Related papers: Monodromy problem and Tangential center-focus prob…

The central fiber of a Gross-Siebert type toric degeneration is known to satisfy homological mirror symmetry: its category of coherent sheaves is equivalent to the wrapped Fukaya category of a certain exact symplectic manifold. Here we show…

We upgrade the classical operation of \textit{isomonodromic deformations} along a path $\gamma$ to a functor $\mathbb{P}_{\gamma}$ between categories of flat connections with logarithmic singularities along a divisor $D$, which itself…

Working in the Palatini formalism, we describe a procedure for constructing degenerate solutions of general relativity on 4-manifold M from certain solutions of 2-dimensional BF theory on any framed surface Sigma embedded in M. In these…

This paper is a summary of author's results on finite flat commutative group schemes. The properties of the generic fibre functor are discussed. A complete classification of finite local flat commutative group schemes over mixed…

Given a simply connected manifold $M$, we completely determine which rational monomial Pontryagin numbers are attained by fiber homotopy trivial $M$-bundles over the $k$-sphere, provided that $k$ is small compared to the dimension of $M$.…

Let \ $\lambda \in \mathbb{Q}^{*+}$ \ and consider a multivalued formal function of the type $$ \phi(s) : = \sum_{j=0}^k \ c_j(s).s^{\lambda + m_j}.(Log\, s)^j $$ where \ $c_j \in \C[[s]], m_j \in \mathbb{N}$ \ for \ $j \in [0,k-1]$. The…

Let f be a hypersurface surface local singularity whose zero set has 1-dimensional singular locus. We develop an explicit procedure that provides the boundary of the Milnor fibre of f as an oriented plumbed 3-manifold. The method provides…

We compute the monodromy of the Hitchin fibration for the moduli space of $L$-twisted $SL(n,\mathbb{C})$ and $GL(n,\mathbb{C})$-Higgs bundles for any $n$, on a compact Riemann surface of genus $g>1$. We require the line bundle $L$ to either…

We consider monodromy groups of the generalized hypergeometric equation \begin{equation*} \big[z(\theta+\alpha_{1})\cdots (\theta+\alpha_{n})-(\theta+\beta_{1}-1)\cdots (\theta+\beta_{n}-1)\big]f(z) = 0\text{, where }\theta = z d/dz,…

We compute the subgroup of the monodromy group of a generalized Kummer variety associated to equivalences of derived categories of abelian surfaces. The result was previously announced in arXiv:1201.0031. Mongardi showed that the subgroup…

We use pluriharmonic maps to study representations of fundamental groups of algebraic manifolds. This approach is functorial in the sense that the restriction of such a map to a fiber of a fibration remains pluriharmonic, and on this basis,…

We show that the generic fiber of a family of smooth $\mathbb{A}^{1}$-ruled affine surfaces always carries an $\mathbb{A}^{1}$-fibration, possibly after a finite extension of the base. In the particular case where the general fibers of the…

In this paper we use motivic integration and non-archimedean analytic geometry to study the singularities at infinity of the fibers of a polynomial map $f\colon \mathbb A^d_\mathbb C \to \mathbb A^1_\mathbb C$. We show that the motive…

We classify the orientably regular maps which are elementary abelian regular branched coverings of Platonic maps M, in the case where the covering group and the rotation group G of M have coprime orders. The method involves studying the…

The current article studies certain problems related to complex cycles of holomorphic foliations with singularities in the complex plane. We focus on the case when polynomial differential one-form gives rise to a foliation by Riemann…

We classify fibrations by integral plane projective rational quartic curves whose generic fibre is regular but admits a non-smooth point that is a canonical divisor. These fibrations can only exist in characteristic two. The geometric…

We present the expressions for the monodromy matrix elements of the six-vertex model in the F-basis for arbitrary Boltzmann weights. The results rely solely on the property of unitarity and Yang-Baxter relations, avoiding any specific…

We establish a connection between the representation theory of certain noncommutative singular varieties and two-dimensional lattice models. Specifically, we consider noncommutative biparametric deformations of the fiber product of two…

We consider some algebraic aspects of the dynamics of an automorphism on a family of polarized abelian varieties parameterized by the complex unit disk. When the action on the cohomology of the generic fiber has no cyclotomic factor, we…

For $\Gamma$ a finite subgroup of $\mathrm{SL}_2(\mathbb{C})$ and $n \geq 1$, we study the fibers of the Procesi bundle over the $\Gamma$-fixed points of the Hilbert scheme of $n$ points in the plane. For each irreducible component of this…