arXiv research

The paper studies semistability in polarized toric manifolds and their divisors.

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

Let $P$ be a Delzant polytope. We show that the quantization of the corresponding toric manifold $X_{P}$ in toric Kähler polarizations and in the toric real polarization are related by analytic continuation of Hamiltonian flows evaluated at time $t = \sqrt{-1} s$. We relate the quantization of $X_{P}$ in two different …

New method shows unitarity in quantization for toric manifolds.

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

Quantizes symplectic manifolds with toric singularities using Toeplitz operators.

The paper solves a uniform Yau-Tian-Donaldson conjecture for toric manifolds.

Study explores quantum spaces on toric varieties and their limiting behavior.

The paper explores stability and coercivity for toric polarizations, linking them to K-energy.

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…

Study Mabuchi rays on toric Kähler manifolds to understand quantization.

Characterizes toroidal and semi-toric compactifications as log minimal models and applies to weak K-moduli.

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…

The paper studies equivariant sheaves on toric varieties and their quotients.

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.

In this note, we prove that on polarized toric manifolds the relative $K$-stability with respect to Donaldson's toric degenerations is a necessary condition for the existence of Calabi's extremal metrics, and also we show that the modified $K$-energy is proper in the space of $G_0$-invariant Kähler metrics for the case…

Proves weight polytope matches with energy vectors in toric varieties.

Let Δ\subset \mathbb{R}^n be an n-dimensional Delzant polytope. It is well-known that there exist the n-dimensional compact toric manifold X_Δand the very ample (\mathbb{C}^\times)^n-equivariant line bundle L_Δon X_Δassociated with Δ. In the present paper, we show that if (X_Δ,L_Δ^i) is Chow semistable then the sum of …

K-energy is not strictly convex on certain complex manifolds, but under specific conditions, it is.

It's well-known in \kahler geometry that the infinite dimensional symmetric space $\hcal$ of smooth \kahler metrics in a fixed \kahler class on a polarized \kahler manifold is well approximated by finite dimensional submanifolds $\bcal_k \subset \hcal$ of Bergman metrics of height $k$. Then it's natural to ask whether …

We consider the metric space of all toric Kähler metrics on a compact toric manifold; when "looking at it from infinity" (following Gromov), we obtain the tangent cone at infinity, which is parametrized by equivalence classes of complete geodesics. In the present paper, we study the associated limit for the family of m…

The paper simplifies K-stability conditions for spherical varieties.

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

The global holomorphic α-invariant introduced by Tian is closely related with the study in the existence of Kahler-Einstein metric. We apply the result of Tian, Lu and Zelditch on polarized Kahler metrics to approximate plurisubharmonic functions and compute the α-invariant of toric Fano manifolds.

We study the half-form Kaehler quantization of a smooth symplectic toric manifold $(X,ω)$, such that $[ω/2π]-c_{1}(X)/2 \in H^{2}(X,{\mathbb{Z}})$ and is nonnegative. We define the half-form corrected quantization of $(X,ω)$ to be given by holomorphic sections of a certain hermitian line bundle $L\rightarrow X$ with Ch…

The paper characterizes complex projective spaces using Ehrhart polynomials.

We classify irreducible representations of connected compact Lie groups whose orbit space is isometric to the orbit space of a representation of a finite extension of (positive dimensional) toric group. They turn out to be exactly the non-polar irreducible representations preserving an isoparametric submanifold and act…

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

Given a polarized manifold there are obstructions for asymptotic Chow semistability described as integral invariants. One of them is an obstruction to the existence for the first Chern class of the polarization to admit a constant scalar curvature Kähler (cscK) metric. A natural question is whether or not the other obs…

The paper generalizes K-stability results to singular and weighted settings.

Let $(L, h)\to (X, ω)$ denote a polarized toric Kähler manifold. Fix a toric submanifold $Y$ and denote by $\hatρ_{tk}:X\to \mathbb{R}$ the partial density function corresponding to the partial Bergman kernel projecting smooth sections of $L^k$ onto holomorphic sections of $L^k$ that vanish to order at least $tk$ along…

Let $Δ\subset \mathbb{R}^n$ be an $n$-dimensional integral Delzant polytope. It is well-known that there exist the $n$-dimensional compact toric manifold $X_Δ$ and the very ample $(\mathbb{C}^\times)^n$-equivariant line bundle $L_Δ$ on $X_Δ$ associated with $Δ$. In the present paper, we give a necessary and sufficient …

Theory and existence of extremal Kähler metrics on 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-…

The paper describes the Picard group and quantization in toric orbifolds.

The paper proposes a noncommutative deformation of toric varieties.

G. Tian and S.K. Donaldson formulated a conjecture relating GIT stability of a polarized algebraic variety to the existence of a Kahler metric of constant scalar curvature. In [Don02] Donaldson partially confirmed it in the case of projective toric varieties. In this paper we extend Donaldson's results and computations…

We show that the classical Szasz analytic function $S_N(f)(x)$ is obtained by applying the pseudo-differential operator $f(N^{-1}D_θ)$ to the Bergman kernels for the Bargmann-Fock space. The expression generalizes immediately to any smooth polarized noncompact complete toric \kahler manifold, defining the generalized S…

Geodesic rays prove key aspects of cscK metrics existence and stability.

It is shown that any, possibly singular, Fano variety X admitting a Kahler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of Q-Fano varieties equipped with their anti-canonical polarization. The proof exploits convexity properties of the Ding functiona…

Quantum Kirwan maps between K-theories of G-varieties and GIT quotients.

Extends geometric quantization to singular spaces.

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.

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…

The inverse Monge-Ampere flow helps find Kahler-Einstein metrics.

In this paper I construct, using off the shelf components, a compact symplectic manifold with a non-trivial Hamiltonian circle action that admits no Kaehler structure. The non-triviality of the action is guaranteed by the existence of an isolated fixed point. The motivation for this work comes from the program of class…

Constructs scalar-flat Kähler metrics on toric symplectic manifolds.