Mathematics

We study the ∂ ¯ -Laplacian on forms taking values in L k , a high power of a nef line bundle on a compact complex manifold, and give an estimate of the number of the eigenforms whose corresponding eigenvalues smaller than or equal to λ . In particular, the λ=0 case gives an asymptotic estimate for the order of the corresponding cohomology groups. It helps to generalize the Grauert--Riemenschneider conjecture. At last, we discuss the λ=0 case on a pseudo-effective line bundle.

Kalantari's Geometric Modulus Principle describes the local behavior of the modulus of a polynomial. Specifically, if p(z)= a 0 + ??n j=k a j (z??z 0 ) j , a 0 a k a n ?? , then the complex plane near z= z 0 comprises 2k sectors of angle ? k , alternating between arguments of ascent (angles θ where |p( z 0 +t e iθ )|>|p( z 0 )| for small t ) and arguments of descent (where the opposite inequality holds). In this paper, we generalize the Geometric Modulus Principle to holomorphic and harmonic functions. As in Kalantari's original paper, we use these extensions to give succinct, elegant new proofs of some classical theorems from analysis.

We solve fundamental problems in Oka theory by establishing an implicit function theorem for sprays. As the first application of our implicit function theorem, we obtain an elementary proof of the fact that approximation yields interpolation. This proof and Lárusson's elementary proof of the converse give an elementary proof of the equivalence between approximation and interpolation. The second application concerns the Oka property of a blowup. We prove that the blowup of an algebraically Oka manifold along a smooth algebraic center is Oka. In the appendix, equivariantly Oka manifolds are characterized by the equivariant version of Gromov's condition Ell 1 , and the equivariant localization principle is also given.

The aim of this paper is to give a proof of improving of Zalcman's lemma.

We give an exposition of results from a crossroad between geometric function theory, harmonic analysis, boundary value problems and approximation theory, which characterize quasicircles. We will specifically expose the interplay between the jump decomposition, singular integral operators and approximation by Faber series. Our unified point of view is made possible by the the concept of transmission.

The hyperbolicity statements for subvarieties and complements of hypersurfaces in abelian varieties admit arithmetic analogues, due to Faltings (and Vojta for the semi-abelian case). In Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018) by the second author, an analogy between the analytic and arithmetic theories was shown to hold also at proof level, namely in a proof of Raynaud's theorem (Manin-Mumford Conjecture). The first aim of this paper is to extend to the relative setting the above mentioned hyperbolicity results. We shall be concerned with analytic sections of a relative (semi-)abelian scheme over an affine algebraic curve. These sections form a group; while the group of rational sections (the Mordell-Weil group) has been widely studied, little investigation has been pursued so far on the group of the analytic sections. We take the opportunity of developing some basic structure of this apparently new theory, defining a notion of height or order functions for the analytic sections, by means of Nevanlinna theory.

Special atom spaces have been around for quite awhile since the introduction of atoms by R. Coifman in his seminal paper who led to another proof that the dual of the Hardy space H 1 is in fact the space of functions of bounded means oscillations (BMO). Special atom spaces enjoy quite a few attributes of their own, among which the fact that they have an analytic extension to the unit disc. Recently, an extension of special atom spaces to higher dimensions was proposed, making ripe the possible exploration of the above extension in higher dimensions. In this paper we propose an analytic characterization of special atom spaces in higher dimensions.

We find an unifying approach to the analytic representation of the domain bounded by a generalized Pascal snail. Special cases as Pascal snail, Both leminiscate, conchoid of the Sluze and a disc are included. The behavior of functions related to generalized Pascal snail are demonstrated.

Descriptive set theory can be used to prove new results in classical complex analysis. Let A(Ω) be the set of complex analytic functions on an open subset Ω?�C endowed with the usual topology of uniform convergence on compact subsets. A(Ω) is a Polish ring with the operations of point-wise addition and point-wise multiplication. Inspired by Bers' algebraic characterization of the relation of conformality, we show that the topology on A(Ω) is the only Polish topology for which A(Ω) is a Polish ring for arbitrary open Ω . In a different direction, we show that the bounded complex-analytic functions on the unit disk admits no Polish topology for which it is a Polish ring. Along these lines, a corollary of our general result is that the abstract field of meromorphic functions on an open Ω cannot be made into a Polish field. We also study the Lie ring structure on A(Ω) which turns out to be a Polish Lie ring with the usual topology. In this case, we restrict our attention to those domains Ω that are connected. We extend a result of Amemiya to see that the Lie ring structure is determined by the conformal structure of Ω . In a similar vein to our ring considerations, we see that, for certain domains Ω of usual interest, the Lie ring A(Ω) has a unique Polish topology for which it is a Polish Lie ring.

In Mergelyan type approximation we uniformly approximate functions on compact sets K by polynomials or rational functions or holomorphic functions on varying open sets containing K. In the present paper we consider analogous approximation, where uniform convergence on K is replaced by uniform approximation on K of all order derivatives.