Related papers: Toric K\"ahler metrics seen from infinity, quantiz…
The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in…
Let N be a complete hyperbolic 3-manifold that is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We show N is homeomorphic to the interior of a compact 3-manifold, or tame, if one of the following conditions holds: (1) N…
We develop a theory of multi-stage degenerations of toric varieties over finite rank valuation rings, extending the Mumford--Gubler theory in rank one. Such degenerations are constructed from fan-like structures over totally ordered abelian…
We derive the combinatorial representations of Picard group and deformation space of anti-canonical hypersurfaces of a toric variety using techniques in toric geometry. The mirror cohomology correspondence in the context of mirror symmetry…
The notion of a tropical vector bundle on a toric variety was recently introduced by Khan-Maclagan and Kaveh-Manon. In this paper, we study the Euler characteristic and rank of global sections for tropical vector bundles. We associate a…
We present a local classification of conformally equivalent but oppositely oriented 4-dimensional Kaehler metrics which are toric with respect to a common 2-torus action. In the generic case, these "ambitoric" structures have an intriguing…
We study anisotropic scaling limits of topological field theories using tropical geometry. The resulting topological field theories are characterized by foliated geometries and are invariant under foliation-preserving gauge transformations.…
We establish a regularity result for the metric on any 4-dimensional extremal K\"ahler manifold, and a weak compactness theorem on the space of such metrics. Specifically, the sectional curvature at a point is bounded when the quantity…
We show a bijective correspondence between compact toric locally conformally symplectic manifolds which admit a compatible complex structure and pairs $(C,a)$, where $C$ is a good cone in the dual Lie algebra of the torus and $a$ is a…
We show that Martin Olsson's compactification of moduli space of polarized abelian varieties in \cite{ols08} can be interpreted in terms of KSBA stable pairs. We find that there is a canonical set of divisors $S(K_2)$ associated with each…
We prove a regularity result for Monge-Amp\`ere equations degenerate along smooth divisor on Kaehler manifolds in Donaldson's spaces of $\beta$-weighted functions. We apply this result to study the curvature of Kaehler metrics with conical…
We introduce extended tropicalizations for closed subvarieties of toric varieties and show that the analytification of a quasprojective variety over a nonarchimedean field is naturally homeomorphic to the inverse limit of the…
Complete hyperk\"ahler 4-manifolds of finite energy are grouped into ALE, ALF, ALG$^{(*)}$, ALH$^{(*)}$, each of these being further classified according to the Dynkin type of their noncompact end. A family of ALG-$D_4$ spaces are modeled…
Let $\mathbf{TB}$ be the category of totally bounded, locally compact metric spaces with the $C_0$ coarse structures. We show that if $X$ and $Y$ are in $\mathbf{TB}$ then $X$ and $Y$ are coarsely equivalent if and only if their Higson…
Let $G$ be a connected reductive algebraic group. We develop a Gr\"obner theory for multiplicity-free $G$-algebras, as well as a tropical geometry for subschemes in a spherical homogeneous space $G/H$. We define the notion of a spherical…
We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric K\"ahler-Einstein and…
In this Thesis, I investigate how Fano manifolds equipped with a Kahler-Einstein metric can degenerate as metric spaces (in the Gromov-Hausdorff topology) and some of the relations of this question with Algebraic Geometry, in particular in…
We prove the existence of canonical tubular neighbourhoods around complex submanifolds of K\"ahler manifolds that are adapted to both the holomorphic and symplectic structure. This is done by solving the complex Homogeneous Monge-Amp\`ere…
We begin by defining a type of K\"ahler metric near the zero section of a trivial holomorphic open disk bundle $N$ over a compact K\"ahler manifold $X$ by incorporating flows generated by holomorphic vector fields on $X$. These metrics are…
The simplicial wedge construction on simplicial complexes and simple polytopes has been used by a variety of authors to study toric and related spaces, including non-singular toric varieties, toric manifolds, intersections of quadrics and…