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𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More...

A deformation of commutative polynomial algebras in even numbers of variables

Wenhua Zhao (2010)

Open Mathematics

We introduce and study a deformation of commutative polynomial algebras in even numbers of variables. We also discuss some connections and applications of this deformation to the generalized Laguerre orthogonal polynomials and the interchanges of right and left total symbols of differential operators of polynomial algebras. Furthermore, a more conceptual re-formulation for the image conjecture [18] is also given in terms of the deformed algebras. Consequently, the well-known Jacobian conjecture...

A Hilbert Lemniscate Theorem in 2

Thomas Bloom, Norman Levenberg, Yu. Lyubarskii (2008)

Annales de l’institut Fourier

For a regular, compact, polynomially convex circled set K in C 2 , we construct a sequence of pairs { P n , Q n } of homogeneous polynomials in two variables with deg P n = deg Q n ...

A note on Berezin-Toeplitz quantization of the Laplace operator

Alberto Della Vedova (2015)

Complex Manifolds

Given a Hodge manifold, it is introduced a self-adjoint operator on the space of endomorphisms of the global holomorphic sections of the polarization line bundle. Such operator is shown to approximate the Laplace operator on functions when composed with Berezin-Toeplitz quantization map and its adjoint, up to an error which tends to zero when taking higher powers of the polarization line bundle.

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 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 χ ( X , ω ) . This is done by establishing...

A Riemann-Roch-Hirzebruch formula for traces of differential operators

Markus Engeli, Giovanni Felder (2008)

Annales scientifiques de l'École Normale Supérieure

Let D be a holomorphic differential operator acting on sections of a holomorphic vector bundle on an n -dimensional compact complex manifold. We prove a formula, conjectured by Feigin and Shoikhet, giving the Lefschetz number of D as the integral over the manifold of a differential form. The class of this differential form is obtained via formal differential geometry from the canonical generator of the Hochschild cohomology H H 2 n ( 𝒟 n , 𝒟 n * ) of the algebra of differential operators on a formal neighbourhood of a...

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