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

We study log Calabi--Yau pairs of the form $(\mathbb{P}^3,\Delta)$, where $\Delta$ is a quartic surface, and classify all such pairs of coregularity less than or equal to one, up to volume preserving equivalence. In particular, if…

We describe, for a few small examples, the Kauffman bracket skein algebra of a surface crossed with an interval. If the surface is a punctured torus the result is a quantization of the symmetric algebra in three variables (and an algebra…

We study degenerations of cluster type varieties and pairs. Our first theorem proves that degenerations of toric pairs are finite quotients of toric pairs. In a similar vein, under some mild conditions, we prove that degenerations of…

A Calabi-Yau pair of index one and complexity zero is toric. Furthermore, a Calabi-Yau pair of index one and complexity one is of cluster type. In this article, we study Calabi-Yau pairs of index one and complexity two. We develop machinery…

For an unpunctured marked surface $\Sigma$, we consider a skein algebra $\mathscr{S}_{\mathfrak{sl}_{3},\Sigma}^{q}$ consisting of $\mathfrak{sl}_3$-webs on $\Sigma$ with the boundary skein relations at marked points. We construct a quantum…

In this expository note, we review the standard formulation of mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, and compare this construction to a description of mirror symmetry for K3 surfaces which relies on a sublattice…

In this article, we introduce the generalized complexity of a generalized Calabi--Yau pair $(X,B,\textbf{M})$. This invariant compares the dimension of $X$ and Picard rank of $X$ with the sum of the coefficients of $B$ and $\textbf{M}$. It…

This paper aims to study the birational geometry of log Calabi-Yau pairs$(\mathbb{P}^3, D)$ of coregularity 2, where in this case $D$ is an irreducible normal quartic surface with canonical singularities. We completely classify which toric…

Let $K/k$ be a finite Galois extension and $\pi = \fn{Gal}(K/k)$. An algebraic torus $T$ defined over $k$ is called a $\pi$-torus if $T\times_{\fn{Spec}(k)} \fn{Spec}(K)\simeq \bm{G}_{m,K}^n$ for some integer $n$. The set of all algebraic…

We prove a flat torus theorem for quadric complexes. In particular, we show that if a non-cyclic free abelian group $G$ acts metrically properly on a quadric complex $X$, then $G \cong \mathbb{Z}^2$ and $X$ contains a $G$-invariant…

By a K3-surface with nine cusps I mean a compact complex surface with nine isolated double points $A_2$, but otherwise smooth, such that its minimal desingularisation is a K3-surface. In an earlier paper I showd that each such surface is a…

We give a local analytic characterization that a minimal surface in the 3-sphere $\, \ES^3 \subset \R^4$ defined by an irreducible cubic polynomial is one of the Lawson's minimal tori. This provides an alternative proof of the result by…

We study periodic torus orbits on spaces of lattices. Using the action of the group of adelic points of the underlying tori, we define a natural equivalence relation on these orbits, and show that the equivalence classes become uniformly…

In this paper, we construct a new even constrained B(C) type Toda hierarchy and derive its B(C) type Block type additional symmetry. Also we generalize the B(C) type Toda hierarchy to the $N$-component B(C) type Toda hierarchy which is…

We show that under a suitable transversality condition, the intersection of two rational subtori in an algebraic torus $(\C^*)^n$ is a finite group which can be determined using the torsion part of some associated lattice. Applications are…

We study the symplectic topology of certain K3 surfaces (including the "mirror quartic" and "mirror double plane"), equipped with certain K\"ahler forms. In particular, we prove that the symplectic Torelli group may be infinitely generated,…

A very interesting problem in the classical theory of minimal surfaces consists of the classification of such surfaces under some geometrical and topological constraints. In this short paper, we give a brief summary of the known…

We give a geometric interpretation of cluster varieties in terms of blowups of toric varieties. This enables us to provide, among other results, an elementary geometric proof of the Laurent phenomenon for cluster algebras (of geometric…

Every Maslov-zero Lagrangian torus in a K3 surface has non-trivial homology class. This note aims to extend this result to Lagrangian tori with Maslov indices congruent to zero modulo 4. Conversely, we show that every homologically…

We construct and study a family of toric degenerations of the algebra of conformal blocks for a stable marked curve $(C, \vec{p})$ with structure group $SL_3(\mathbb{C}).$ We find that this algebra is Gorenstein. For the genus $0, 1$ cases…