Infinitely stably extendable vector bundles on projective spaces

Matrices of linear forms of constant rank from vector bundles on projective spaces

This paper establishes and investigates a relationship between the space of solutions of a system of constant coefficient partial differential equations and the cohomology ( H 1 {H^1} in particular) of an associated vector bundle/reflexive sheaf on complex projective space. Using results of Grothendieck and Shatz on vector bundles over projective one-space, the case of partial differential equations in two variables is completely analyzed. The final section applies results about vector bundles on higher-dimensional projective spaces to the case of three or more variables.

The study of vector bundles over algebraic varieties has been stimulated over the last few years by successive waves of migrant concepts, largely from mathematical physics, whilst retaining its roots in old questions concerning subvarieties of projective space. The 1993 Durham Symposium on Vector Bundles in Algebraic Geometry brought together some of the leading researchers in the field to explore further these interactions. This book is a collection of survey articles by the main speakers at the symposium and presents to the mathematical world an overview of the key areas of research involving vector bundles. Topics covered include those linking gauge theory and geometric invariant theory such as augmented bundles and coherent systems; Donaldson invariants of algebraic surfaces; Floer homology and quantum cohomology; conformal field theory and the moduli spaces of bundles on curves; the Horrocks–Mumford bundle and codimension 2 subvarieties in P4 and P5; exceptional bundles and stable sheaves on projective space.

A basis is presented for the RP" classes in y\n(BO).This basis is used to prove that every smooth involution (M, T) on a closed manifold bounds if its fixed point set is a disjoint union of odd-dimensional projective spaces of constant dimension.Given any real vector bundle f -* RP" , let (RP" , <*) 6 Wn(BO) denote the stable bordism class determined by the map RP" -> BO(c) <- BO classifying .With n fixed, the classes (RP" ,,) as t, ranges over all vector bundles span a Z2-vector subspace of yi"(BO), where the sum (RP", <*) + (RP" , n) represents the disjoint union.We will exhibit a basis for this subspace.Letting A - RP" denote the canonical line bundle, the KO-theory of real vector bundles tells us that any -* RP" is stably isomorphic to kX for some (possibly large) k , so that (RP", ) = (RP" , kX).The main result of this paper is that up to bordism one need only consider those k < n .Theorem.A basis for the RP" classes in y\n(BO) is {(RP",;A)|0 <;<} ifn = 2m, {(RP" , (2) + 1)A)|0 < j < m} if n = 2m + 1.Thus, given a bundle kX - RP" with k > n , the theorem tells us that (rp" , kX) = Yi"1 > M j with each j < n .As an application we include the following Proposition.Given a closed manifold with differentiate involution (M, T) with fixed point set any disjoint union F = |JRP2"+I (n fixed), then (M, ) bounds.Capobianco [1] has shown this is true when = Rp2m+1.Royster [4] proved it when = RP2"+1 u RP2m+1 .No such result is possible for = IJRP2" Indeed, Stong [5] characterized all involutions fixing RP2".

Decoding by rank-2 bundles over plane quartics

We study aspects of the A^1-homotopy classification problem in dimensions >= 3 and, to this end, we investigate the problem of computing A^1-homotopy groups of some A^1-connected smooth varieties of dimension >=. Using these computations, we construct pairs of A^1-connected smooth proper varieties all of whose A^1-homotopy groups are abstractly isomorphic, yet which are not A^1-weakly equivalent. The examples come from pairs of Zariski locally trivial projective space bundles over projective spaces and are of the smallest possible dimension. Projectivizations of vector bundles give rise to A^1-fiber sequences, and when the base of the fibration is an A^1-connected smooth variety, the associated long exact sequence of A^1-homotopy groups can be analyzed in detail. In the case of the projectivization of a rank 2 vector bundle, the structure of the A^1-fundamental group depends on the splitting behavior of the vector bundle via a certain obstruction class. For projective bundles of vector bundles of rank >=, the A^1-fundamental group is insensitive to the splitting behavior of the vector bundle, but the structure of higher A^1-homotopy groups is influenced by an appropriately defined higher obstruction class.

Let {Xt ↪→ Y | t ∈ M} be a family of compact complex submanifolds Xt of a complex manifold Y parametrised by a complex manifold M . In this paper we study vector bundles and linear connections on the moduli space M induced from holomorphic vector bundles on the space Y . The origin of this approach lies in the Ward transform [W] of holomorphic vector bundles E on Y which are trivial when restricted to each submanifoldXt of the family {Xt ↪→ Y | t ∈M}. We consider a more general class of holomorphic vector bundles E on twistor spaces Y which, when restricted to a submanifold Xt of the family, have the same integer h(Xt, E|Xt) for all t ∈M . With such a vector bundle E on Y there is an associated vector bundle E on M whose fibre at t ∈ M is isomorphic to H(Xt, E|Xt). It is shown that provided the cohomology groups H ( Xt, E|Xt ⊗N ∗ t ) and H ( Xt, E|Xt ⊗N ∗ t ) vanish (where Nt is the normal bundle of Xt ↪→ Y ), the vector bundle E induced on the parameter space M comes equipped with a distinguished linear connection ∇ satisfying some natural integrability conditions (cf. [L, E, B-E]). The curvature tensor of ∇ is represented, at each t ∈ M , by a cohomology class in H ( Xt, E|Xt ⊗ 2N∗ ) ⊗ ( H(Xt, E|Xt) )∗ . This theorem-construction gives a very simple way to estimate the curvature tensor of an induced connection directly from the original twistor data. As an application, we consider a pair X ↪→ Y consisting of a complex contact 3-fold (Y, L) and a Legendre submanifold X = CP such that the contact line bundle L restricts to X as O(3). Then the moduli space M of all holomorphic Legendre deformations of X inside Y is a 4-dimensional manifold which comes equipped with an induced 1-flat G3structure [Br2], and any such structure arises locally in this way. Suppose that there exists a line bundle E → Y such that L = E⊗3 (such an E always exists on a sufficiently small tubular neighbourhood of X in Y ). Since the normal bundle of X is isomorphic to JO(3) = C ⊗ O(2), the above cohomology conditions on E|Xt ⊗ N ∗ t are satisfied. Then the induced rank 2 vector bundle E comes equipped with a distinguished linear connection which, as easily follows from its integrability properties, is a “spinor” version of the torsion-free connection on M with holonomy in G3. In fact, any torsion-free connection with holonomy in G3 can be constructed, at least locally, in this way (cf. [Me1]). The point is that the way we prove the above mentioned theorem-construction gives a rather

Algebraic Vector Bundles

We study the (k, s)-positivity for holomorphic vector bundles on compact complex manifolds. (0, s)-positivity is exactly the Demailly s-positivity and a (k, 1)-positive line bundle is just a k-positive line bundle in the sense of Sommese. In this way we get a unified theory for all kinds of positivities used for semipositive vector bundles. Several new vanishing theorems for (k, s)-positive vector bundles are proved and the vanishing theorems for k-ample vector bundles on projective algebraic manifolds are generalized to k-positive vector bundles on compact Kähler manifolds.

Let X be a smooth projective curve defined over an algebraically closed field k, and let FX denote the absolute Frobenius morphism of X when the characteristic of k is positive. A vector bundle over X is called virtually glob- ally generated if its pull back, by some finite morphism to X from some smooth projective curve, is generated by its global sections. We prove the following. If the characteristic of k is positive, a vector bundle E over X is virtually globally generated if and only if (F m X ) ∗ E ∼ Ea ⊕ E f for some m, where Ea is some ample vector bundle and E f is some finite vector bundle over X (either of Ea and E f are allowed to be zero). If the characteristic of k is zero, a vector bundle E over X is virtually globally generated if and only if E is a direct sum of an ample vector bundle and a finite vector bundle over X (either of them are allowed to be zero).

We introduce the notion of hyperbolic equivalence for quadric bundles and quadratic forms on vector bundles and show that hyperbolic equivalent quadric bundles share many important properties: they have the same Brauer data; moreover, if they have the same dimension over the base, they are birational over the base and have equal classes in the Grothendieck ring of varieties. Furthermore, when the base is a projective space we show that two quadratic forms are hyperbolic equivalent if and only if their cokernel sheaves are isomorphic up to twist, their fibers over a fixed point of the base are Witt equivalent, and, in some cases, certain quadratic forms on intermediate cohomology groups of the underlying vector bundles are Witt equivalent. For this we show that any quadratic form over $\mathbb{P}^n$ is hyperbolic equivalent to a quadratic form whose underlying vector bundle has many cohomology vanishings; this class of bundles, called VLC bundles in the paper, is interesting by itself.

We construct equivariant vector bundles over quantum projective spaces making use of parabolic Verma modules over the quantum general linear group. Using an alternative realization of the quantized coordinate ring of projective space as a subalgebra in the algebra of functions on the quantum group, we reformulate quantum vector bundles in terms of quantum symmetric pairs. In this way, we prove complete reducibility of modules over the corresponding coideal stabilizer subalgebras, via the quantum Frobenius reciprocity.

The purpose of this paper is to study extendibility and stable extendibility of vector bundles over real projective spaces. We determine a necessary and sufficient condition that a vector bundle ζ over the real projective n-space RPn is extendible (or stably extendible) to RPm for every m>n in the case where ζ is the complexification of the tangent bundle of RPn and in the case where ζ is the normal bundle associated to an immersion of RPn in the Euclidean (n+k)-space Rn+k or its complexification, and give examples of the normal bundle which is extendible to RPN but is not stably extendible to RPN+1.

We construct a 10-dimensional symplectic moduli space of torsion sheaves on the cubic 4-fold in the 5-dimensional projective space. It parametrizes the stable rank 2 vector bundles on hyperplane sections of the cubic 4-fold which are obtained by the Serre construction from normal elliptic quintics. The natural projection of this moduli space onto the dual projective 5-space is a Lagrangian fibration. The symplectic structure is closely related (and conjecturally equal) to the quasi-symplectic structure induced by the Yoneda pairing on the moduli space.

On the vector bundles over rationally connected varieties