Log Picard Algebroids and Meromorphic Line Bundles

We introduce logarithmic Picard algebroids, a natural class of Lie algebroids adapted to a simple normal crossings divisor on a smooth projective variety. We show that such algebroids are classified by a subspace of the de Rham cohomology of the divisor complement determined by its mixed Hodge structure. We then solve the prequantization problem, showing that under the appropriate integrality condition, a log Picard algebroid is the Lie algebroid of symmetries of what is called a meromorphic line bundle, a generalization of the usual notion of line bundle in which the fibers degenerate along the divisor. We give a geometric description of such bundles and establish a classification theorem for them, showing that they correspond to a subgroup of the holomorphic line bundles on the divisor complement. Importantly, these holomorphic line bundles need not be algebraic. Finally, we provide concrete methods for explicitly constructing examples of meromorphic line bundles, such as for a smooth cubic divisor in the projective plane.

Irreducible Unitary Representations Concerning Homogeneous Holomorphic Line Bundles Over Elliptic Orbits

In this paper we consider a homogeneous holomorphic line bundle over an elliptic adjoint orbit of a real semisimple Lie group, and set a continuous representation of the Lie group on a certain complex vector subspace of the complex vector space of holomorphic cross-sections of the line bundle. Then, we demonstrate that the representation is irreducible unitary.

Correspondences between convex geometry and complex geometry

We present several analogies between convex geometry and the theory of holomorphic line bundles on smooth projective varieties or K\"ahler manifolds. We study the relation between positive products and mixed volumes. We define and study a Blaschke addition for divisor classes and mixed divisor classes, and prove new geometric inequalities for divisor classes. We also reinterpret several classical convex geometry results in the context of algebraic geometry: the Alexandrov body construction is the convex geometry version of divisorial Zariski decomposition; Minkowski's existence theorem is the convex geometry version of the duality between the pseudo-effective cone of divisors and the movable cone of curves. Comment: EpiGA Volume 1 (2017), Article Nr. 6

Holomorphic Line Bundles Over Domains in Cousin Groups and the Algebraic Dimension of Oeljeklaus-Toma Manifolds

In this paper we extend results due to Vogt on line bundles over Cousin groups to the case of domains stable by the maximal compact subgroup. This is used to show that the algebraic dimension of Oeljeklaus—Toma manifolds (OT-manifolds) is 0. In the last part we establish that certain Cousin groups, in particular those arising from the construction of OT-manifolds, have finite-dimensional irregularity.

A CHARACTERIZATION OF NONCOMPACT KOISO-TYPE SOLITONS

We construct complete gradient Kähler–Ricci solitons of various types on the total spaces of certain holomorphic line bundles over compact Kähler–Einstein manifolds with positive scalar curvature. Those are noncompact analogues of the compact examples found by Koiso [On rotationally symmetric Hamilton's equations for Kähler–Einstein metrics, in Recent Topics in Differential and Analytic Geometry, Advanced Studies in Pure Mathematics, Vol. 18-I (Academic Press, Boston, MA, 1990), pp. 327–337]. Our examples can be viewed a generalization of previous examples by Cao [Existense of gradient Kähler–Ricci solitons, in Elliptic and Parabolic Methods in Geometry (Minneapolis, MN, 1994), pp. 1–16], Chave and Valent [On a class of compact and non-compact quasi-Einstein metrics and their renormalizability properties, Nuclear Phys. B 478 (1996) 758–778], Pedersen, Tønnesen-Friedman, and Valent [Quasi-Einstein Kähler metrics, Lett. Math. Phys. 50(3) (1999) 229–241], and Feldman, Ilmanen and Knopf [Rotationally symmetric shrinking and expanding gradient Kähler–Ricci solitons, J. Differential Geom. 65 (2003) 169–209]. We also prove a uniformization result on complete steady gradient Kähler–Ricci solitons with non-negative Ricci curvature under additional assumptions.