Modular deformations of analytic polyhedra

Deformations of period lattices of flexible polyhedral surfaces

In the end of the 19th century Bricard discovered a phenomenon of flexible polyhedra, that is, polyhedra with rigid faces and hinges at edges that admit non-trivial flexes. One of the most important results in this field is a theorem of Sabitov asserting that the volume of a flexible polyhedron is constant during the flexion. In this paper we study flexible polyhedral surfaces in the 3-space two-periodic with respect to translations by two non-colinear vectors that can vary continuously during the flexion. The main result is that the period lattice of a flexible two-periodic surface homeomorph…

Monodromy zeta-function of a polynomial on a complete intersection and Newton polyhedra

For a generic (polynomial) one-parameter deformation of a complete intersection, there is defined its monodromy zeta-function. We provide explicit formulae for this zeta-function in terms of the corresponding Newton polyhedra in the case the deformation is non-degenerate with respect to its Newton polyhedra. Using this result we obtain the formula for the monodromy zeta-function at the origin of a polynomial on a complete intersection, which is an analog of the Libgober--Sperber theorem.

Hessian polyhedra, invariant theory and Appell hypergeometric partial differential equations

It is well-known that Klein's lectures on the icosahedron and the solution of equations of fifth degree is one of the most important and influential books of 19th-century mathematics. In the present paper, we will give the complex counterpart of Klein's book, i.e., a story about complex regular polyhedra. We will show that the following four apparently disjoint theories: the symmetries of the Hessian polyhedra (geometry), the resolution of some system of algebraic equations (algebra), the system of partial differential equations of Appell hypergeometric functions (analysis) and the modular equ…

On a $q$-deformation of modular forms

There are many instances known when the Fourier coefficients of modular forms are congruent to partial sums of hypergeometric series. In our previous work arXiv:1803.01830, such partial sums are related to the radial asymptotics of infinite $q$-hypergeometric sums at roots of unity. Here we combine the two features to construct a hypergeometric $q$-deformation of two CM modular forms of weight 3 and discuss the corresponding $q$-congruences.

形变广义Kummer簇模空间的一般型结果

General type results for moduli of deformation generalised Kummer varieties

本文证明了形变广义Kummer簇模空间相关的正交模簇在2d足够大时为一般型，并去除了先前结果中的无平方因子条件。

One-parameter families containing three-dimensional toric Gorenstein singularities

For affine toric varieties, the vector space T1 (containing the infinitesimal deformations) will be interpreted via Minkowski summands of cross cuts of the defining polyhedral cone. This result will be applied to study the deformation theory of (in particular non-isolated) three-dimensional Gorenstein singularities.