) . Let E be a semistable vector bundle over X of rank n 0 and degree d 0 . The generalised Picard bundle W ξ ( E ) is by definition the vector bundle over M ξ defined by the direct image p M ξ * ( U ξ ⊗ p X * E ) where U ξ is a universal vector bundle over X × M ξ . We obtain an inversion formula allowing us to recover E from W ξ ( E ) and show that the space of infinitesimal deformations of W ξ ( E ) is isomorphic to H 1 ( X , End ( E ) ) . This construction gives a locally complete family of vector bundles over M ξ parametrised by the moduli space M ( n 0 , d 0 ) of stable bundles of rank n 0 and degree d 0 over X. If ( n 0 , d 0 ) = 1 and W ξ ( E ) is stable for all E ∈ M ( n 0 , d 0 ) , the construction determines an isomorphism from M ( n 0 , d 0 ) to a connected component M 0 of a moduli space of stable sheaves over M ξ . This applies in particular when n 0 = 1 , in which case M 0 is isomorphic to the Jacobian J of X as a polarised variety. The paper as a whole is a generalisation of results of Kempf and Mukai on Picard bundles over J, and is also related to a paper of Tyurin on the geometry of moduli of vector bundles.

GUT PARTICLE SPECTRUM FROM HETEROTIC COMPACTIFICATION

This review intends to introduce the reader to the general methodology of computing GUT particle spectrum from compactifying the E8 × E8 heterotic string on smooth Calabi–Yau threefolds endowed with stable vector bundles. We use the spectral construction for the bundle and show, with illustrative examples, how one computes the requisite cohomology groups that determine the spectrum. Moreover, physical constraints lead to an algorithmic classification scheme. Finally, we will see the fascinating moduli-dependence, a so-called jumping phenomenon, of the particle spectra.

Stable geometric dimension of vector bundles over even-dimensional real projective spaces

In (6), the geometric dimension of all stable vector bundles over real projective spaceP n was determined ifn is even and sufficiently large with respect to the order 2 e of the bundle in f

On Poincaré bundles of vector bundles on curves

Let M denote the moduli space of stable vector bundles of rank n and fixed determinant of degree coprime to n on a non-singular projective curve X of genus g≥2. Denote by a universal bundle on X×M. We show that, for x,y ∈ X, x≠y, the restrictions |{x}×M and |{y}×M are stable and non-isomorphic when considered as bundles on M. Vector bundle, Poincaré bundle, moduli space

Stable Bundles on Non-Kähler Elliptic Surfaces

In this paper, we study the moduli spaces of stable rank-2 vector bundles on non-Kähler elliptic surfaces, thus giving a classification of these bundles; in the case of Hopf and Kodaira surfaces, these moduli spaces admit the structure of an algebraically completely integrable Hamiltonian system.

Minimal rational curves on moduli spaces of stable bundles

Let C be a smooth projective curve of genus g ≥ 2 and L be a line bundle on C of degree d. Let M := UC(r,L) be the moduli space of stable vector bundles on C of rank r and with the fixed determinant L. Assume that (r, d) = 1, then M is a smooth projective Fano variety with Picard number 1. For any projective curve on M , we can define its degree with respect to the ample anti-canonical line bundle −KM . The first result of this paper determines all rational curves of minimal degree passing through a generic point of M , which answers a question of Jun-Muk Hwang (see Question 1 in [Hw]).

Stable bundles on positive principal elliptic fibrations

Let $M\stackrel\pi \arrow X$ be a principal elliptic fibration over a Kaehler base $X$. We assume that the Kaehler form on $X$ is lifted to an exact form on $M$ (such fibrations are called positive). Examples of these are regular Vaisman manifolds (in particular, the regular Hopf manifolds) and Calabi-Eckmann manifolds. Assume that $\dim M > 2$. Using the Kobayashi-Hitchin correspondence, we prove that all stable bundles on $M$ are flat on the fibers of the elliptic fibration. This is used to show that all stable vector bundles on $M$ take form $L\otimes \pi^* B_0$, where $B_0$ is a stable bundle on $X$, and $L$ a holomorphic line bundle. For $X$ algebraic this implies that all holomorphic bundles on $M$ are filtrable (that is, obtained by successive extensions of rank-1 sheaves). We also show that all positive-dimensional compact subvarieties of $M$ are pullbacks of complex subvarieties on $X$.

Canonical metrics on stable vector bundles

Canonical metrics on stable vector bundles