We investigate the relationship between affine and Stein varieties in the context of rigid geometry. We show that the two concepts are much more closely related than in complex geometry, e.g. they are equivalent for surfaces. This rests on the density of algebraic functions in analytic functions. One key ingredient to prove such a density statement is an extension result for Cartier divisors.

We define the Schur–Agler class in infinite variables to consist of functions whose restrictions to finite-dimensional polydisks belong to the Schur–Agler class. We show that a natural generalization of an Agler decomposition holds and the functions possess transfer function realizations that allow us to extend the functions to the unit ball of $\ell ^\infty $. We also give a Pick interpolation type theorem which displays a subtle difference with finitely many variables. Finally, we make a brief connection to Dirichlet series derived from the Schur–Agler class in infinite variables via the Bohr correspondence.

In this paper, we first prove that the totally real discs lying in certain Levi-flat hypersurfaces are polynomially convex. We also studied the polynomial convexity of totally real discs lying in the regular part of certain singular Levi-flat hypersurfaces. In particular, a necessary and sufficient condition for polynomial convexity of totally real discs lying in the non-singular part of the boundary of the Hartogs triangle is achieved. Sufficient conditions on general compact subsets lying on those hypersurfaces for polynomial convexity are also reported here.

In this article, we discuss the topology of varieties over $\mathbb {C}$, viz., their homology and homotopy groups. We show that the fundamental group of a quasi-projective variety has negative deficiency under a certain hypothesis on its second homology and therefore a large class of groups cannot arise as fundamental groups of varieties. For a smooth projective surface admitting a fibration over a curve, we give a detailed analysis of the homology and homotopy groups of their universal cover via a case-by-case analysis, depending on the nature of the singular fibers. For smooth, projective surfaces whose universal cover is holomorphically convex (conjecturally always true), we show that the second and third homotopy groups are free abelian, often of infinite rank.

In this paper, we present an extension theorem which is equivalent to an injectivity theorem on the cohomology groups of vector bundles equipped with singular Hermitian metrics over holomorphically convex Kähler manifolds.

In this article, we present a concavity property of the minimal $L^{2}$ integrals related to multiplier ideal sheaves with Lebesgue measurable gain. As applications, we give necessary conditions for our concavity degenerating to linearity, characterizations for 1-dimensional case, and a characterization for the holding of the equality in optimal $L^2$ extension problem on open Riemann surfaces with weights may not be subharmonic.

A subset ${\mathcal D}$ of a domain $\Omega \subset {\mathbb C}^d$ is determining for an analytic function $f:\Omega \to \overline {{\mathbb D}}$ if whenever an analytic function $g:\Omega \rightarrow \overline {{\mathbb D}}$ coincides with f on ${\mathcal D}$, equals to f on whole $\Omega $. This note finds several sufficient conditions for a subset of the symmetrized bidisk to be determining. For any $N\geq 1$, a set consisting of $N^2-N+1$ many points is constructed which is determining for any rational inner function with a degree constraint. We also investigate when the intersection of the symmetrized bidisk intersected with some special algebraic varieties can be determining for rational inner functions.

In 1955, Lehto showed that, for every measurable function $\psi $ on the unit circle ${\mathbb T}$, there is a function f holomorphic in the unit disc ${{\mathbb D}}$, having $\psi $ as radial limit a.e. on ${\mathbb T}$. We consider an analogous boundary value problem, where the unit disc is replaced by a Stein domain on a complex manifold and radial approach to a boundary point p is replaced by (asymptotically) total approach to p.

In this paper, we present a version of Guan-Zhou’s optimal $L^{2}$ extension theorem and its application. As a main application, we show that under a natural condition, the question posed by Ohsawa in his series paper VIII on the extension of $L^{2}$ holomorphic functions holds. We also give an explicit counterexample which shows that the question fails in general.

We strengthen, in various directions, the theorem of Garnett that every $\unicode[STIX]{x1D70E}$-compact, completely regular space $X$ occurs as a Gleason part for some uniform algebra. In particular, we show that the uniform algebra can always be chosen so that its maximal ideal space contains no analytic discs. We show that when the space $X$ is metrizable, the uniform algebra can be chosen so that its maximal ideal space is metrizable as well. We also show that for every locally compact subspace $X$ of a Euclidean space, there is a compact set $K$ in some $\mathbb{C}^{N}$ so that $\widehat{K}\backslash K$ contains a Gleason part homeomorphic to $X$, and $\widehat{K}$ contains no analytic discs.

We consider smooth, complex quasiprojective varieties $U$ that admit a compactification with a boundary, which is an arrangement of smooth algebraic hypersurfaces. If the hypersurfaces intersect locally like hyperplanes, and the relative interiors of the hypersurfaces are Stein manifolds, we prove that the cohomology of certain local systems on $U$ vanishes. As an application, we show that complements of linear, toric, and elliptic arrangements are both duality and abelian duality spaces.

Let $\unicode[STIX]{x1D6FA}$ be a member of a certain class of convex ellipsoids of finite/infinite type in $\mathbb{C}^{2}$ . In this paper, we prove that every holomorphic function in $L^{p}(\unicode[STIX]{x1D6FA})$ can be approximated by holomorphic functions on $\bar{\unicode[STIX]{x1D6FA}}$ in $L^{p}(\unicode[STIX]{x1D6FA})$ -norm, for $1\leq p<\infty$ . For the case $p=\infty$ , the continuity up to the boundary is additionally required. The proof is based on $L^{p}$ bounds in the additive Cousin problem.

In this paper we study holomorphic Legendrian curves in the standard holomorphic contact structure on $\mathbb{C}^{2n+1}$ for any $n\in \mathbb{N}$ . We provide several approximation and desingularization results which enable us to prove general existence theorems, settling some of the open problems in the subject. In particular, we show that every open Riemann surface $M$ admits a proper holomorphic Legendrian embedding $M{\hookrightarrow}\mathbb{C}^{2n+1}$ , and we prove that for every compact bordered Riemann surface $M={M\unicode[STIX]{x0030A}}\,\cup \,bM$ there exists a topological embedding $M{\hookrightarrow}\mathbb{C}^{2n+1}$ whose restriction to the interior is a complete holomorphic Legendrian embedding ${M\unicode[STIX]{x0030A}}{\hookrightarrow}\mathbb{C}^{2n+1}$ . As a consequence, we infer that every complex contact manifold $W$ carries relatively compact holomorphic Legendrian curves, normalized by any given bordered Riemann surface, which are complete with respect to any Riemannian metric on $W$ .

In this paper we study the classification of holomorphic flows on Stein spaces of dimension two. We assume that the flow has periodic orbits, not necessarily with a same period. Then we prove a linearization result for the flow, under some natural conditions on the surface.

Let E be a totally real set on a Stein open set Ω on a complete noncompact Kähler manifold (M,g) with nonnegative holomorphic bisectional curvature such that (Ω,g) has bounded geometry at E. Then every function f in a Cp class with compact support on Ω and -flat on E up to order p−1,p≥2 (respectively, in a Gevrey class of order s>1, with compact support on Ω and -flat on E up to infinite order) can be approximated on compacts subsets of E by holomorphic functions fk on Ω with degree of approximation equal k−p/2 (respectively, exp (−c(s)k1/2(s−1)) ).