Related papers: Meromorphic Line Bundles and Holomorphic Gerbes

Consider the holomorphic bundle with connection on $\mathbb P^1-\{0,1,\infty\}$ corresponding to the regular hypergeometric differential operator \[ \prod_{j=1}^h(D-\alpha_j)-z\prod_{j=1}^h(D-\beta_j), \qquad D=z\frac{d}{dz}. \] If the…

In this paper we construct explicit examples of both closed and non-compact finite volume hyperbolic manifolds which provide counterexamples to the conjecture that the co-rank of a 3-manifold group (also known as the cut number) is bounded…

We prove an inequality between the sum of the Betti numbers of a complex projective manifold and its total curvature, and we characterize the complex projective manifolds whose total curvature is minimal. These results extend the classical…

We prove a central limit theorem for smooth linear statistics associated with zero divisors of standard Gaussian holomorphic sections in a sequence of holomorphic line bundles with Hermitian metrics of class $\mathscr{C}^{3}$ over a compact…

We study relations among characteristic classes of smooth manifold bundles with highly-connected fibers. For bundles with fiber the connected sum of $g$ copies of a product of spheres $S^d \times S^d$ and an odd $d$, we find numerous…

A Bott manifold is a closed smooth manifold obtained as the total space of an iterated $\C P^1$-bundle starting with a point, where each $\C P^1$-bundle is the projectivization of a Whitney sum of two complex line bundles. A…

We call a sheaf on an algebraic variety immaculate if it lacks any cohomology including the zero-th one, that is, if the derived version of the global section functor vanishes. Such sheaves are the basic tools when building exceptional…

We construct a category of flat vector bundles on an elliptic curve. It arises in the representation theory of quantum affine algebras and carries meromorphic braided structure with singularities on the diagonal of the square of the curve.

We show that there exist non-formal compact oriented manifolds of dimension $n$ and with first Betti number $b_1=b\geq 0$ if and only if $n\geq 3$ and $b\geq 2$, or $n\geq (7-2b)$ and $0\leq b\leq 2$. Moreover, we present explicit examples…

Let D = {D_{1},...,D_{l}} be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space P^n and let \Omega^{1}_{P^n}(log D) be the logarithmic bundle attached to it. Following [1], we show that…

We study the existence and regularity of invariant graphs for bundle maps (or bundle correspondences with generating bundle maps motivated by ill-posed differential equations) having some relative partial hyperbolicity on non-trivial and…

For homogeneous simply connected Hodge manifolds it is proved that the set of coherent vectors orthogonal to a given one is the divisor responsible for the homogeneous holomorphic line bundle of the coherent vectors. In particular, for…

We construct examples of flat fiber bundles over the Hopf surface such that the total spaces have no pseudoconvex neighborhood basis, admit a complete K\"ahler metric, or are hyperconvex but have no nonconstant holomorphic functions. For…

The families of morphisms of vector fibre bundle (\cite{Mill1}) defined by the linear systems of differential equations with non-negative coefficients is considered. Authors proved that the specified families of morphisms is not saturated…

The notion of a higher bundle gerbe is introduced to give a geometric realization of the higher degree integral cohomology of certain manifolds. We consider examples using the infinite dimensional spaces arising in gauge theories.

Let $X$ be a complex manifold and $L$ be a holomorphic line bundle on $X$. Assume that $L$ is semi-positive, namely $L$ admits a smooth Hermitian metric with semi-positive Chern curvature. Let $Y$ be a compact K\"ahler submanifold of $X$…

The notion of a gerbe with connection is conveniently reformulated in terms of the simplicial deRham complex. In particular the usual Chern-Weil and Chern-Simons theory is well adapted to this framework and rather easily gives rise to…

The character varieties of representations of a surface group into the Lie groups SL(m,H), SO(2m,H) and Sp(m,m) have a holomorphic description in terms of the moduli space of Higgs bundles. We show that the fibres of the integrable system…

We study a novel type of braid groups on a closed orientable surface $\Sigma$. These are fundamental groups of certain manifolds that are hybrids between symmetric products and configuration spaces of points on $\Sigma$; a class of examples…

We develop a holomorphic equivalence between on one hand the space of pairs (stable bundle, flat connection on the bundle) and the "sheaf of holomorphic connections" (the sheaf of splittings of the one-jet sequence) for the determinant…