Related papers: Meromorphic Convexity on Complex Manifolds

We define a new class of orthogonal polyhedra, called orthogrids, that can be unfolded without overlap with constant refinement of the gridded surface.

The richly developed theory of complex manifolds plays important roles in our understanding of holomorphic functions in several complex variables. It is natural to consider manifolds that will play similar roles in the theory of holomorphic…

The classical theory of elliptic curves with complex multiplication is a fundamental tool for studying the arithmetic of abelian extensions of imaginary quadratic fields. While no direct analogue is available for real quadratic fields, a…

On a complex symplectic manifold, we construct the stack of quantization-deformation modules, that is, (twisted) modules of microdifferential operators with an extra central parameter, a substitute to the lack of homogeneity. We also…

The main topic is the development of a Fredholm theory in a new class of spaces called M-polyfolds. In the subsequent Volume II the theory will be generalized to an even larger class of spaces called polyfolds, which can also incorporate…

We consider closed symplectically aspherical manifolds, i.e. closed symplectic manifolds $(M,\omega)$ satisfying the condition $[\omega]|_{\pi_2M}=0$. Rudyak and Oprea [RO] remarked that such manifolds have nice and controllable homotopy…

We extend Matveev's complexity of 3-manifolds to PL compact manifolds of arbitrary dimension, and we study its properties. The complexity of a manifold is the minimum number of vertices in a simple spine. We study how this quantity changes…

Round fold maps are smooth maps on closed manifolds which are locally represented as the product maps of Morse functions and identity maps on open disks and whose singularity is realized as concentrically embedded spheres. The author…

In this paper, a new class of convex functions as a generalization of convexity which is called (h-m)-convex functions and some properties of this class is given. We also prove some Hadamard's type inequalities.

A general problem in complex cobordism theory is to find useful representatives for cobordism classes. One particularly convenient class of complex manifolds consists of smooth projective toric varieties. The bijective correspondence…

Stable fold maps are fundamental tools in a generalization of the theory of Morse functions on smooth manifolds and its application to studies of topological properties of smooth manifolds. Round fold maps were introduced as stable fold…

Fold maps are fundamental tools in the theory of singularities of differentiable maps and its applications to geometry. They are higher dimensional variants of Morse functions. Classes of special generic maps and round fold maps are…

Thirty research questions on meromorphic functions and complex differential equations are listed and discussed. The main purpose of this paper is to make this collection of problems available to everyone.

It is often noted that many of the basic concepts of differential geometry, such as the definition of connection, are purely algebraic in nature. Here, we review and extend existing work on fully algebraic formulations of differential…

A general invariant manifold theorem is needed to study the topological classes of smooth dynamical systems. These classes are often invariant under renormalization. The classical invariant manifold theorem cannot be applied, because the…

We introduce a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. We prove for instance that if the fundamental group of a compact aspherical manifold M has FDC, and if N is…

We introduce the notion of symplectic microfolds and symplectic micromorphisms between them. They form a monoidal category, which is a version of the "category" of symplectic manifolds and canonical relations obtained by localizing them…

We argue for more widespread use of manifold-like polyfolds (M-polyfolds) as differential geometric objects. M-polyfolds possess a distinct advantage over differentiable manifolds, enabling a smooth and local change of dimension. To…

The notions of holomorphic symplectic structures and hypercomplex structures on Courant algebroids are introduced and then proved to be equivalent. These generalize hypercomplex triples and holomorphic symplectic 2-forms on manifolds…

We investigate the meromorphic quasi-modular forms and their $L$-functions. We study the space of meromorphic quasi-modular forms. Then we define their $L$-functions by using the technique of regularized integral. Moreover, we give an…