General Vector Bundle Research Articles

Generic Beauville’s Conjecture

Abstract Let $\alpha \colon X \to Y$ be a finite cover of smooth curves. Beauville conjectured that the pushforward of a general vector bundle under $\alpha $ is semistable if the genus of Y is at least $1$ and stable if the genus of Y is at least $2$ . We prove this conjecture if the map $\alpha $ is general in any component of the Hurwitz space of covers of an arbitrary smooth curve Y.

Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited

In this manuscript, we present a coherent rigorous overview of the main properties of Sobolev-Slobodeckij spaces of sections of vector bundles on compact manifolds; results of this type are scattered through the literature and can be difficult to find. A special emphasis has been put on spaces with noninteger smoothness order, and a special attention has been paid to the peculiar fact that for a general nonsmooth domain Ω in Rn, 0<t<1, and 1<p<∞, it is not necessarily true that W1,p(Ω)↪Wt,p(Ω). This has dire consequences in the multiplication properties of Sobolev-Slobodeckij spaces and subsequently in the study of Sobolev spaces on manifolds. We focus on establishing certain fundamental properties of Sobolev-Slobodeckij spaces that are particularly useful in better understanding the behavior of elliptic differential operators on compact manifolds. In particular, by introducing notions such as “geometrically Lipschitz atlases” we build a general framework for developing multiplication theorems, embedding results, etc. for Sobolev-Slobodeckij spaces on compact manifolds. To the authors’ knowledge, some of the proofs, especially those that are pertinent to the properties of Sobolev-Slobodeckij spaces of sections of general vector bundles, cannot be found in the literature in the generality appearing here.

Sobolev spaces and $$\nabla $$-differential operators on manifolds I: basic properties and weighted spaces

We study covariant Sobolev spaces and $$\nabla $$ -differential operators with coefficients in general Hermitian vector bundles on Riemannian manifolds, stressing a coordinate-free approach that uses connections (which are typically denoted $$\nabla $$ ). These concepts arise naturally from geometric partial differential equations, including some that are formulated on plain Euclidean domains, for instance, from problems formulated on the boundary of smooth domains or in relation to the weighted Sobolev spaces used to study PDEs on polyhedral domains. We prove several basic properties of the covariant Sobolev spaces and of the $$\nabla $$ -differential operators on general manifolds. For instance, we prove mapping properties for our differential operators and the independence of the covariant Sobolev spaces on the choices of the connection $$\nabla $$ , as long as the new connection is obtained using a totally bounded perturbation. We also introduce the Fréchet finiteness condition (FFC) for totally bounded vector fields, which is satisfied, for instance, by open subsets of manifolds with bounded geometry. When (FFC) is satisfied, we provide several equivalent definitions of our covariant Sobolev spaces and of our $$\nabla $$ -differential operators. We also introduce and study the notion of a $$\nabla $$ -bidifferential operator (a bilinear version of differential operators), obtaining results similar to those obtained for $$\nabla $$ -differential operators. Bilinear differential operators are necessary for a global, geometric discussion of variational problems. We tried to write the paper so that it is accessible to a large audience.

Stability Conditions for Restrictions of Vector Bundles on Projective Surfaces

Using Bridgeland stability conditions, we give sufficient criteria for a stable vector bundle on a smooth complex projective surface to remain stable when restricted to a curve. We give a stronger criterion when the vector bundle is a general vector bundle on the plane. As an application, we compute the cohomology of such bundles for curves that lie in the plane or on Hirzebruch surfaces.

Extending the geometry of heterotic spectral cover constructions

In this work we extend the well-known spectral cover construction first developed by Friedman, Morgan, and Witten to describe more general vector bundles on elliptically fibered Calabi-Yau geometries. In particular, we consider the case in which the Calabi-Yau fibration is not in Weierstrass form, but can rather contain fibral divisors or multiple sections (i.e. a higher rank Mordell-Weil group). In these cases, general vector bundles defined over such Calabi-Yau manifolds cannot be described by ordinary spectral data. To accomplish this we employ well established tools from the mathematics literature of Fourier-Mukai functors. We also generalize existing tools for explicitly computing Fourier-Mukai transforms of stable bundles on elliptic Calabi-Yau manifolds. As an example of these new tools we produce novel examples of chirality changing small instanton transitions. The goal of this work is to provide a geometric formalism that can substantially increase the understood regimes of heterotic/F-theory duality.

A weighted topological quantum field theory for Quot schemes on curves

We study Quot schemes of vector bundles on algebraic curves. Marian and Oprea gave a description of a topological quantum field theory (TQFT) studied by Witten in terms of intersection numbers on Quot schemes of trivial bundles. Since these Quot schemes can have the wrong dimension, virtual classes are required. But Quot schemes of general vector bundles always have the right dimension. Using the degree of the general vector bundle as an additional parameter, we construct a weighted TQFT containing both Witten's TQFT and the small quantum cohomology TQFT of the Grassmannian. This weighted TQFT is completely geometric (no virtual classes are needed), can be explicitly computed, and recovers known formulas enumerating the points of finite Quot schemes.

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.

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 $g$.

Petri map for vector bundles near good bundles

Let C be a projective nonsingular curve of genus g and let Br,dk be the locus of stable rank r degree d vector bundles with at least k linearly independent sections on C. It is known that when C is general and under certain conditions on g, d, r, and k, there exists a component B of Br,dk of the expected dimension. In this paper we show that for a general vector bundle E of B, the Petri map is injective.

Nonlinear generalized sections and vector bundle homomorphisms in Colombeau spaces of generalized functions

AbstractWe define and characterize spaces of manifold‐valued generalized functions and generalized vector bundle homomorphisms in the setting of the full diffeomorphism‐invariant vector‐valued Colombeau algebra. Furthermore, we establish point value characterizations for these spaces.

A note on theta divisors of stable bundles

Let C be a smooth complex irreducible projective curve of genus g \geq 3 . We show that if C is a Petri curve with g \geq 4 , a general stable vector bundle E on C , with integer slope, admits an irreducible and reduced theta divisor \Theta_E , whose singular locus has dimension g-4 . If C is non-hyperelliptic of genus 3 , then actually \Theta_E is smooth and irreducible for a general stable vector bundle E with integer slope on C .

Twisted K-Theory and Finite-Dimensional Approximation

We provide a finite-dimensional model of the twisted K-group twisted by any degree three integral cohomology class of a CW complex. One key to the model is Furuta’s generalized vector bundle, and the other is a finite-dimensional approximation of Fredholm operators.

Sheaves of nonlinear generalized functions and manifold-valued distributions

This paper is part of an ongoing program to develop a theory of generalized differential geometry. We consider the space G[X, Y ] of Colombeau generalized functions defined on a manifold X and taking values in a manifold Y . This space is essential in order to study concepts such as flows of generalized vector fields or geodesics of generalized metrics. We introduce an embedding of the space of continuous mappings C(X, Y ) into G[X, Y ] and study the sheaf properties of G[X, Y ]. Similar results are obtained for spaces of generalized vector bundle homomorphisms. Based on these constructions we propose the definition of a space D 0 [X, Y ] of distributions on X taking values in Y . D 0 [X, Y ] is realized as a quotient of a certain subspace of G[X, Y ].

Vector bundles on an infinitesimal neighborhood of a negative and split plane

Let W be an n-dimensional complex manifold or a smooth alge- braic variety and Z ⊂ W. Assume Z ∼ P 2 , n ≥ 3 and that the normal bundle NZ,W of Z in W is a direct sum of line bundles with negative degree. Let Z (∞) denote the formal completion of W along Z. Here we compute the dimension of the deformation space of several general vector bundles A on Z (∞) in terms of the splitting type of NZ,W and the integers c1(A|Z) and c2(A|Z).

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.

Intrinsic Characterization of Manifold-Valued Generalized Functions

The concept of generalized functions taking values in a differentiable manifold is extended to a functorial theory. We establish several characterization results which allow a global intrinsic formulation both of the theory of manifold-valued generalized functions and of generalized vector bundle homomorphisms. As a consequence, a characterization of equivalence that does not resort to derivatives (analogous to scalar-valued cases of Colombeau's construction) is achieved. These results are employed to derive a point value description of all types of generalized functions valued in manifolds and to show that composition can be carried out unrestrictedly. Finally, a new concept of association adapted to the present setting is introduced. 2000 Mathematics Subject Classification 46T30 (primary), 46F30, 53B20 (secondary).

Generalized Functions Valued in a Smooth Manifold

Based on Colombeau’s theory of algebras of generalized functions we introduce the concepts of generalized functions taking values in differentiable manifolds as well as of generalized vector bundle homomorphisms. We study their basic properties, in particular with respect to some new point value concepts for generalized functions and indicate applications of the resulting theory in general relativity.

Curves in Grassmannians and Plücker Embeddings: The Injectivity Range

Fix integers g, r, d, v with g ≥ 2, r ≥ 2 and r < v ≤ d + r – rg. Let X be a smooth curve of genus g and E a general stable vector bundle on X with rank (E) = r and deg (E) = d. Assume E spanned and take a general linear subspace V of H0 (X, E) with dim (V) = v. Let hV : X → G(r, V) be the induced map into a Grassmannian. Here we give conditions on d and v which assure that the natural map (X, det (E)) is injective. Use the Plücker embedding to see G(r, V) as a subvariety of P. According to M. Teixidor i Bigas this means that hV(X) ⊂ P spans P

On the Symmetric Algebra of Stable Vector Bundles on Curves

Let X be a smooth projective curve of genus g ≥ 2 and E a general stable vector bundle on X with rank(E) = r and deg(E) = d > 0. Let z be the smallest integer with zd > 2gr. Here we study the graded algebra H0 (X, Sym(E)) := ⌖n≥0H0 (X, Sn (E)) and prove that it is generated by elements of degree ≤ z.

A super-twisted Dirac operator and Novikov inequalities

A super-twisted Dirac operator is constructed and deformed suitably. Following Shubin’s approach to Novikov inequalities associated to the deformed de Rham-Hodge operator, we give a for mula for the index of the super-twisted Dirac operator, and Novikov type inequalities for the deformed operator. In particular, we obtain a purely analytic proof of the Hopf index theorem for general vector bundles.