Related papers: A remark on toric foliations
This paper describes the structure of singular codimension one foliations with numerically trivial canonical bundle on projective manifolds.
Given a complex projective manifold $X$ and a divisor $D$ with normal crossings, we say that the logarithmic tangent bundle $T_X(-\log D)$ is R-flat if its pull-back to the normalization of any rational curve contained in $X$ is the trivial…
Toric geometry provides a bridge between the theory of polytopes and algebraic geometry: one can associate to each lattice polytope a polarized toric variety. In this thesis we explore this correspondence to classify smooth lattice…
To any compact K\"ahler manifold $(X, \omega)$ one may associate a bundle of affine spaces $Z_X\rightarrow X$ called a \emph{canonical extension} of $X$. In this paper we prove that if the tangent bundle of $X$ is nef, then the total space…
We consider classes of noncompact n-folds with trivial canonical bundle, that are linear foliations on nonsingular projective varieties, in general without a projection to the base. We obtain them as first-order deformations of total spaces…
We show that every smooth toric variety (and many other algebraic spaces as well) can be realized as a moduli space for smooth, projective, polarized varieties. Some of these are not quasi--projective. This contradicts a recent paper…
We introduce a collection of convex polytopes associated to a torus-equivariant vector bundle on a smooth complete toric variety. We show that the lattice points in these polytopes correspond to generators for the space of global sections…
We define the notion of a piecewise linear map from a fan $\Sigma$ to $\tilde{\mathfrak{B}}(G)$, the cone over the Tits building of a linear algebraic group $G$. Let $X_\Sigma$ be a toric variety with fan $\Sigma$. We show that when $G$ is…
In this paper, we investigate birational toric morphisms between quantum toric stacks -- namely, toric (analytic) stacks associated with fans whose cones may be irrational -- focusing on two primary classes of examples: weighted blow-ups…
Some diophantine aspects of projective toric varieties: We present several faces of projective toric varieties, of interest from the point of view of diophantine geometry. We make explicit the theory on a number of meaningful examples and…
Over a smooth projective toric variety we study toric sheaves, that is, reflexive sheaves equivariant with respect to the acting torus, from a polyhedral point of view. One application is the explicit construction of the torus invariant…
A toric variety is a normal complex variety which is completely described by combinatorial data, namely by a fan of strongly convex rational (with respect to a lattice) cones. Due to this rationality condition, toric varieties are…
We show that an equivariant vector bundle on a complete toric variety is nef or ample if and only if its restriction to every invariant curve is nef or ample, respectively. Furthermore, we show that nef toric vector bundles have a…
In this paper we prove the scalar curvature extremality and rigidity for a class of warped product spaces that are possibly degenerate at the two ends. The leaves of these warped product spaces can be any closed Riemannian manifolds with…
We conclude the classification of isoparametric (or equivalently, polar) foliations of complex and quaternionic projective spaces. This is done by investigating the projections of certain inhomogeneous isoparametric foliations of the…
The goal of this paper is to introduce a construction of a vector bundle on a tropical variety. When the base is a tropical toric variety these tropicalize toric vector bundles, and are described by the data of a valuated matroid and some…
Let X be a complete toric variety and let Y be a smooth projective variety with Picard number one. We prove that, if there exists a surjective morphism from X to Y, then Y is a projective space.
We establish faithful tropicalisation for point configurations on algebraic tori. Building on ideas from enumerative geometry, we introduce tropical scaffolds and use them to construct a system of modular fan structures on the tropical…
If $X$ is a smooth toric variety over an algebraically closed field of positive characteristic and $L$ is an invertible sheaf on $X$, it is known that $F_* L$, the push-forward of $L$ along the Frobenius morphism of $X$, is a direct sum of…
This paper proves that every projective toric variety is the fine moduli space for stable representations of an appropriate bound quiver. To accomplish this, we study the quiver $Q$ with relations $R$ corresponding to the finite-dimensional…