A. Sukhov, A. Tumanov
We prove a result on removing singularities of almost complex structures pulled back by a non-diffeomorphic map. As an application we prove the existence of global J-holomorphic discs with boundaries attached to real tori.
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Under an assumption of normal genericity, we show that a stable J-holomorphic curve has, in the space of homologous curves of the same genus, a locally Euclidean neighbourhood of the expected dimension given by Riemann-Roch. In dimension 4, the normal genericity condition is satisfied in by every curve in CP2 (for an almost complex structure homotopic with the standard one) which has only nodes as singularities. This leads in particular to a solution of the symplectic isotopy problem for surfaces of degree 3.
We provide a local approximation result of non-holomorphic discs with small d-bar by pseudoholomorphic ones. As an application, we provide a certain gluing construction.
In our previous work, for a given Riemann surface $Y_0$ with marked handle, we investigated geometric properties of the set of marked once-holed tori $X$ allowing holomorphic mappings of $X$ into $Y_0$. It turned out that it is a closed domain with Lipschitz boundary. In the present paper we show that the boundary is never smooth. Also, we evaluate the critical extremal length for the existence of holomorphic mappings in terms of hyperbolic lengths.
We prove that some holomorphic functions on the moduli space of tori have only simple zeros. Instead of computing the derivative with respect to the moduli parameter $τ$, we introduce a conceptual proof by applying Painlevé VI\ equation. As an application of this simple zero property, we obtain the smoothness of all the degeneracy curves of trivial critical points for some multiple Green function.
Given a generic totally real torus unknotted in the unit sphere of the complex plane, we prove the following alternative : either there exists a filling of the torus by holomorphic discs and the torus is rationally convex, or its rational hull contains a holomorphic annulus.
The purpose of this paper is to develop the theory of holomorphic gerbes on complex tori in a manner analogous to the classical theory for line bundles. In contrast to past studies on this subject, we do not restrict to the case where these gerbes are torsion or topologically trivial. We give an Appell-Humbert type description of all holomorphic gerbes on complex tori. This gives an explicit, simple, cocycle representative (and hence gerbe) for each equivalence class of holomorphic gerbes. We also prove that a gerbe on the fiber product of four spaces over a common base is trivial as long as i…
The Index theorem for holomorphic line bundles on complex tori asserts that some cohomology groups of a line bundle vanish according to the signature of the associated hermitian form. In this article, this theorem is generalized to quasi-tori, i.e. connected complex abelian Lie groups which are not necessarily compact. In view of the Remmert-Morimoto decomposition of quasi-tori as well as the Künneth formula, it suffices to consider only Cousin-quasi-tori, i.e. quasi-tori which have no non-constant holomorphic functions. The Index theorem is generalized to holomorphic line bundles, both linear…
As in the case of irreducible holomorphic symplectic manifolds, the period domain $Compl$ of compact complex tori of even dimension $2n$ contains twistor lines. These are special $2$-spheres parametrizing complex tori whose complex structures arise from a given quaternionic structure. In analogy with the case of irreducible holomorphic symplectic manifolds, we show that the periods of any two complex tori can be joined by a {\em generic} chain of twistor lines. We also prove a criterion of twistor path connectivity of loci in $Compl$ where a fixed second cohomology class stays of Hodge type (1…
The configuration space C^n of unordered n-tuples of distinct points on a torus T^2 is a non-singular complex algebraic variety. We study holomorphic self-maps of C^n and prove that for n>4 any such map F either carries the whole of C^n into an orbit of the diagonal Aut(T^2) action in C^n or is of the form F(x)=T(x)x for some holomorphic map T:C^n-->Aut(T^2). We also prove that for n>4 any endomorphism of the torus braid group B_n(T^2) with a non-abelian image preserves the pure torus braid group PB_n(T^2).
Universal holomorphic maps with slow growth I. An Algorithm
本文提出一种算法，可在任意复欧几里得空间间构造满足预设超越增长率的通用全纯映射，并得到到复射影空间与复环面的缓慢增长映射。
We study the global invariants of real analytic manifolds in the complex space with respect to the group of holomorphic unimodular transformations. We consider only totally real manifolds which admits a certain fibration over the circle. We find a complete set of invariants for totally real tori in C^n which are close to the standard torus. The invariants are obtained by an analogous classification of complex-valued analytic $n$-forms on the standard torus. We also study the realization of certain exact complex-valued analytic $n$-forms on the standard torus through non-critical totally real e…
A holomorphic (2, 2)-Theorem for Abelian varieties of CM-type
该研究针对CM型阿贝尔簇，证明了第二Milnor K群解析层的第二上同调群到对应Hodge型有理上同调类的映射在不计分母时为满射。
Oka complements in algebraic tori and their logarithmic covers
该研究证明代数环面中紧多项式凸集K对应的补集X_K及其对数覆盖Y_K均为Oka集，Y_K还允许指定全纯截面的归一化全纯嵌入族，相关方法可应用于构造多种Oka集的补集。
Ueda foliation problem for complex tori
研究复环面嵌入复流形时的 Ueda 叶状结构问题，在法丛为酉平坦等条件下，证明了其邻域内存在以该复环面为叶的非奇异全纯叶状结构。
Let D be a relatively compact strongly pseudoconvex domain in a Stein manifold, and let Y be a complex manifold. We prove that the set A(D,Y), consisting of all continuous maps from the closure of D to Y which are holomorphic in D, is a complex Banach manifold. When D is the unit disc in C (or any other topologically trivial strongly pseudoconvex domain in a Stein manifold), A(D,Y) is locally modeled on the Banach space A(D,C^n)=A(D)^n with n=dim Y. Analogous results hold for maps which are holomorphic in D and of class C^r up to the boundary for any positive integer r. We also establish the O…
We study the existence of topologically closed complex curves normalized by bordered Riemann surfaces in complex spaces. Our main result is that such curves abound in any noncompact complex space admitting an exhaustion function whose Levi form has at least two positive eigenvalues at every point outside a compact set, and this condition is essential. The proof involves a lifting method for the boundary of the curve and a newly developed technique of gluing holomorphic sprays over Cartan pairs in Stein manifolds whose value lie in a complex space, with control up to the boundary of the domains…
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