Related papers: Algebraic tori in the complement of quartic surfac…

Let $\k$ be a characteristic zero PID, $S$ be a $\k$-algebra and $T\subseteq S$ be a full rank subalgebra. Suppose the algebra $T$ is symmetric. It is important to know when $T$ is a {\em maximal symmetric subalgebra} of $S$, i.e. no…

We give a classification of the equivariant principal $G$-bundles on a nonsingular toric variety when $G$ is a closed Abelian subgroup of $GL_k(\mathbb{C})$. We prove that any such bundle splits, that is, admits a reduction of structure…

We prove two results regarding Hodge structures appearing in the cohomology of complex tori. First, we prove that if a polarizable Hodge structure appears in the cohomology of a complex torus $T$, it appears in the cohomology of an abelian…

Given a strictly unbounded toric symplectic 4-manifold, we explicitly construct complete toric scalar-flat K\"ahler metrics on the complement of a toric divisor. These symplectic 4-manifolds correspond to a specific class of non-compact…

We study the solvability of a class of fully nonlinear equations on the flat torus. The equations arise in the study of some Calabi-Yau type problems in torus bundles.

We classify semi-algebraic surfaces in $\mathbb{R}^n$ with isolated singularities up to bi-Lipschitz homeomorphisms with respect to the inner distance. In particular, we obtain complete classifications for the Nash surfaces and the complex…

We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra does not depend on the choice of…

We extract a nonnegative integer-valued invariant, which we call the "order of algebraic torsion", from the Symplectic Field Theory of a closed contact manifold, and show that its finiteness gives obstructions to the existence of symplectic…

Using the theory of cohomology support locus, we give a necessary condition for the Albanese map of a smooth projective surface being a submersion. More precisely, assuming the cohomology support locus of any finite abelian cover of a…

We compute the integral homology (including torsion), the topological K-theory, and the Hodge structure on cohomology of Calabi-Yau threefold hypersurfaces and complete intersections in Gorenstein toric Fano varieties. The methods are…

Using the well-known recognition and structural theorem(s) for root-graded Lie algebras and their universal coverings, we give a finite presentation for the universal covering algebra of a centerless Lie torus of type $X\not=A,C,BC$. We…

In this paper we study a $k$-dimensional analytic subvariety of the complex algebraic torus. We show that if its logarithmic limit set is a finite rational $(k-1)$-dimensional spherical polyhedron, then each irreducible component of the…

We study the geometry and arithmetic of the curves $C \colon y^3 = x^4 + ax^2 + b$ and their associated Prym abelian surfaces $P$. We prove a Torelli theorem in this context and give a geometric proof of the fact that $P$ has quaternionic…

In this article we fully classify regular tubular surfaces in Euclidean, Lorentzian and hyperbolic 3-spaces whose Gaussian and mean curvatures $K$ and $H$ verify a polynomial relation. More precisely, we determine the set $S(Q)$ of all…

We study K3 surfaces with 9 cusps, i.e. 9 disjoint $A_2$ configurations of smooth rational curves, over algebraically closed fields of characteristic $p\neq 3$. Much like in the complex situation studied by Barth, we prove that each such…

Let G be a connected linear algebraic group over a field k. We say that G is toric-friendly if for any field extension K/k and any maximal K-torus T in G the group G(K) has only one orbit in (G/T)(K). Our main result is a classification of…

We characterize isotropic trialitarian triples in terms of the Schur indices of the underlying algebras over a base field $F$ of arbitrary characteristic satisfying $I_q^3 F=0$. We also construct anisotropic trialitarian triples over such…

We prove equivalent numerical conditions for a complete spherical variety to admit a toric structure, and for the smoothness of an arbitrary spherical variety along any given G-orbit. The conditions are in terms of spherical skeletons, a…

For a finite-type surface $\mathfrak{S}$, we study a preferred basis for the commutative algebra $\mathbb{C}[\mathscr{R}_{\mathrm{SL}_3(\mathbb{C})}(\mathfrak{S})]$ of regular functions on the $\mathrm{SL}_3(\mathbb{C})$-character variety,…

We discuss logical links among uniformity conjectures concerning K3 surfaces and abelian varieties of bounded dimension defined over number fields of bounded degree. The conjectures concern the endomorphism algebra of an abelian variety,…