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A priori estimates for weak solutions of complex Monge-Ampère equations

Slimane Benelkourchi, Vincent Guedj, Ahmed Zeriahi (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let X be a compact Kähler manifold and ω be a smooth closed form of bidegree ( 1 , 1 ) which is nonnegative and big. We study the classes ℰ χ ( X , ω ) of ω -plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight χ has fast growth at infinity, the corresponding functions are close to be bounded. We show that if a positive Radon measure is suitably dominated by the Monge-Ampère capacity, then it belongs to the range of the Monge-Ampère operator on some class ℰ χ ( X , ω ) . This is done by establishing...

A proof of the birationality of certain BHK-mirrors

Patrick Clarke (2014)

Complex Manifolds

We generalize and give an elementary proof of Kelly’s refinement [9] of Shoemaker’s result [11] on the birationality of certain BHK-mirrors. Our approach uses a construction that is equivalent to the Krawitz generalization [10] of the duality in Berglund-Hübsch [2].

A proof of the stratified Morse inequalities for singular complex algebraic curves using the Witten deformation

Ursula Ludwig (2011)

Annales de l’institut Fourier

The Witten deformation is an analytic method proposed by Witten which, given a Morse function f : M → R on a smooth compact manifold M , allows to prove the Morse inequalities. The aim of this article is to generalise the Witten deformation to stratified Morse functions (in the sense of stratified Morse theory as developed by Goresky and MacPherson) on a singular complex algebraic curve. In a previous article the author developed the Witten deformation for the model of an algebraic curve with cone-like singularities...

A proof of the valuation property and preparation theorem

Krzysztof Jan Nowak (2007)

Annales Polonici Mathematici

The purpose of this article is to present a short model-theoretic proof of the valuation property for a polynomially bounded o-minimal theory T. The valuation property was conjectured by van den Dries, and proved for the polynomially bounded case by van den Dries-Speissegger and for the power bounded case by Tyne. Our proof uses the transfer principle for the theory T c o n v (i.e. T with an extra unary symbol denoting a proper convex subring), which-together with quantifier elimination-is due to van den...

A propos du problème des arcs de Nash

Camille Plénat (2005)

Annales de l’institut Fourier

Soit ℋ = ⋃ N i la décomposition canonique de l’espace des arcs ℋ passant par une singularité normale de surface. Dans cet article, on propose deux nouvelles conditions qui si elles sont vérifiées permettent de montrer que N i n’est pas inclus dans N j . On applique ces conditions pour donner deux nouvelles preuves du problème de Nash pour les singularités sandwich minimales.

A pseudo-trigonometry related to Ptolemy's theorem and the hyperbolic geometry of punctured spheres

Joachim A. Hempel (2004)

Annales Polonici Mathematici

A hyperbolic geodesic joining two punctures on a Riemann surface has infinite length. To obtain a useful distance-like quantity we define a finite pseudo-length of such a geodesic in terms of the hyperbolic length of its surrounding geodesic loop. There is a well defined angle between two geodesics meeting at a puncture, and our pseudo-trigonometry connects these angles with pseudo-lengths. We state and prove a theorem resembling Ptolemy's classical theorem on cyclic quadrilaterals and three general...

A pure smoothness condition for Radó’s theorem for α -analytic functions

Abtin Daghighi, Frank Wikström (2016)

Czechoslovak Mathematical Journal

Let Ω ⊂ ℂ n be a bounded, simply connected ℂ -convex domain. Let α ∈ ℤ + n and let f be a function on Ω which is separately C 2 α j - 1 -smooth with respect to z j (by which we mean jointly C 2 α j - 1 -smooth with respect to Re z j , Im z j ). If f is α -analytic on Ω ∖ f - 1 ( 0 ) , then f is α -analytic on Ω . The result is well-known for the case α i = 1 , 1 ≤ i ≤ n , even when f a priori is only known to be continuous.

A radial Phragmén-Lindelöf estimate for plurisubharmonic functions on algebraic varieties

Rüdiger Braun, Reinhold Meise, B. Taylor (1999)

Annales Polonici Mathematici

For complex algebraic varieties V, the strong radial Phragmén-Lindelöf condition (SRPL) is defined. It means that a radial analogue of the classical Phragmén-Lindelöf Theorem holds on V. Here we derive a sufficient condition for V to satisfy (SRPL), which is formulated in terms of local hyperbolicity at infinite points of V. The latter condition as well as the extension of local hyperbolicity to varieties of arbitrary codimension are introduced here for the first time. The proof of the main result...

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