Semiample vector bundles and a characterization of étale trivial vector bundles

Let ξ be a stably fibre homotopic trivial vector bundle. A classical result of Thorn states that the Stiefel-Whitney classes of ξ vanish, and one way to prove this is as follows. Letube the Thorn class of ξ in mod 2 cohomology. Thenuis stably spherical by [2] and therefore all stable cohomology operations vanish onu, showing thatwi(ξ)u= Sqiu= 0. In this note we shall apply this same method using complex cobordism and Landweber-Novikov operations to study relations among Chern classes of a stably fibre homotopic trivial complex vector bundle. We will thus obtain in a unified way certain strong modpconditions for every primep.

Let $\mathcal{p : X \to S}$ be a proper Kähler fibration and $\mathcal{E \to X}$ a Hermitian holomorphic vector bundle. As motivated by the work of Berndtsson (Curvature of vector bundles associated to holomorphic fibrations), by using basic Hodge theory, we derive several general curvature formulas for the direct image $\mathcal{p_* (K_{X/S} \otimes E)}$ for general Hermitian holomorphic vector bundle $\mathcal{E}$ in a simple way. A straightforward application is that, if the family $\mathcal{X \to S}$ is infinitesimally trivial and Hermitian vector bundle $\mathcal{E}$ is Nakano-negative along the base $\mathcal{S}$, then the direct image $\mathcal{p_* (K_{X/S} \otimes E)}$ is Nakano-negative. We also use these curvature formulas to study the moduli space of projectively flat vector bundles with positive first Chern classes and obtain that, if the Chern curvature of direct image $p_*(K_X \otimes E)$ —of a positive projectively flat family $(E, h(t))_{t \in \mathbb{D}} \to X$ —vanishes, then the curvature forms of this family are connected by holomorphic automorphisms of the pair $(X,E)$.

The primary focus of this thesis is to consider Kato square root problems for various divergence-form operators on manifolds. This is the study of perturbations of second-order differential operators by bounded, complex, measurable coefficients. In general, such operators are not self-adjoint but uniformly elliptic. The Kato square root problem is then to understand when the square root of such an operator, which exists due to uniform ellipticity, is comparable to its unperturbed counterpart. A remarkably adaptable operator-theoretic framework due to Axelsson, Keith and McIntosh sits in the background of this work. This framework allows us to take a powerful first-order perspective of the problems which we consider in a geometric setting. Through a well established procedure, we reduce these problems to the study of quadratic estimates. Under a set of natural conditions, we prove quadratic estimates for a class of operators on vector bundles over complete measure metric spaces. The first kind of estimates we prove are global, and we establish them on trivial vector bundles when the underlying measure grows at most polynomially. The second kind are local, and there, we allow the vector bundle to be non-trivial but bounded in an appropriate sense. Here, the measure is allowed to grow exponentially. An important consequence of obtaining quadratic estimates on measure metric spaces is that it allows us to consider subelliptic operators on Lie groups. The first-order perspective allows us to reduce the subelliptic problem to a fully elliptic one on a sub-bundle. As a consequence, we are able to solve a homogeneous Kato square root problem for perturbations of subelliptic operators on nilpotent Lie groups. For general Lie groups we solve a similar inhomogeneous problem. In the situation of complete Riemannian manifolds, we consider uniformly elliptic divergence-form operators arising from connections on vector bundles. Under a set of assumptions, we show that the Kato square root problem can be solved for such operators. As a consequence, we solve this problem on functions under the condition that the Ricci curvature and injectivity radius are bounded. Assuming an additional lower bound for the curvature endomorphism on forms, we solve a similar problem for perturbations of inhomogeneous Hodge-Dirac operators. A theorem for tensors is obtained by additionally assuming boundedness of a second-order Riesz transform. Motivated by the study of these Kato problems, where for technical reasons it is useful to know the density of compactly supported functions in the domains of operators, we study connections and their divergence on a vector bundle. Through a first-order formulation, we show that this density property holds for the domains of these operators if the metric and connection are compatible and the underlying manifold is complete. We also show that compactly supported functions are dense in the second-order Sobolev space on complete manifolds under the sole assumption that the Ricci curvature is bounded below, improving a result that previously required an additional lower bound on the injectivity radius.

In the first section, we show that, up to isomorphism, vector bundles are just locally trivial fibre bundles with a finite-dimensional vector space V as fibre and GL(V), the group of automorphisms of V, as a structure group. This is done by examining how trivial bundles are pieced together, using systems of transition functions to define a general locally trivial fibre bundle. We can apply this analysis to prove a theorem which says that any continuous functorial operation on vector spaces determines an operation on vector bundles. This allows construction of tensor products, exterior products, etc., of vector bundles.

Open quotients of trivial vector bundles

Monodromies of algebraic connections on the trivial bundle

The versal deformation space of a reflexive module on a rational cone

We study when a smooth variety X, embedded diagonally in its Cartesian square, is the zero scheme of a section of a vector bundle of rank dim(X) on X × X.We call this the diagonal property (D).It was known that it holds for all flag manifolds SL n /P .We consider mainly the cases of proper smooth varieties, and the analogous problems for smooth manifolds ("the topological case").Our main new observation in the case of proper varieties is a relation between (D) and cohomologically trivial line bundles on X, obtained by a variation of Serre's classic argument relating rank 2 vector bundles and codimension 2 subschemes, combined with Serre duality.Based on this, we have several detailed results on surfaces, and some results in higher dimensions.For smooth affine varieties, we observe that for an affine algebraic group over an algebraically closed field, the diagonal is in fact a complete intersection; thus (D) holds, using the trivial bundle.We conjecture the existence of smooth affine complex varieties for which (D) fails; this leads to an interesting question on projective modules.The arguments in the topological case have a different flavour, with arguments from homotopy theory, topological K-theory, index theory etc.There are 3 variants of the diagonal problem, depending on the type of vector bundle we want (arbitrary, oriented or complex).We obtain a homotopy theoretic reformulation of the diagonal property as an extension problem for a certain homotopy class of maps.We also have detailed results in several cases: spheres, odd dimensional complex projective quadric hypersurfaces, and manifolds of even dimension ≤ 6 with an almost complex structure.

The versal deformation space of a reflexive module on a rational cone

IN MEMORIAM OF ALEXANDER GROTHENDIECK. THE MAN.

We provide unipotent factorizations of vector bundle automorphisms of real and complex vector bundles over finite dimensional locally finite CW-complexes. This generalizes work of Thurston–Vaserstein and Vaserstein for trivial vector bundles. We also address two symplectic cases and propose a complex geometric analog of the problem in the setting of holomorphic vector bundles over Stein manifolds.

We show that a unipotent vector bundle on a non-Kaehler compact complex manifold does not admit a flat holomorphic connection in general. We also construct examples of topologically trivial stable vector bundle on compact Gauduchon manifold that does not admit any unitary flat connection.

Let G be a reductive algebraic group over C, let F be a G-module, and let B be an affine G-variety, i.e., an affine variety with an algebraic action of G. Then B x F is in a natural way a G-vector bundle over B, which we denote by F. (All vector bundles here are algebraic.) A G-vector bundle over B is called trivial if it is isomorphic to F for some G-module F. From the endomorphism ring R of the G-vector bundle S, we construct G-vector bundles over B. The bundles constructed this way have the property that when added to S they are isomorphic to F e S for a fixed G-module F. They are called stably trivial. The set of isomorphism classes of G-vector bundles over B which satisfy this condition is denoted by VEC(B, F; S). For such a bundle E we define an invariant p(E) which lies in a quotient of R. This invariant allows us to distinguish non-isomorphic G-vector bundles. When B is a Gmodule, a G-vector bundle over B defines an action of G on affine space. We give criteria which in certain cases allow us to distinguish the underlying actions. The construction and invariants are applied to the following two problems:

We study partially and totally associative ternary algebras of first and second\nkind. Assuming the vector space underlying a ternary algebra to be a topological\nspace and a~triple product to be continuous mapping, we consider the trivial\nvector bundle over a ternary algebra and show that a triple product induces a\nstructure of binary algebra in each fiber of this vector bundle. We find the\nsufficient and necessary condition for a ternary multiplication to induce a\nstructure of associative binary algebra in each fiber of this vector bundle.\nGiven two modules over the algebras with involutions, we construct a~ternary\nalgebra which is used as a building block for a Lie algebra. We construct\nternary algebras of cubic matrices and find four different totally associative\nternary multiplications of second kind of cubic matrices. It is proved that\nthese are the only totally associative ternary multiplications of second kind in\nthe case of cubic matrices. We describe a ternary analog of Lie algebra of cubic\nmatrices of second order which is based on a notion of j-commutator and find all\ncommutation relations of generators of this algebra.

Gluck and Ziller proved that Hopf vector fields on S3 have minimum volume among all unit vector fields. Thinking of S3 as a Lie group, Hopf vector fields are exactly those with unit length which are left or right invariant, and TS3 is a trivial vector bundle with a connection induced by the adjoint representation. We prove the analogue of the stated result of Gluck and Ziller for the representation given by quaternionic multiplication. The resulting vector bundle over S3, with the Sasaki metric, has as well no parallel unit sections. We provide an application of a double point calibration, proving that the submanifolds determined by the left and right invariant sections minimize volume in their homology classes.