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## Monodromies at infinity of confluent A-hypergeometric functions

Advances in Mathematics. 2015. Vol. 272. P. 1–19.

Esterov A. I., Takeuchi K., Ando K.

We study the monodromies at infinity of confluent A-hypergeometric functions introduced by Adolphson. In particular, we compute the monodromy zeta-function.

Publication based on the results of:

Esterov A. I., Takeuchi K., / Cornell University. Series math "arxiv.org". 2014.

We prove some vanishing theorems for the cohomology groups of local systems associated to Laurent polynomials. In particular, we extend one of the results of Gelfand-Kapranov-Zelevinsky into various directions. ...

Added: October 10, 2014

V. V. Shevchishin, Izvestiya. Mathematics 2009 Vol. 73 No. 4 P. 797–859

In this paper we prove the non-existence of Lagrangian embeddings of the Klein bottle K in R4 and CP2. We exploit the existence of a special embedding of K in a symplectic Lefschetz pencil pr:X→S2 and study its monodromy. As the main technical tool, we develop the combinatorial theory of mapping class groups. The results ...

Added: March 18, 2013

Lvovsky S., / Cornell University. Series math "arxiv.org". 2013. No. 1305.2205.

We show that using an idea from a paper by Van de Ven one may obtain a simple proof of Zak's classification of smooth projective surfaces with zero vanishing cycles. This method of proof allows one to extend Zak's theorem to the case of finite characteristic. ...

Added: October 3, 2013

Vyugin I. V., Левин Р. И., Труды Математического института им. В.А. Стеклова РАН 2017 Т. 297 С. 326–343

An analog of the classical Riemann-Hilbert problem formulated for classes of difference and q-difference systems is considered. We propose some strengthening of Birkhoff's existence theorem. ...

Added: August 18, 2017

Vyugin I. V., Гонцов Р. Р., Успехи математических наук 2012 Т. 67 № 3 (405) С. 183–184

Получено обобщение результата Ильяшенко-Хованского, утверждающего, что разрешимость в квадратурах фуксовой системы с малыми коэффициентами эквивалентна ее треугольности. В работе этот результат обобщен на случай систем с малыми собственными значениями матриц вычетов. ...

Added: February 21, 2013

Esterov A. I., Gusev G. G., Mathematische Annalen 2016 Vol. 365 No. 3 P. 1091–1110

We generalize the Abel–Ruffini theorem to arbitrary dimension, i.e. classify general square systems of polynomial equations solvable by radicals. In most cases, they reduce to systems whose tuples of Newton polytopes have mixed volume not exceeding 4. The proof is based on topological Galois theory, which ensures non-solvability by any formula involving quadratures and single-valued ...

Added: February 27, 2017

Serge Lvovski, / Cornell University. Series arXiv "math". 2017.

We show that the monodromy group acting on $H^1(\cdot,\mathbb Z)$ of a smooth
hyperplane section of a del Pezzo surface over $\mathbb C$ is the entire
group $\mathrm{SL}_2(\mathbb Z)$. For smooth surfaces with $b_1=0$ and hyperplane section
of genus $g>2$, there exist examples in which a similar assertion is
false. Actually, if hyperplane sections of ...

Added: June 14, 2017

Serge Lvovski, Manuscripta Mathematica 2014 Vol. 145 P. 235–242

We show that using an idea from a paper by Van de Ven one may obtain a
simple proof of Zak's classification of smooth projective surfaces
with zero vanishing cycles. This method of proof allows one to extend
Zak's theorem to the case of finite characteristic. ...

Added: October 14, 2014

Gusein-Zade S., Journal of Algebra and its Applications 2018 Vol. 17 No. 10 P. 1–13

In a previous paper, the authors defined an equivariant version of the so-called Saito duality between the monodromy zeta functions as a sort of Fourier transform between the Burnside rings of an abelian group and of its group of characters. Here, a so-called enhanced Burnside ring Bˆ(G) of a finite group G is defined. An ...

Added: October 27, 2020

Yu. Burman, Serge Lvovski, Moscow Mathematical Journal 2015 Vol. 15 No. 1 P. 31–48

Suppose that C ⊂ P^2 is a general enough nodal plane curve
of degree > 2, : \hat C → C is its normalization, and π: C′ → P^1 is a finite
morphism simply ramified over the same set of points as a projection
pr_p ◦ν : \hat C → P1, where p ∈ P^2 ...

Added: January 14, 2015

Serge Lvovski, Moscow Mathematical Journal 2019 Vol. 19 No. 3 P. 597–613

We show that if we are given a smooth non-isotrivial family of curves of genus 1 over C with a smooth base B for which the general fiber of the mapping J : B → A 1 (assigning j-invariant of the fiber to a point) is connected, then the monodromy group of the family (acting ...

Added: August 30, 2019

Khoroshkin S. M., Tsuboi Z., Journal of Physics A: Mathematical and Theoretical 2014 Vol. 47 P. 1–11

We consider the 'universal monodromy operators' for the Baxter Q-operators. They are given as images of the universal R-matrix in oscillator representation. We find related universal factorization formulas in the Uq(\hat{sl}(2)) case. ...

Added: December 8, 2014

Takeuchi K., Esterov A. I., Lemahieu A., / Cornell University. Series math "arxiv.org". 2016. No. arXiv:1309.0630v4.

Recently the second author and Van Proeyen proved the monodromy conjecture on topological zeta functions for all non-degenerate surface singularities. In this paper, we obtain higher-dimensional analogues of their results, which, in particular, prove the conjecture for all isolated singularities of 4 variables, as well as for many classes of non-isolated and higher-dimensional singularities. One ...

Added: September 18, 2017

Brav C. I., Thomas H., Compositio Mathematica 2014 Vol. 150 No. 3 P. 343–333

We show that some hypergeometric monodromy groups in Sp(4,Z) split as free or amalgamated products and hence by cohomological considerations give examples of Zariski dense, non-arithmetic monodromy groups of real rank 2. In particular, we show that the monodromy of the natural quotient of the Dwork family of quintic threefolds in P^{4} splits as Z*Z/5. ...

Added: September 29, 2014

Esterov A. I., Takeuchi K., American Journal of Mathematics 2015 Vol. 137 No. 2 P. 365–409

We study confluent A-hypergeometric functions introduced by Adolphson. In particular, we give their integral representations by using rapid decay homology cycles of Hien and obtain a formula for the asymptotic expansions at infinity of confluent A-hypergeometric functions. ...

Added: October 10, 2014

Esterov A. I., Compositio Mathematica 2019 Vol. 155 No. 2 P. 229–245

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables.
In particular, our result proves the multivariate ...

Added: February 5, 2019

Vyugin I. V., Дудникова Л. А., Математический сборник 2024 Т. 215 № 2 С. 3–20

The paper is devoted to the study of holomorphic vector bundles with logarithmic connections on a compact Riemann surface and the application of the results obtained to the study of the question of positive solvability of the Riemann–Hilbert problem on a Riemann surface. We give an example of a representation of the fundamental group of a ...

Added: March 5, 2024

Lvovsky S., / Cornell University. Series arXiv "math". 2018.

We show that if we are given a smooth non-isotrivial family of elliptic curves over ℂ with a smooth base B for which the general fiber of the mapping J:B→𝔸^1 (assigning j-invariant of the fiber to a point) is connected, then the monodromy group of the family (acting on H1(⋅,ℤ) of the fibers) coincides with SL(2,ℤ); if the general fiber has m≥2 connected components, then the ...

Added: December 5, 2018

Бухштабер В. М., Glutsyuk A., Труды Математического института им. В.А. Стеклова РАН 2017 Т. 297 С. 62–104

Abstract—We study a family of double confluent Heun equations of the form LE = 0, where
L = L(λ,μ,n) is a family of second-order differential operators acting on germs of holomorphic
functions of one complex variable. They depend on complex parameters λ, μ, and n. The
restriction of the family to real parameters satisfying the inequality λ + μ^2>0 ...

Added: June 29, 2018

Frolenkov D., Быковский В. А., Дальневосточный математический журнал 2015 Т. 15 № 1 С. 38–40

Two integral representations for hypergeometric function are obtained. ...

Added: August 27, 2016

Kudryashov Y., Goncharuk N. B., Bulletin of the Brazilian Mathematical Society 2017 No. 1

In this article we prove in a new way that a generic polynomial vector field in ℂ² possesses countably many homologically independent limit cycles. The new proof needs no estimates on integrals, provides thinner exceptional set for quadratic vector fields, and provides limit cycles that stay in a bounded domain. ...

Added: April 15, 2016

Alexander Esterov, Lang L., Geometry and Topology 2021 Vol. 25 No. 6 P. 3053–3077

Let C_d be the space of non-singular, univariate polynomials of degree d. The Viète map V sends a polynomial to its unordered set of roots. It is a classical fact that the induced map V_∗ at the level of fundamental groups realises an isomorphism between π_1(C_d) and the Artin braid group B_d. For fewnomials, or equivalently for the intersection C of C_d with a collection of coordinate ...

Added: October 27, 2020

Esterov A. I., Takeuchi K., Lemahieu A., Journal of the European Mathematical Society 2021

The monodromy conjecture is an umbrella term for several conjectured relationships between poles of zeta functions, monodromy eigenvalues and roots of Bernstein-Sato polynomials in arithmetic geometry and singularity theory. Even the weakest of these relations --- the Denef--Loeser conjecture on topological zeta functions --- is open for surface singularities. We prove it for a wide ...

Added: November 28, 2020

Verbitsky M., Mehrotra S., Markman E., European Journal of Mathematics 2019 Vol. 5 No. 3 P. 964–1012

Let S be a K3 surface and M a smooth and projective 2n-dimensional moduli space of stable coherent sheaves on S. Over 𝑀×𝑀 there exists a rank 2𝑛−2 reflexive hyperholomorphic sheaf 𝐸_𝑀, whose fiber over a non-diagonal point (𝐹_1, 𝐹_2) is Ext^1_𝑆 (𝐹_1, 𝐹_2). The sheaf 𝐸_𝑀 can be deformed along some twistor path to a sheaf 𝐸_𝑋 over the Cartesian square 𝑋×𝑋 of every Kähler manifold X deformation equivalent to M. We prove that 𝐸_𝑋 is ...

Added: March 11, 2019