arXiv research

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…

Cylindrical contact homology linked to Ehrhart polynomials and Chen-Ruan cohomology.

Almost toric manifolds form a class of singular Lagrangian fibered symplectic manifolds that is a natural generalization of toric manifolds. Notable examples include the K3 surface, the phase space of the spherical pendulum and rational balls useful for symplectic surgeries. The main result of the paper is a complete c…

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

Extends Kähler metrics theory to symplectic manifolds with toric actions.

Classifies almost-toric systems in four dimensions.

The paper constructs toric vector bundles using spectral networks and non-abelianization.

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…

In this short note, based on the work of Wang-Zhu, we determine the greatest lower bounds on Ricci curvature for all toric Fano manifolds.

We introduce the cutting construction of possibly non-compact symplectic toric manifolds, in particular, toric symplectic cones that correspond to a weakly convex good cone. Since the symplectization of a toric contact manifold is a toric symplectic cone, we can also construct toric contact manifolds that correspond to…

Proves Yau-Tian-Donaldson conjecture for toric manifolds and bundles.

We determine the greatest lower bounds on the transverse Ricci curvature of compact toric Sasaki manifolds with positive basic first Chern class and with the first Chern class of the contact bundle being trivial. This is based on Wang-Zhu's and Futaki-Ono-Wang's works, and is an analogue of C. Li's work on toric Fano m…

Constructs scalar-flat Kähler metrics on toric symplectic manifolds.

Uniqueness of Kähler Ricci shrinkers proven on toric orbifolds.

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…

We prove a GAGA-style result for toric vector bundles with smooth base and give an algebraic construction of the Frölicher approximating vector bundle that has recently been introduced by Dan Popovici using analytic techniques.

Toric quasifolds extend toric geometry to non-rational polytopes.

Vaisman manifolds are strongly related to Kähler and Sasaki geometry. In this paper we introduce toric Vaisman structures and show that this relationship still holds in the toric context. It is known that the so-called minimal covering of a Vaisman manifold is the Riemannian cone over a Sasaki manifold. We show that if…

Classifies toric fibers in $S^2 imes S^2$.

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…

Book teaches how Lagrangian torus fibration base geometry can be read off.

Paper generalizes toric concepts to nonrational settings.

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…

New SKT manifolds created using toric geometry.

Study $G_2$-manifolds from symplectic $SU(3)$-manifolds with $T^2$-symmetry.

Three-dimensional smooth, compact toric varieties (SCTV), when viewed as real six-dimensional manifolds, can admit G-structures rendering them suitable for internal manifolds in supersymmetric flux compactifications. We develop techniques which allow us to systematically construct G-structures on SCTV and read off thei…

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

Paper describes holomorphic polyvector fields on toric varieties.

We prove that the mean Euler characteristic of a Gorenstein toric contact manifold, i.e. a good toric contact manifold with zero first Chern class, is equal to half the normalized volume of the corresponding toric diagram and give some applications. A particularly interesting one, obtained using a result of Batyrev and…

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…

We prove that a compact toric locally conformally Kähler manifold which is not Kähler admits a toric Vaisman structure, a fact which was conjectured in \cite{mmp}. This is the final step leading to the classification of compact toric locally conformally Kähler manifolds started in \cite{p} and \cite{mmp}. We also show,…

In \cite{btoric}, Guillemin et al. proved a Delzant-type theorem which classifies $b$-symplectic toric manifolds. More generally, in \cite{torus} they proved a similar convexity result for general Hamiltonian torus action on $b$-symplectic manifolds. In this paper, we provide a new way to construct $b$-symplectic toric…

Moment polytope of toric exponential families is a projection of a simplex.

Study shows certain toric arrangements have minimal topological complements.

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…

New measure for nonrationality of toric quasifolds.

A series of examples of toric Sasaki-Einstein 5-manifolds is constructed. These are submanifolds of toric 3-Sasaki 7-manifolds and such a Sasaki-Einstein 5-manifold corresponds uniquely to a toric 3-Sasaki 7-manifold. This produces examples of quasi-regular Sasaki-Einstein structures on every #k(S^2 xS^3), for k odd. T…

Study of special Kato manifolds derived from toric geometry.

Generalized Laurent monomials for nonrational spaces.

The paper derives a formula for Chow weights of toric blow-ups.

Existence of Kähler-Einstein metrics on toric varieties proven.

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

Optimal family of Calabi-Yau cone metrics found for toric Kähler cones.

New toric Fano manifolds found without extremal Kähler metrics.

Study of Hermitian structures on toric suspensions of balanced manifolds.

We show that compact toric cosymplectic manifolds are mapping tori of equivariant symplectomorphisms of toric symplectic manifolds.

We give criterions for the existence of toric conical Kahler-Einstein and Kahler-Ricci soliton metrics on any toric manifold in relation to the greatest Ricci and Bakry-Emery-Ricci lower bound. We also show that any two toric manifolds with the same dimension can be joined by a continuous path of toric manifolds with c…

Researchers describe ECC of concave lens spaces and compute symplectic capacities.