arXiv research

We introduce the fibred toric varieties as equivariant $\mathbb{C}P^r$ bundles over lower dimensional toric varieties. An equivalent characterization is that the natural morphisms on them degenerate to bundle projections in the context of variation of toric varieties as GIT quotients. Our main observation is that these…

Paper describes holomorphic polyvector fields on toric varieties.

Toric hyperk{ä}hler manifolds are quaternion analog of toric varieties. Bielawski pointed out that they can be glued by cotangent bundles of toric varieties. Following his idea, viewing both toric varieties and toric hyperk{ä}her manifolds as GIT quotients, we first establish geometrical criteria for the semi-stable po…

Extending work of Bielawski-Dancer and Konno, we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study of semi-projective toric varieties, meaning GIT quotients of affine spaces by torus actions, and specifically, of Lawren…

Existence of Kähler-Einstein metrics on toric varieties proven.

We extend a recent result of Burns, Guillemin and Uribe on the asymptotics of the spectral measure for the reduction metric on a toric variety to any toric metric on a toric variety. We show how this extended result together with the Tian-Yau-Zelditch asymptotic expansion can be used to deduce Abreu's formula for the s…

Study co-Higgs sheaves on toric varieties, finding explicit examples.

Defines and classifies toric co-Higgs bundles on projective toric varieties.

Uniformly K-stable toric varieties are asymptotically Chow stable if their Futaki-Ono invariant vanishes.

Study BV operators on holomorphic polyvector fields on toric varieties.

We establish a natural and geometric 1-1 correspondence between projective toric varieties of dimension $n$ and horofunction compactifications of $\mathbb{R}^n$ with respect to rational polyhedral norms. For this purpose, we explain a topological model of toric varieties. Consequently, toric varieties in algebraic geom…

Survey various symmetry notions for toric varieties.

The paper defines and proves conditions for numerical semistability of smooth toric varieties.

Proves weight polytope matches with energy vectors in toric varieties.

Non-exact Poisson structures found on toric varieties.

New SKT manifolds created using toric geometry.

By using the equivariant localization formula of toric varieties. We prove the vanishing of the Witten genus of some string complete intersections in smooth toric varieties.

We survey recent developments in the study of SYZ mirror symmetry for compact toric and toric Calabi-Yau varieties, with a special emphasis on works of the author and his collaborators.

Given a smooth projective toric variety X, we construct an A-infinity category of Lagrangians with boundary on a level set of the Landau-Ginzburg mirror of X. We prove that this category is quasi-equivalent to the DG category of line bundles on X. This establishes part of the Homological Mirror Conjecture for toric var…

Contact manifolds' momentum polytopes are convex.

Chow stability is one notion of Mumford's Geometric Invariant Theory for studying the moduli space of polarized varieties. Kapranov, Sturmfels and Zelevinsky detected that Chow stability of polarized toric varieties is determined by its inherent {\it secondary polytope}, which is a polytope whose vertices correspond to…

In this paper we calculate fundamental groups (and some of their quotients) of complements of four toric varieties branch curves. For these calculations, we study properties and degenerations of these toric varieties and the braid monodromies of the branch curves in $\mathbb{CP}^2$. The fundamental groups related to th…

Study on non-archimedean μ-entropy for toric varieties, proving existence and uniqueness.

In this paper we start the program of constructing generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric variety near the large complex limit, with respect to the restriction of a toric metric on the toric variety to the Calabi-Yau hypersurface. The construction is based on the deformati…

This note proves combinatorially that the intersection pairing on the middle dimensional compactly supported cohomology of a smooth toric hyperkaehler variety is always definite, providing a large number of non-trivial L^2 harmonic forms for toric hyperkaehler metrics on these varieties. This is motivated by a result o…

In this paper we prove that for toric varieties the uniform K-stability is the necessary condition for the existence of extremal metrics.

Using the short time existence of the Calabi flow, we prove that any extremal Kaehler metric on a product toric variety is a product extremal Kaehler metric.

In this paper, we prove that a Gorenstein toric Fano variety $(X, -K_{X})$ is asymptotically Chow semistable then it is Ding polystable with respect to toric test configurations (Theorem 1.3). This extends the known result obtained by others (Theorem 1.2) to the case where $X$ admits Gorenstein singularity. We also sho…

Sharp bounds on K-semistable Fano varieties for low dimensions.

Inspired by recent work of S. K. Donaldson on constant scalar curvature metrics on toric complex surfaces, we study obstructions to the extension of the Calabi flow on a polarized toric variety. Under some technical assumptions, we prove that the Calabi flow can be extended for all time.

Strong formal properties for toric and homogeneous Kähler manifolds.

In this paper we construct monodromy representing generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties near the large complex limit.

Study of Kähler-Ricci flow on toric Fano varieties with preserved symplectic condition.

Generalized Laurent monomials for nonrational spaces.

In this paper we give a construction of Lagrangian torus fibration for Calabi-Yau hypersurface in toric variety via the method of gradient flow. Using our construction of Lagrangian torus fibration, we are able to prove the symplectic topological version of SYZ mirror conjecture for generic Calabi-Yau hypersurface in t…

In this paper we construct all smooth torus fibres of the generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties near the large complex limit.

New conical metrics found on toric varieties with convex cones.

The equivariant cohomology of a space with a group action is not only a ring but also an algebra over the cohomology ring of the classifying space of the acting group. We prove that toric manifolds (i.e. compact smooth toric varieties) are isomorphic as varieties if and only if their equivariant cohomology algebras are…

The paper proposes a noncommutative deformation of toric varieties.

This is an extended example of the study of mirror symmetry via log schemes and the discrete Legendre transform on affine manifolds, introduced by myself and Bernd Siebert in "Mirror Symmetry via Logarithmic Degeneration Data I" (math.AG/0309070). In this paper, I consider the construction as it applies to the Batyrev-…

On a K-unstable toric variety we show the existence of an optimal destabilising convex function. We show that if this is piecewise linear then it gives rise to a decomposition into semistable pieces analogous to the Harder-Narasimhan filtration of an unstable vector bundle. We also show that if the Calabi flow exists f…

Study moduli space of cscK surfaces around toric ones, introducing foldable surfaces.

We extend Guillemin's formula for Kaehler potentials on toric manifolds to singular quotients of C^N and CP^N.

In this survey article we describe the geometry of toric hyperkähler varieties, which are hyperkähler quotients of the quaternionic vector spaces by tori. In particular, we discuss the Betti numbers, the cohomology ring, and variation of hyperkähler structures of these spaces with many improved results and proofs.

Generalizes soft noncommutative schemes to flag varieties.

In this paper we construct a family of complex structures on a complex flag manifold that converge to the real polarization coming from the Gelfand-Cetlin integrable system, in the sense that holomorphic sections of a prequantum line bundle converge to delta-function sections supported on the Bohr-Sommerfeld fibers. Ou…

Let $\mathcal{O}(D)$ be an equivariant line bundle which is big and nef on a complex projective nonsingular toric variety $X$. Given a continuous toric metric $\|\cdot\|$ on $\mathcal{O}(D)$, we define the energy at equilibrium of $(X,φ_{\bar{D}})$ where $φ_{\bar{D}}$ is the weight of the metrized toric divisor $\bar{D…

The paper studies how Kähler polarizations degenerate to mixed polarizations on toric varieties.