Stable Vector Bundles Research Articles

Stable Higgs bundles over positive principal elliptic fibrations

Abstract Let M be a compact complex manifold of dimension at least three and Π : M → X a positive principal elliptic fibration, where X is a compact Kähler orbifold. Fix a preferred Hermitian metric on M. In [14], the third author proved that every stable vector bundle on M is of the form L⊕ Π ⃰ B0, where B0 is a stable vector bundle on X, and L is a holomorphic line bundle on M. Here we prove that every stable Higgs bundle on M is of the form (L ⊕ Π ⃰B0, Π ⃰ ɸX), where (B0, ɸX) is a stable Higgs bundle on X and L is a holomorphic line bundle on M.

Moduli space of parabolic vector bundles over hyperelliptic curves

Let X be a smooth projective hyperelliptic curve of arbitrary genus g. In this article, we will classify the rank 2 stable vector bundles with parabolic structure along a reduced divisor of degree 4.

On the stability of flat complex vector bundles over parallelizable manifolds

We investigate the flat holomorphic vector bundles over compact complex parallelizable manifolds G/Γ, where G is a complex connected Lie group and Γ is a cocompact lattice in it. The main result proved here is a structure theorem for flat holomorphic vector bundles Eρ associated with any irreducible representation ρ:Γ⟶GL(r,C). More precisely, we prove that Eρ is holomorphically isomorphic to a vector bundle of the form E⊕n, where E is a stable vector bundle. All the rational Chern classes of E vanish, in particular, its degree is zero.We deduce a stability result for flat holomorphic vector bundles Eρ of rank 2 over G/Γ. If an irreducible representation ρ:Γ⟶GL(2,C) satisfies the condition that the induced homomorphism Γ⟶PGL(2,C) does not extend to a homomorphism from G, then Eρ is proved to be stable.

Brill–Noether loci of rank 2 vector bundles on a general $\nu $-gonal curve

In this paper we study the Brill Noether locus of rank 2, (semi)stable vector bundles with at least two sections and of suitable degrees on a general ν-gonal curve. We classify its irreducible components having at least expected dimension. We moreover describe the general member F of such components just in terms of extensions of line bundles with suitable minimality properties, providing information on the birational geometry of such components as well as on the very-ampleness of F.

Conformal invariance of (0, 2) sigma models on Calabi-Yau manifolds

Long ago, Nemeschansky and Sen demonstrated that the Ricci-flat metric on a Calabi-Yau manifold could be corrected, order by order in perturbation theory, to produce a conformally invariant (2, 2) nonlinear sigma model. Here we extend this result to (0, 2) sigma models for stable holomorphic vector bundles over Calabi-Yaus.

Very stable bundles and properness of the Hitchin map

Let $X$ be a smooth complex projective curve of genus $g\geq 2$ and let $K$ be its canonical bundle. In this note we show that a stable vector bundle $E$ on $X$ is very stable, i.e. $E$ has no non-zero nilpotent Higgs field, if and only if the restriction of the Hitchin map to the vector space of Higgs fields $H^0(X, \mathrm{End}(E) \otimes K)$ is a proper map.

Brauer group of the moduli spaces of stable vector bundles of fixed determinant over a smooth curve

Let X be an irreducible smooth projective curve, defined over an algebraically closed field k, of genus at least three and L a line bundle on X. Let MX(r,L) be the moduli space of stable vector bundles on X of rank r and determinant L with r≥2. We prove that the Brauer group Br(MX(r,L)) is cyclic of order g.c.d.(r,degree(L)). We also prove that Br(MX(r,L)) is generated by the class of the projective bundle obtained by restricting the universal projective bundle. These results were proved earlier in [1] under the assumption that k=C.

Embedding of a compactified Jacobian and theta divisors

Using a general stable vector bundle, we give an embedding $$\alpha _Y$$ of the compactified Jacobian $$\bar{J}(Y)$$ of an integral nodal curve Y into the moduli space $$U_Y(r,d)$$ of semistable torsion free sheaves of rank r and degree d on Y. We also give an embedding of the normalisation $$\tilde{J}(Y)$$ of $$\bar{J}(Y)$$ in the normalisation P(r, d) of $$U_Y(r,d)$$ . We determine a relation between the restriction of the theta line bundle on P(r, d) to $$\tilde{J}(Y)$$ and the theta line bundle on $$\tilde{J}(Y)$$ . We show that the restriction of the Picard bundle $$E_{r,d}$$ on $$U_Y(r,d)$$ to $$\bar{J}(Y)$$ is stable with respect to any theta divisor $$\theta _{\bar{J}(Y)}$$ on $$\bar{J}(Y)$$ if $$d > r(2g-1)$$ and it is semistable if $$d= r(2g-1)$$ . Finally we prove some results on natural cohomology.

Irreducibility of the moduli space of stable vector bundles of rank two and odd degree on a very general quintic surface

The moduli space $M(c_2)$, of stable rank two vector bundles of degree one on a very general quintic surface $X\subset {\mathbb P}^3$, is irreducible for all $c_2\geq 4$ and empty otherwise.

Rank-two vector bundles on non-minimal ruled surfaces

We continue previous works by various authors and study the birational geometry of moduli spaces of stable rank-two vector bundles on surfaces with Kodaira dimension $-\infty$. To this end, we express vector bundles as natural extensions, by using two numerical invariants associated to vector bundles, similar to the invariants defined by Brinzanescu and Stoia in the case of minimal surfaces. We compute explicitly these natural extensions on blowups of general points on a minimal surface. In the case of rational surfaces, we prove that any irreducible component of a moduli space is either rational or stably rational.

Matrix divisors on Riemann surfaces and Lax operator algebras

Matrix divisors are introduced in the work by A.Weil (1938) which is considered as a starting point of the theory of holomorphic vector bundles on Riemann surfaces. In this theory matrix divisors play the role similar to the role of usual divisors in the theory of line bundles. Moreover, they provide explicit coordinates (Tyurin parameters) in an open subset of the moduli space of stable vector bundles. These coordinates turned out to be helpful in integration of soliton equations. We would like to gain attention to one more relationship between matrix divisors of vector G-bundles (where G is a complex semi-simple Lie group) and the theory of integrable systems, namely to the relationship with Lax operator algebras. The result we obtain can be briefly formulated as follows: the moduli space of matrix divisors with certain discrete invariants and fixed support is a homogeneous space. Its tangent space at the unit is naturally isomorphic to the quotient space of M-operators by L-operators, both spaces essentially defined by the same invariants (the result goes back to Krichever, 2001). We give one more description of the same space in terms of root systems.

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.

Stable maps of genus zero in the space of stable vector bundles on a curve

Let [Formula: see text] be a smooth projective curve with genus [Formula: see text]. Let [Formula: see text] be the moduli space of stable rank two vector bundles on [Formula: see text] with a fixed determinant [Formula: see text] for [Formula: see text]. In this paper, as a generalization of Kiem and Castravet’s works, we study the stable maps in [Formula: see text] with genus [Formula: see text] and degree [Formula: see text]. Let [Formula: see text] be a natural closed subvariety of [Formula: see text] which parametrizes stable vector bundles with a fixed subbundle [Formula: see text] for a line bundle [Formula: see text] on [Formula: see text]. We describe the stable map space [Formula: see text]. It turns out that the space [Formula: see text] consists of two irreducible components. One of them parameterizes smooth rational cubic curves and the other parameterizes the union of line and smooth conics.

Tangent cones to generalised theta divisors and generic injectivity of the theta map

Let $C$ be a Petri general curve of genus $g$ and $E$ a general stable vector bundle of rank $r$ and slope $g-1$ over $C$ with $h^{0}(C,E)=r+1$. For $g\geqslant (2r+2)(2r+1)$, we show how the bundle $E$ can be recovered from the tangent cone to the generalised theta divisor $\unicode[STIX]{x1D6E9}_{E}$ at ${\mathcal{O}}_{C}$. We use this to give a constructive proof and a sharpening of Brivio and Verra’s theorem that the theta map $\mathit{SU}_{C}(r){\dashrightarrow}|r\unicode[STIX]{x1D6E9}|$ is generically injective for large values of $g$.

A Metric for Heterotic Moduli

Heterotic vacua of string theory are realised, at large radius, by a compact threefold with vanishing first Chern class together with a choice of stable holomorphic vector bundle. These form a wide class of potentially realistic four-dimensional vacua of string theory. Despite all their phenomenological promise, there is little understanding of the metric on the moduli space of these. What is sought is the analogue of special geometry for these vacua. The metric on the moduli space is important in phenomenology as it normalises D-terms and Yukawa couplings. It is also of interest in mathematics, since it generalises the metric, first found by Kobayashi, on the space of gauge field connections, to a more general context. Here we construct this metric, correct to first order in $${\alpha^{\backprime}}$$ , in two ways: first by postulating a metric that is invariant under background gauge transformations of the gauge field, and also by dimensionally reducing heterotic supergravity. These methods agree and the resulting metric is Kähler, as is required by supersymmetry. Checking the metric is Kähler is intricate and the anomaly cancellation equation for the H field plays an essential role. The Kähler potential nevertheless takes a remarkably simple form: it is the Kähler potential of special geometry with the Kähler form replaced by the $${\alpha^{\backprime}}$$ -corrected hermitian form.

On exact triangles consisting of stable vector bundles on tori

In this paper, we consider the exact triangles consisting of stable vector bundles on one-dimensional complex tori, and give a geometric interpretation of them in terms of the corresponding Fukaya category via the homological mirror symmetry.

Theta divisors and the geometry of tautological model

Let E be a stable vector bundle of rank r and slope \(2g-1\) on a smooth irreducible complex projective curve C of genus \(g \ge 3\). In this paper we show a relation between theta divisor \(\Theta _E\) and the geometry of the tautological model \(P_E\) of E. In particular, we prove that for \(r > g-1\), if C is a Petri curve and E is general in its moduli space then \(\Theta _E\) defines an irreducible component of the variety parametrizing \((g-2)\)-linear spaces which are g-secant to the tautological model \(P_E\). Conversely, for a stable, \((g-2)\)-very ample vector bundle E, the existence of an irreducible non special component of dimension \(g-1\) of the above variety implies that E admits theta divisor.

Derived categories of moduli spaces of vector bundles on curves

Let X be a smooth projective curve of genus g≥4 over C and M the moduli space of stable vector bundles of rank 2 and determinant isomorphic to a fixed line bundle of degree 1. We prove in this paper that Db(X) is embedded (via ΦE) in Db(M) as a full subcategory (if g≥4).

Finite group actions on moduli spaces of vector bundles

We study actions of finite groups on moduli spaces of stable holomorphic vector bundles and relate the fixed-point sets of those actions to representation varieties of orbifold fundamental groups.

Stability of Picard sheaves for vector bundles on curves

Abstract For a stable vector bundle E of slope μ ( E ) > 2 g − 1 on a smooth, projective curve of genus g , we show that the Picard sheaf E ˆ on the Picard variety of the curve is stable. We introduce a homological tool for testing semistability of Picard sheaves. We also obtain the semistability of the general Picard sheaf if μ ( E ) ∈ [ g − 2 , g ] , μ ( E ) ≠ g − 1 .