Related papers: Rank 2 integrable systems of Prym varieties

Using previous results concerning the rank two and rank three cases, all connected simply connected Cartan schemes for which the real roots form a finite irreducible root system of arbitrary rank are determined. As a consequence one obtains…

The contributions of Sophya Kowalewski to the integrability theory of the equations for the heavy top extend to a larger class of Hamiltonian systems on Lie groups; this paper explains these extensions, and along the way reveals further…

A complex integrable system determines a family of complex tori over a Zariski-open and dense subset in its base. This family in turn yields an integral variation of Hodge structures of weight $\pm 1$. In this paper, we study the converse…

We define relative Ruan invariants that count embedded connected symplectic submanifolds which contact a fixed stable symplectic hypersurface V in a symplectic 4-manifold (X,w) at prescribed points with prescribed contact orders (in…

Let $\mathcal{X}\rightarrow C$ be a dominant morphism between smooth irreducible varieties over a finitely generated field $k$ such that the generic fiber $X$ is smooth, projective and geometrically connected. Assuming that $C$ is a curve…

Construction and classification of 2D superintegrable systems (i.e. systems admitting, in addition to two global integrals of motion guaranteeing the Liouville integrability, the third global and independent one) defined on 2D spaces of…

This is the second paper of a series dedicated to the study of Poisson structures of compact types (PMCTs). In this paper, we focus on regular PMCTs, exhibiting a rich transverse geometry. We show that their leaf spaces are integral affine…

For a Lagrangian torus A in a simply-connected projective symplectic manifold M, we prove that M has a hypersurface disjoint from a deformation of A. This implies that a Lagrangian torus in a compact hyperk\"ahler manifold is a fiber of an…

One can associate to many of the well known algebraically integrable systems of Jacobians (generalized Hitchin systems, Sklyanin) a ruled surface which encodes much of its geometry. If one looks at the classification of such surfaces, there…

It is known (work of Galluzzi, Lombardo, Dolgachev and Naruki) that there is a unique K3 surface X which corresponds to a genus 2 curve C such that X has a Shioda-Inose structure with quotient birational to the Kummer surface of the…

Abundant second-order maximally conformally superintegrable Hamiltonian systems are re-examined, revealing their underlying natural Weyl structure and offering a clearer geometric context for the study of St\"ackel transformations (also…

We study limiting cases of the two known integrable chiral-type models with tree-dimensional configuration space. One of the initial models is the non-Abelian Toda $A_2^{(1)}$ model and the other was found by means of the symmetry approach…

Let $G=H\ltimes K$ denote a semidirect product Lie group with Lie algebra $\mathfrak g=\mathfrak h \oplus \mathfrak k$, where $\mathfrak k$ is an ideal and $\mathfrak h$ is a subalgebra of the same dimension as $\mathfrak k$. There exist…

Let $k$ be a field of characteristic zero containing a primitive fifth root of unity. Let $X/k$ be a smooth cubic threefold with an automorphism of order five, then we observe that over a finite extension of the field actually the dihedral…

The classical Liouville Theorem on conformal transformations determines local conformal transformations on the Euclidean space of dimension $\geq 3$. Its natural adaptation to the general framework of Riemannian structures is the 2-rigidity…

In a talk at the Banff International Research Station in 2015 Asher Auel asked questions about genus one curves in Severi-Brauer varieties $SB(A)$. More specifically he asked about the smooth cubic curves in Severi-Brauer surfaces, that is…

The paper is devoted to the study of exact curves on Arborealized Liouville surfaces. We introduce the notion of a generating family for such curves. Our main statement is a hamiltonian lifting property: the set of curves admitting a…

Let E be a transitive Courant algebroid with scalar product of neutral signature. A generalized almost complex structure \mathcal J on E is a skew-symmetric smooth field of endomorphisms of E which squares to minus the identity. We say that…

Let $k$ be a field of characteristic zero containing a primitive $n$-th root of unity. Let $C^0_n$ be a singular plane curve of degree $n$ over $k$ admitting an order $n$ automorphism, $n$ nodes as the singularities, and $C_n$ be its…

We introduce orthogonal ring patterns in the 2-sphere and in the hyperbolic plane, consisting of pairs of concentric circles, which generalize circle patterns. We show that their radii are described by a discrete integrable system. This is…