Moduli Space Of Stable Bundles Research Articles

Multiplicity algebras for rank 2 bundles on curves of small genus

In [11], Hausel introduced a commutative algebra — the multiplicity algebra — associated to a fixed point of the [Formula: see text]-action on the Higgs bundle moduli space. Here we describe this algebra for a fixed point consisting of a very stable rank 2 vector bundle and zero Higgs field for a curve of low genus. Geometrically, the relations in the algebra are described by a family of quadrics and we focus on the discriminant of this family, providing a new viewpoint on the moduli space of stable bundles. The discriminant in our examples demonstrates that as the bundle varies, we obtain a continuous variation in the isomorphism class of the algebra.

Extension groups of tautological bundles on symmetric products of curves

We provide a spectral sequence computing the extension groups of tautological bundles on symmetric products of curves. One main consequence is that, if Ene mathcal O_X is simple, then the natural map {{,mathrm{mathsf {Ext}},}}^1(E,E)rightarrow {{,mathrm{mathsf {Ext}},}}^1(E^{[n]},E^{[n]}) is injective for every n. Along with previous results, this implies that Emapsto E^{[n]} defines an embedding of the moduli space of stable bundles of slope mu notin [-1,n-1] on the curve X into the moduli space of stable bundles on the symmetric product X^{(n)}. The image of this embedding is, in most cases, contained in the singular locus. For line bundles on a non-hyperelliptic curve, the embedding identifies the Brill–Noether loci of X with the loci in the moduli space of stable bundles on X^{(n)} where the dimension of the tangent space jumps. We also prove that E^{[n]} is simple if E is simple.

Disconnected moduli spaces of stable bundles on surfaces

We use hypersurfaces containing unexpected linear spaces to construct interesting vector bundles on complete intersection surfaces in projective space. We discover examples of moduli spaces of rank 2 stable bundles on surfaces of Picard rank one with arbitrarily many connected components.

Stable vector bundles on the families of curves

We offer a new approach to proving the Chen–Donaldson–Sun theorem which we demonstrate with a series of examples. We discuss the existence of a construction of a special metric on stable vector bundles over the surfaces formed by a family of curves and its relation to the one-dimensional cycles in the moduli space of stable bundles on curves.

A NOTE ON RANK TWO STABLE BUNDLES OVER SURFACES

Abstract.Let π : X → C be a fibration with integral fibers over a curve C and consider a polarization H on the surface X. Let E be a stable vector bundle of rank 2 on C. We prove that the pullback π*(E) is a H-stable bundle over X. This result allows us to relate the corresponding moduli spaces of stable bundles $${{\mathcal M}_C}(2,d)$$ and $${{\mathcal M}_{X,H}}(2,df,0)$$ through an injective morphism. We study the induced morphism at the level of Brill–Noether loci to construct examples of Brill–Noether loci on fibered surfaces. Results concerning the emptiness of Brill–Noether loci follow as a consequence of a generalization of Clifford’s Theorem for rank two bundles on surfaces.

An analytic application of Geometric Invariant Theory

Given a compact Kähler manifold, Geometric Invariant Theory is applied to construct analytic GIT-quotients that are local models for a classifying space of (poly)stable holomorphic vector bundles containing the coarse moduli space of stable bundles as an open subspace. For local models invariant generalized Weil-Petersson forms exist on the parameter spaces, which are restrictions of symplectic forms on smooth ambient spaces. If the underlying Kähler manifold is of Hodge type, then the Weil-Petersson form on the moduli space of stable vector bundles is known to be the Chern form of a certain determinant line bundle equipped with a Quillen metric. It gives rise to a holomorphic line bundle on the classifying GIT space together with a continuous hermitian metric.

Hitchin Fibration on Moduli of Symplectic and Orthogonal Parabolic Higgs Bundles

Let $X$ be a compact Riemann surface of genus $g \geq 2$, and let $D \subset X$ be a fixed finite subset. Let $\mathcal{M}(r,d,\alpha)$ denote the moduli space of stable parabolic $G$-bundles (where $G$ is a complex orthogonal or symplectic group) of rank $r$, degree $d$ and weight type $\alpha$ over $X$. Hitchin discovered that the cotangent bundle of the moduli space of stable bundles on an algebraic curve is an algebraically completely integrable system fibered, over a space of invariant polynomials, either by a Jacobian or a Prym variety of spectral curves. In this paper we study the Hitchin fibers for $\mathcal{M}(r,d,\alpha)$.

Rationality of moduli spaces of stable bundles on curves over [formula omitted

Let C be a smooth, projective, geometrically irreducible curve defined over R such that C(R)=∅. Let r>0 and d be integers which are coprime. Let L be a line bundle on C which corresponds to an R point of PicC/Rd. Let Mr,L be the moduli space of stable bundles on the complexification of C of rank r and determinant L. We classify birational types of Mr,L over R.

Non-emptiness of Brill–Noether loci over very general quintic hypersurface

In this article we study Brill–Noether loci of moduli space of stable bundles over smooth surfaces. We define Petri map as an analogy with the case of curves. We show the non-emptiness of certain Brill–Noether loci over very general quintic hypersurface in P3, and use the Petri map to produce components of expected dimension.

Stability of secant bundles on the second symmetric power of curves

Given a rank $r$ stable bundle over a smooth irreducible projective curve $C,$ there is an associated rank $2r$ bundle over $S^2(C),$ the second symmetric power of $C.$ In this article we study the stability of this bundle. As a consequence we get an immersion from the moduli space of stable bundles over $C$ to the associated moduli space of stable bundles over $S^2(C).$

Rational curves on the moduli spaces of stable bundles

In this paper, we give an account of rational curves in moduli spaces of stable vector bundles on a smooth curve.

Geometry of moduli stacks of(k,l)-stable vector bundles over algebraic curves

We study the geometry of the moduli stack of vector bundles of fixed rank and degree over an algebraic curve by introducing a filtration made of open substacks build from ( k , l ) -stable vector bundles. The concept of ( k , l ) -stability was introduced by Narasimhan and Ramanan to study the geometry of the coarse moduli space of stable bundles. We will exhibit the stacky picture and analyse the geometric and cohomological properties of the moduli stacks of ( k , l ) -stable vector bundles. For particular pairs ( k , l ) of integers we also show that these moduli stacks admit coarse moduli spaces and we discuss their interplay.

Remarks on minimal rational curves on moduli spaces of stable bundles

Let C be a smooth projective curve of genus g≥2 over an algebraically closed field of characteristic zero, and M be the moduli space of stable bundles of rank 2 and with fixed determinant L of degree d on the curve C. When g=3 and d is even, we prove that, for any point [W]∈M, there is a minimal rational curve passing through [W], which is not a Hecke curve. This complements a theorem of Xiaotao Sun.

G2–instantons over twisted connected sums

We introduce a method to construct $G_2$-instantons over compact $G_2$-manifolds arising as the twisted connected sum of a matching pair of building blocks [Kov03,KL11,CHNP12]. Our construction is based on gluing $G_2$-instantons obtained from holomorphic bundles over the building blocks via the first named author's work [SE11]. We require natural compatibility and transversality conditions which can be interpreted in terms of certain Lagrangian subspaces of a moduli space of stable bundles on a K3 surface.

Postnikov-stability versus semistability of sheaves

We present a novel notion of stable objects in a triangulated category. This Postnikov-stability is preserved by equivalences. We show that for the derived category of a projective variety this notion includes the case of semistable sheaves. As one application we compactify a moduli space of stable bundles using genuine complexes via Fourier-Mukai transforms. MSC 2000: 14F05, 14J60, 14D20

On the Hilbert scheme of the moduli space of vector bundles over an algebraic curve

Let M(n, ξ) be the moduli space of stable vector bundles of rank n ≥ 3 and fixed determinant ξ over a complex smooth projective algebraic curve X of genus g ≥ 4. We use the gonality of the curve and r-Hecke morphisms to describe a smooth open set of an irreducible component of the Hilbert scheme of M(n, ξ), and to compute its dimension. We prove similar results for the scheme of morphisms \({M or_P (\mathbb{G}, M(n, \xi))}\) and the moduli space of stable bundles over \({X \times \mathbb{G}}\), where \({\mathbb{G}}\) is the Grassmannian \({\mathbb{G}(n - r, \mathbb{C}^n)}\). Moreover, we give sufficient conditions for \({M or_{2ns}(\mathbb{P}^1, M(n, \xi))}\) to be non-empty, when s ≥ 1.

Seminatural bundles of rank two, degree one, and c2=10 on a quintic surface

In this paper we continue our study of the moduli space of stable bundles of rank two and degree 1 on a very general quintic surface. The goal in this paper is to understand the irreducible components of the moduli space in the first case in the "good" range, which is $c_2=10$. We show that there is a single irreducible component of bundles which have seminatural cohomology, and conjecture that this is the only component for all stable bundles.

STABLE ULRICH BUNDLES

The existence of stable ACM vector bundles of high rank on algebraic varieties is a challenging problem. In this paper, we study stable Ulrich bundles (that is, stable ACM bundles whose corresponding module has the maximum number of generators) on nonsingular cubic surfaces X ⊂ ℙ3. We give necessary and sufficient conditions on the first Chern class D for the existence of stable Ulrich bundles on X of rank r and c1= D. When such bundles exist, we prove that the corresponding moduli space of stable bundles is smooth and irreducible of dimension D2- 2r2+ 1 and consists entirely of stable Ulrich bundles (see Theorem 1.1). We are also able to prove the existence of stable Ulrich bundles of any rank on nonsingular cubic threefolds in ℙ4, and we show that the restriction map from bundles on the threefold to bundles on the surface is generically étale and dominant.

POSTNIKOV-STABILITY FOR COMPLEXES ON CURVES AND SURFACES

We present a novel notion of stable objects in the derived category of coherent sheaves on a smooth projective variety. As one application we compactify a moduli space of stable bundles using genuine complexes.

Moduli spaces of stable bundles on Calabi–Yau varieties and Donaldson–Thomas invariants

Let Y be a smooth Calabi–Yau hypersurface of P 1 × P where P stands for a P d -bundle over P 1 . We will prove that for many ample line bundles L and certain Chern characters c , the moduli space M ¯ L ( c ) (resp. M L ( c ) ) of L -Gieseker semistable (resp. L -stable ) rank two torsion free sheaves (resp. vector bundles) on Y with Chern character c are smooth and irreducible and we will compute its dimension. Moreover, we will prove that both moduli spaces coincide. As a byproduct of the geometrical description of these moduli spaces, we will compute the Donaldson–Thomas invariants of some Calabi–Yau 3-folds.