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In this paper, we prove that for the oper stratification of the de Rham moduli space $M_{\mathrm{dR}}(X,r)$, the closed oper stratum is the unique minimal stratum with dimension $r^2(g-1)+g+1$, and the open dense stratum consisting of irreducible flat bundles with stable underlying vector bundles is the unique maximal stratum., Comment: This paper comes from the last section of arXiv:1905.10765v1 as an independent paper. Comments are welcome! To appear in manuscripta mathematica

Let G be $$A_5$$ , $$A_4$$ , or a finite group with cyclic Sylow 2 subgroup. We show that every closed smooth G manifold M has a strongly algebraic model. This means, there exist a nonsingular real algebraic G variety X which is equivariantly diffeomorphic to M and all G vector bundles over X are strongly algebraic. Making use of improved blow-up techniques and the literature on equivariant bordism theory, we are extending older algebraic realization results.

Let X be a smooth irreducible projective curve with an involution $$\sigma $$ . A vector bundle E over X is called anti-invariant if there exists an isomorphism $$\sigma ^*E\rightarrow E^*$$ . In this paper, we give a construction of the moduli spaces of anti-invariant vector bundles over X.

The main goal of this short paper is to prove that any non-arithmetically Cohen–Macaulay polarized scheme $$(X,\mathscr {O}_X(1))$$ of dimension $${{\mathrm{dim}}}(X)\ge 2$$ , under mild conditions on $$\mathscr {O}_X(1)$$ , supports arbitrarily large families of non-isomorphic indecomposable aCM vector bundles with respect to $$\mathscr {O}_X(l)$$ , $$l\ge 3$$ . Namely, they are of wild representation type.

We introduce Dolbeault cohomology valued characteristic classes of Higgs bundles over complex manifolds. Flat vector bundles have characteristic classes lying in odd degree de Rham cohomology and a theorem of Reznikov says that these must vanish in degrees three and higher over compact Kahler manifolds. We provide a simple and independent proof of Reznikov’s result and show that our characteristic classes of Higgs bundles and the characteristic classes of flat vector bundles are compatible via the nonabelian Hodge theorem.

For any simple Lie algebra, a positive integer, and n-tuple of compatible weights, the conformal blocks bundle is a globally generated vector bundle on the moduli space of pointed rational curves. We classify all vector bundles of conformal blocks for \({\mathfrak{sl}_n}\), with Sn-invariant weights, which have rank one. We show that the cone generated by their base point free first Chern classes is polyhedral, generated by level one divisors.

Let \({c : C \rightarrow X \times X}\) be a correspondence with C and X quasi-projective schemes over an algebraically closed field k. We show that if \({u_\ell : c_1^*\mathbb{Q}_\ell \rightarrow c_2^!\mathbb{Q}_\ell}\) is an action defined by the localized Chern classes of a c2-perfect complex of vector bundles on C, where l is a prime invertible in k, then the local terms of ul are given by the class of an algebraic cycle independent of l. We also prove some related results for quasi-finite correspondences. The proofs are based on the work of Cisinski and Deglise on triangulated categories of motives.

Let C be a generic curve, E a generic vector bundle on C. Then, for every line bundle on C the twisted Petri map $$P_{E}:H^0(C,L\otimes E)\otimes H^0(C, K\otimes L^*\otimes E^{*})\rightarrow H^0(C, K)$$ is injective.

We consider an analogue of the notion of instanton bundle on the projective 3-space, consisting of a class of rank-2 vector bundles defined on smooth Fano threefolds X of Picard number one, having even or odd determinant according to the parity of K X . We first construct a well-behaved irreducible component of their moduli spaces. Then, when the intermediate Jacobian of X is trivial, we look at the associated monads, hyperdeterminants and nets of quadrics. We also study one case where the intermediate Jacobian of X is non-trivial, namely when X is the intersection of two quadrics in $${\mathbb{P}^5}$$ , relating instanton bundles on X to vector bundles of higher rank on a the curve of genus 2 naturally associated with X.

In this article we give a vanishing result for Dolbeault cohomology groups \({H^{p,q}(X, S^{\nu}E\otimes L)}\), where ν is a positive integer, E is a vector bundle generated by sections and L is an ample line bundle on a smooth projective variety X. We also give a condition for Hp,q(X, SνE) to vanish when E is s-ample and generated by sections. We also give an application related to a result of Barth-Lefschetz type. A general nonvanishing result under the same hypothesis is given to prove the optimality of the vanishing result for some parameter values.

We describe the Chern classes of holomorphic vector bundles on non-algebraic complex torus of dimension 2.

We study growth of holomorphic vector bundles E over smooth affine manifolds. We define Finsler metrics of finite order on E by estimates on the holomorphic bisectional curvature. These estimates are very similar to the ones used by Griffiths and Cornalba to define Hermitian metrics of finite order. We then generalize the Vanishing Theorem of Griffiths and Cornalba to the Finsler context. We develop a value distribution theory for holomorphic maps from the projectivization of E to projective space. We show that the projectivization of E can be immersed into a projective space of sufficiently large dimension via a map of finite order., Comment: version 2 has some typos corrected; to appear in Manuscripta Mathematica

Let Open image in new window = [X/G] be the quotient stack of a scheme X by an affine group scheme G over a field k. Assume that there is a line bundle on Open image in new window whose underlying line bundle on X is very ample. Let VB(Open image in new window) be the category of vector bundles on Open image in new window.We show that Open image in new window is canonically isomorphic to the stack of fiber functors on VB(Open image in new window). This is an analogue of the Tannaka duality for affine groups.

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, Poincare bundle, moduli space

In the first part, we give some necessary conditions for a torsion free sheaf on a smooth threefold to be a "reduced" limit of vector bundles. In a second part of the article, we illustrate these results by reinterpreting a condition described by G. Ellingsrud and S.A. Stromme as a condition obtained in the first part. This enables us to identify, in a well known too big familly, the torsion free sheaves that are limit of vector bundles.

In this work, using elementary transformations and prioritary sheaves, we establish birational maps between certain moduli spaces of stable vector bundles over ℙ2 with the same rank and different Chern classes. As an application we give a simple proof of the rationality of the moduli spaces M(r; c 1, c 2) of rank r stable vector bundles over ℙ2 with given Chern classes for a huge families of the triples (r; c 1, c 2).

Let ℰ be a vector bundle on Pn. There is a strong relationship between ℰ and its intermediate cohomology modules. In the case where ℰ has low rank, we exploit this relationship to provide various splitting criteria for ℰ. In particular, we give a splitting criterion for ℰ in terms of the vanishing of certain intermediate cohomology modules. We also show that the Horrocks-Mumford bundle is the only non-split rank two bundle on P4 with a Buchsbaum second cohomology module.

By a result of Mukai, the non-abelian Brill-Noether locus X=MC(2, K : 3F) of type II, defined by a stable rank 2 vector bundle F of invariant 3 over a plane quartic curve C, is a prime Fano 3-fold X=X16 of degree 16. The associate ruled surface SX=P(F) is uniquely defined by X, and we see that for the general X=X16, SX is isomorphic to the Fano surface of conics on X. The argument uses the geometry of the Sp3-grassmannian and the double projection from a line on X16.

Let $X$ be a complex projective bundle. We prove that $X$ admits an endomorphism of degree $>1$ and commuting with the projection to the base, if and only if $X$ trivializes after a finite covering. When $X$ is the projectivization of a vector bundle $E$ of rank 2, we prove that it has an endomorphism of degree $>1$ on a general fiber only if $E$ splits after a finite base change., LaTeX 2.09, 13 pages

Let C be a curve of genus g and L a line bundle of degree 2g on C. Let ML be the kernel of the evaluation map \(\). We show that when L is general enough, the rank g bundle ML and its exterior powers are stable, but admit a reducible theta divisor.

The 3-dimensional Hopf vector field is shown to be a stable harmonic section of the unit tangent bundle. In contrast, higher dimensional Hopf vector fields are unstable harmonic sections; indeed, there is a natural variation through smooth unit vector fields which is locally energy-decreasing, and whose asymptotic limit is a singular vector field of finite energy. This energy is explicitly calculated, and conjectured to be the infimum of the energy functional over all smooth unit vector fields.

We show that over some smooth projective varieties every semistable Higgs logarithmic vector bundle is semistable in the ordinary sense, hence satisfies Bogomolov inequality. More generaly, we prove that semistable Higgs parabolic vector bundles of rank two over smooth projective varieties of dimension ≥ 2 satisfy the “parabolic” 'Bogomolov inequality

Let be a ruled Fano 3-fold. The goal of this paper is to compute the dimension, prove the irreducibility and smoothness and describe the structure of the moduli space M L (2;c 1,c 2) of L-stable, rank 2 vector bundles E on X with certain Chern classes and for a suitable polarization L closely related to c 2. More precisely, we will cover the study of some moduli spaces M L (2;c 1,c 2) such that the generic point is given as a non-trivial extension of line bundles. This work nicely reflects the general philosophy that moduli spaces inherits a lot of geometrical properties of the underlying variety.

Let K be a number field and \(\) its ring of integers. Let \(\) be a Hermitian vector bundle over \(\). In the first part of this paper we estimate the number of points of bounded height in \(\) (generalizing a result by Schanuel). We give then some applications: we estimate the number of hyperplanes and hypersurfaces of degree d>1 in \(\) of bounded height and containing a fixed linear subvariety and we estimate the number of points of height, with respect to the anticanonical line bundle, less then T (when T goes to infinity) of ℙNK blown up at a linear subspace of codimension two.

The purpose of this paper is to characterise the invariant sections-distributions by a proper action. More precisely, we show that if G is a connected Lie group acting on a differentiable vector bundle E→V such that the induced action on V is proper, then the topological vector space of the G-invariant linear functionals (on the space of C ∞ sections with compact support) equipped with the induced weak-topology (resp. the strong-topology), is isomorphic to the weak (resp. strong) topological dual of the space (of all G-invariant sections σ with compact quotient supp(σ)/G) equipped with a suitable topology; this coincides with the usual C ∞-topology if the orbit space is compact, and with the Schwartz-topology if the group G is compact.

Necessary and sufficient conditions for the existence of a complex structure in an 8-dimensional spin vector bundle over a closed connected spin manifold of dimension 8 are given in terms of characteristic classes. The result completes the papers by Heaps [H] and Thomas [T] on the same topic.

One of our two main results exhibits for a vector bundle over a compact Hausdorff spaceX an interplay between its span, its possible splittings, and the Lyusternik-Shnirel’man category ofX. The other main result, also on vector bundles and the Lyusternik-Shnirel’man category, enables us to derive certain inequalities connecting the immersion codimension, the stable span, and the Lyusternik-Shnirel’man category of a smooth closed manifold which is not stably parallelizable. Our results are applicable in various situations of general interest.

Let X be a smooth manifold with boundary of dimension n > 1. The operators of order −n and type zero in Boutet de Monvel's calculus form a subset of Dixmier's trace ideal \(\) for the Hilbert space \(\) of L2 sections in vector bundles E over X, F over ∂X.

We construct an explicit Cech cocycle representing the k-th Stiefel-Whitney class of a vector bundle. This construction involves only the transition functions of the bundle. We also give local formulae for the secondary Stiefel-Whitney classes. These may be useful in determining whether the Stiefel-Whitney numbers of a flat bundle are zero.

In this note we use the monodromy argument to prove a Noether-Lefschetz theorem for vector bundles.

If π:X→B is a non-Kahlerian elliptic surface with generic fibreF, the moduli space of stable holomorphic vector bundles with torsion Chern classes onX has an induced fibred structure with base Pico(F) and the moduli space of stable parabolic bundles onB orb as fibre. This is specific to the non-Kahler case.

We derive in a simple way some higher estimates of the number of every-where linearly independent cross-sections for even multiples of real vector bundles, and, more generally, for a certain class of homomorphism bundles, supposing that there exists at least one nowhere vanishing cross-section.

In this paper, we study the compact Kahler manifolds whose tangent bundles are numerically effective and whose anti-Kodaira dimensions are equal to one. LetX be a compact Kahler manifold with nef tangent bundle and semiample anti-canonical bundle. We prove that κ(−K X )=1 if and only if there exists a finite etale coverY→X such thatY≅ℙ1×A, whereA is a complex torus. As a consequence, we are able to improve upon a result of T. Fujiwara [3, 4].

In this paper we prove that an Enriques surfaceX has a smooth rational curve if and only if there exists an exceptional bundleE t of rank 2 withc 2 (E t )=t for any integer t onX. We describe all exceptional bundles of rank 2 on Enriques surfaces and show that they are all stable with respect to any ample divisor.

In this article I describe construction methods for smooth subvarieties of codimension 3 in projective spaces or other ambient spaces. The methods include liaison of 3-folds in ℙ6, sections in smooth reflexive sheaves, and Pfaffians of twisted skew-symmetric vector bundle morphisms. I use these methods to construct new families of 3-folds in ℙ6, and new codimension 3 submanifolds in ℙ8 and ℙ9.

In this paper we investigate projective 4-dimensional manifolds X whose tangent bundles TX are numerically effective and give an almost complete classification. An important technical tool is the “Mori theory” of projective manifolds X whose canonical bundles KX are not numerically effective.

From an irreducible complete immersed curveX in a projective space ℙ other than a line, one obtains a curveX ′ in a Graasmann manifoldG of lines in ℙ that is the image ofX under the Gauss map, which is defined by the embedded tangents ofX. The main result of this article clarifies in case of positive characteristic what curvesX have the sameX′: It is shown thatX is uniquely determined byX′ ifX, or equivalentlyX′, has geometric genus at least two, and that for curvesX 1 andX 2 withX 1 ≠X 2 in ℙ, ifX′1 =X′2 inG and eitherX 1 orX 2 is reflexive, then bothX 1 andX 2 are rational or supersingular elliptic; moreover, examples of smoothX 1 andX 2 in that case are given.

Denote by Wn,dr the set of stable vector bundles of rank n and degree d with at least r+1 sections. We compute the singular locus of Wn,d1. We show that W2,d2 is irreducible and reduced and we characterize the singular locus of Wn,d2.

O. INTRODUCTION Let K be a compact subset of p,+k. Suppose that we want to recognize K just by using information that comes from some of its n-sections. When can we conclude that K has some property just from the fact that a sufficiently big collection of its n-sections have the same property and how to measure this collection? More precisely, suppose that we have a collection of n-sections of K which are solid spheres [respectively, are convex, have constant width[. We would like to measure this collection and conclude that if it is big enough then K is a solid sphere [is convex, has constarut width]. The purpose of this paper is to prove that the homology and cohomology of a collection of n-sections of K which are solid spheres [respectively, are convex, have constant width], as a subset of the corresponding Grassmannian manifold, contains enough information which can be used as a delicate measure to decide if K is a solid sphere [is convex, has constaxlt width[. As an application, we will prove that if for every n-plane H which contains B. "-2 we can choose continuously an n-plane H' parallel to H such that H' f3 K is a solid sphere [respectively, is convex, has constant width], then K is a solid sphere [is convex, has constant width].

We show that h0(X, L)≥2k for every k-spanned line bundle on a smooth complete complex surface X.

In this paper, we give several existence theorems of rank two special stable vector bundles on smooth complex projective curves of genusg≥2.

In this paper, we study the classification theory of uniruled varieties by means of the adjoint system for vector bundles on the varieties. We prove that ifE is an ample vector bundle on a smooth projective varietyX with rank(E)=dimX-2, thenKX+C1(E) is numerically effective except in a few cases. In all of the exceptional cases,X is a uniruled variety. As consequences, we generalized a result of Fujita [Fu3] and Ionescu [Io] and improve upon a theorem of Wiśniewski [Wi1].

Here we prove in positive characteristic the spannedness or very ampleness or k-ampleness of the adjuntion bundle KS⊕L with S surface of Kodaira dimension ≤0 L∈Pic(S), L ample, under numerical conditions on L which are similar to the ones required in characteristic 0 by Reider method.

In the following paper homotopy representations (h.r.) for metacyclic groups of the form $$G = G_{p,q} = \left\langle {h,k|h^p = k^q = 1, khk^{ - 1} = h^d } \right\rangle $$ with dq≡1 modp, d≡1 mod p, H= , K= , odd primes p and q are studied. (Look at Gp.q as the easiest example of a non-abelian group).