相关论文: The tropicalisation of a $(-2,0)$-flop
In tropical geometry, one studies algebraic curves using combinatorial techniques via the tropicalization procedure. The tropicalization depends on a map to an algebraic torus and the combinatorial methods are most useful when the…
We study the mirror operation of the Atiyah flop in symplectic geometry. We formulate the operation for a symplectic manifold with a Lagrangian fibration. Furthermore we construct geometric stability conditions on the derived Fukaya…
This note discusses recent new approaches to studying flopping curves on 3-folds. In a joint paper, the author and Michael Wemyss introduced a 3-fold invariant, the contraction algebra, which may be associated to such curves. It…
Using tropical geometry we propose a mirror construction for monomial degenerations of Calabi-Yau varieties in toric Fano varieties. The construction reproduces the mirror constructions by Batyrev for Calabi-Yau hypersurfaces and by Batyrev…
Given a polyhedral cone sigma with smooth two-dimensional faces and, moreover, a lattice point R in the dual cone of sigma, we describe the part of the versal deformation of the associated toric variety TV(sigma) that is built from the…
The mirror of a toric orbifold is an affine curve called the mirror curve. In this paper, firstly, we recall the basic tools in tropical geometry and give a definition of the mirror curve. Then we calculate the tropical spine of the mirror…
We consider a toric degeneration of Calabi--Yau complete intersections of Batyrev--Borisov in the Gross--Siebert program. One can associate two types of tropical spaces with it. One is a tropical variety obtained by tropicalization. The…
In [7], Donovan and Wemyss introduced the contraction algebra of flop- ping curves in 3-folds. When the flopping curve is smooth and irreducible, we prove that the contraction algebra together with its A_\infty-structure recovers various…
We consider type II superstring compactifications on the singular Spin(7) manifold constructed as a cone on SU(3)/U(1). Based on a toric realization of the projective space CP^2, we discuss how the manifold can be viewed as three…
We give a method to construct singular Lagrangian 3-torus fibrations over certain a priori given integral affine manifolds with singularities, which we call simple. The main result of this article is the proof that the topological…
This is an outline of work in progress concerning an algebro-geometric form of the Strominger-Yau-Zaslow conjecture. We introduce a limited type of degeneration of Calabi-Yau manifolds, which we call toric degenerations. For these, the…
Mirror symmetry of Calabi-Yau manifolds can be understood via a Legendre duality between a pair of certain affine manifolds with singularities called tropical manifolds. In this article, we study conifold transitions from the point of view…
We consider D3-brane gauge theories at an arbitrary toric Calabi-Yau 3-fold cone singularity that are then further compactified on a Riemann surface $\Sigma_g$, with an arbitrary partial topological twist for the global $U(1)$ symmetries.…
We present a geometric framework for reconstruction problems based on Vaisman foliations and Atiyah--Molino sequences. Independent projections induce transverse foliations and dual connections; vanishing torsion and curvature duality…
Let $X$ be a smooth threefold over an algebraically closed field of positive characteristic. We prove that an arbitrary flop of $X$ is smooth. To this end, we study Gorenstein curves of genus one and two-dimensional elliptic singularities…
Let $k$ be a perfect field and let $C_0:f=0$ be a smooth curve in the torus $\mathbb{G}_{m,k}^2$. Let $\mathbb{T}_\Delta$ be the toric variety associated to the Newton polygon of $f$. Extending the toric resolution of $C_0$ on…
We prove transformation formulae for generating functions of Gromov-Witten invariants on general toric Calabi-Yau threefolds under flops. Our proof is based on a combinatorial identity on the topological vertex and analysis of fans of toric…
For an affine, toric Q-Gorenstein variety Y (given by a lattice polytope Q) the vector space T^1 of infinitesimal deformations is related to the complexified vector spaces of rational Minkowski summands of faces of Q. Moreover, assuming Y…
We present tools and definitions to study abstract tropical manifolds in dimension 2, which we call simply tropical surfaces. This includes explicit descriptions of intersection numbers of 1-cycles, normal bundles to some curves and…
We discuss the symplectic topology of the Stein manifolds obtained by plumbing two 3-dimensional spheres along a circle. These spaces are related, at a derived level and working in a characteristic determined by the specific geometry, to…