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We consider smooth, complex quasiprojective varieties
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
vanishes. As an application, we show that complements of linear, toric, and elliptic arrangements are both duality and abelian duality spaces.
be a member of a certain class of convex ellipsoids of finite/infinite type in
. In this paper, we prove that every holomorphic function in
can be approximated by holomorphic functions on
. For the case
, the continuity up to the boundary is additionally required. The proof is based on
bounds in the additive Cousin problem.
In this paper we study holomorphic Legendrian curves in the standard holomorphic contact structure on
. 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
admits a proper holomorphic Legendrian embedding
, and we prove that for every compact bordered Riemann surface
there exists a topological embedding
whose restriction to the interior is a complete holomorphic Legendrian embedding
. As a consequence, we infer that every complex contact manifold
carries relatively compact holomorphic Legendrian curves, normalized by any given bordered Riemann surface, which are complete with respect to any Riemannian metric on
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)) ).
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