Complex Geometry and Dynamics

Inspired by constructions of automorphisms on rational surfaces and a recent paper of Perroni and Zhang (Mathematische Annalen 359(1–2):189–209, 2014), we give a concrete construction of pseudoautomorphisms of higher dimensional rational surfaces that have an invariant cuspidal curve and first dynamical degree larger than one. Taking advantage of the group structure on the smooth points of the curve and elementary projective geometry, we arrive at explicit formulas for these pseudoautomorphisms. Though it is not used in our construction, we further indicate a relationship between aspects of our construction with certain coxeter groups.

We discuss recent versions of the Brunn-Minkowski inequality in the complex setting, and use it to prove the openness conjecture of Demailly and Kollár.

We discuss some recently obtained \(\bar{\partial }\)-estimates and their relations to the classical ones, as well as to the Ohsawa-Takegoshi extension theorem. We also show that the constants obtained earlier in estimates due to Donnelly-Fefferman and Berndtsson are optimal.

The goal of this survey is to present various results concerning the cohomology of pseudoeffective line bundles on compact Kähler manifolds, and related properties of their multiplier ideal sheaves. In case the curvature is strictly positive, the prototype is the well known Nadel vanishing theorem, which is itself a generalized analytic version of the fundamental Kawamata-Viehweg vanishing theorem of algebraic geometry. We are interested here in the case where the curvature is merely semipositive in the sense of currents, and the base manifold is not necessarily projective. In this situation, one can still obtain interesting information on cohomology, e.g. a Hard Lefschetz theorem with pseudoeffective coefficients, in the form of a surjectivity statement for the Lefschetz map. More recently, Junyan Cao, in his PhD thesis defended in Grenoble, obtained a general Kähler vanishing theorem that depends on the concept of numerical dimension of a given pseudoeffective line bundle. The proof of these results depends in a crucial way on a general approximation result for closed (1, 1)-currents, based on the use of Bergman kernels, and the related intersection theory of currents. Another important ingredient is the recent proof by Guan and Zhou of the strong openness conjecture. As an application, we discuss a structure theorem for compact Kähler threefolds without nontrivial subvarieties, following a joint work with F. Campana and M. Verbitsky. We hope that these notes will serve as a useful guide to the more detailed and more technical papers in the literature; in some cases, we provide here substantially simplified proofs and unifying viewpoints.

In this paper we survey some recent contributions by the authors (Alarcón and Forstnerič, Math Ann 357:1049–1070, 2013; Invent Math 196:733–771, 2014; Math Ann, in press, http://​link.​springer.​com/​10.​1007/​s00208-015-1189-9. arXiv:1308.0903) to the theory of null holomorphic curves in the complex Euclidean space \(\mathbb{C}^{3}\), as well as their applications to null holomorphic curves in the special linear group \(SL_{2}(\mathbb{C})\), minimal surfaces in the Euclidean space \(\mathbb{R}^{3}\), and constant mean curvature one surfaces (Bryant surfaces) in the hyperbolic space \(\mathbb{H}^{3}\). The paper is an expanded version of the lecture given by the second named author at the Abel Symposium 2013 in Trondheim.

In our recent works (Grushevsky and Krichever, The universal Whitham hierarchy and the geometry of the moduli space of pointed Riemann surfaces. In: Surveys in differential geometry. Vol. XIV. Geometry of Riemann surfaces and their moduli spaces. Volume 14 of surveys in differential geometry. International Press, Somerville, pp 111–129, 2009; Grushevsky and Krichever, Foliations on the moduli space of curves, vanishing in cohomology, and Calogero-Moser curves, arXiv:1108.4211, part 1, under revision) we have used meromorphic differentials on Riemann surfaces all of whose periods are real to study the geometry of the moduli spaces of Riemann surfaces. In this paper we survey the relevant constructions and show how they are related to and motivated by the spectral theory of the elliptic Calogero-Moser integrable system.

Let M _ℓ be a smooth Levi-nondegenerate hypersurface of signature ℓ in C ⁿ with n ≥ 3, and write H _ℓ ^N for the standard hyperquadric of the same signature in C ^N with \(N - n <\frac{n-1} {2}\). Let F be a holomorphic map sending M _ℓ into \(H_{\ell}^{N}\). Assume F does not send a neighborhood of M _ℓ in C ⁿ into \(H_{\ell}^{N}\). We show that F is necessarily CR transversal to M _ℓ at any point. Equivalently, we show that F is a local CR embedding from M _ℓ into \(H_{\ell}^{N}\).

Parametric Cartan theory of exterior differential systems, and explicit cohomology of projective manifolds reveal united rationality features of differential algebraic geometry.

Works on Levi flat hypersurfaces in complex manifolds will be reviewed with an emphasis on the cases in \(\mathbb{C}\mathbb{P}^{n}\), complex tori and Hopf surfaces. Related results on locally pseudoconvex domains whose complements are analytic sets will also be presented.

By a beautiful theorem of Harvey and Lawson, strongly pseudoconvex connected compact embeddable CR manifolds are the boundaries of subvarieties in \(\mathbb{C}^{N}\) with only normal isolated singularities. This leads to a natural question of how to determine the properties of the interior singularities from the CR manifolds and vice versa. In this paper, we give a survey on the interplay between CR Geometry and Algebraic Geometry for the last 30 years or so.

We will give an overview on the use of Fatou coordinates in the study of (semi-)parabolic fixed points and their bifurcations.

The purpose of this article is explain some aspects in complex hyperbolicity, through discussions of examples. We would focus our discussions on some recent results of Wing-Keung To and myself on Kobayashi hyperbolicity of some moduli space of polarized varieties, but would also mention some related results in complex hyperbolicity, as well as some examples for arithmetic problems related to hyperbolicity.

In the present paper, we’ll give a survey of our recent results on the L ² extension problem with optimal estimate. We’ll consider the problem in various settings according to Ohsawa’s series papers, and present our optimal versions of Ohsawa’s L ² extension theorems. We’ll discuss the problem in a general setting and present a solution of the problem in the general setting. We’ll give some applications of our results including a solution of the equality part of Suita’s conjecture. Finally, we present our recent solutions of Demailly’s strong openness conjecture on multiplier ideal sheaf and related problems.