Existence of Kähler-Einstein metrics on toric varieties proven.
problem Existence of Kähler-Einstein metrics on toric varieties.
method Characterization of K-stability using log Cox ring and universal orbifold cover.
result Every Q-factorial normal projective toric variety allows an orbifold Kähler-Einstein metric.
Non-exact Poisson structures found on toric varieties.
problem Existence of exact Poisson structures on toric varieties.
method Geometric criterion for non-exactness of Poisson structures with finite symplectic leaves.
result Non-exactness of Poisson structures on projective toric varieties.
We introduce the fibred toric varieties as equivariant CPr 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…
Defines and classifies toric co-Higgs bundles on projective toric varieties.
problem Classifying co-Higgs bundles on toric varieties.
method Using Klyachko's fan filtration and studying the co-Higgs bundle fiber at a closed point.
result Provides a Lie-theoretic classification of toric co-Higgs bundles.
Study shows how the energy of a metric on a toric variety relates to the volume of holomorphic sections.
problem Understanding the volume of holomorphic sections on projective toric varieties.
method Defined energy at equilibrium and showed its asymptotic behavior as a function of the volume of L2-norm unit balls. result The energy of a metric on a toric variety describes the asymptotic behavior of the volume of holomorphic sections.
Geometrically connects toric varieties to normed spaces.
problem Connecting algebraic geometry with convex analysis.
method Establishes a 1-1 correspondence using topological models.
result Toric varieties correspond to horofunction compactifications of polyhedral norms.
The paper defines and proves conditions for numerical semistability of smooth toric varieties.
problem Understanding the numerical semistability of smooth toric varieties.
method Analyzing the Chow/Hurwitz forms and applying toric degenerations.
result A necessary and sufficient condition for a smooth toric variety to be numerically semistable.
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…
New SKT manifolds created using toric geometry.
problem Creating SKT manifolds.
method Using toric geometry and J-construction. result Infinite families of SKT manifolds produced.
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…
Sharp bounds on K-semistable Fano varieties for low dimensions.
problem Establishing bounds on the height of K-semistable Fano varieties.
method Analyzing canonical integral models of toric Fano varieties, using the gap hypothesis and Donaldson's modular height.
result Sharp lower bounds on the height of toric Fano varieties, with applications to Mabuchi functional and Odaka's modular height.
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…
New proof shows coherent sheaves quasi-equivalent to constructible sheaves on torus.
problem Establishing equivalence between coherent and constructible sheaves on toric varieties.
method Using non-characteristic deformation of sheaves to find twisted polytope sheaves.
result Quasi-equivalence of coherent and constructible sheaves on smooth projective toric varieties.
The paper derives a formula for Chow weights of toric blow-ups.
problem Chow weights of toric blow-ups.
method Combinatorial formula derived from toric manifold and Delzant polytope.
result Explicit formula for Chow weights of blow-ups.
The paper studies equivariant sheaves on toric varieties and their quotients.
problem Understanding stability of sheaves on toric GIT quotients.
method Defining equivariant sheaves and showing stability preservation under certain conditions.
result Stability of sheaves on toric GIT quotients is related to combinatorial criteria.
Non-archimedean SYZ fibration constructed for Calabi-Yau hypersurfaces.
problem Analyzing Calabi-Yau hypersurfaces using non-archimedean geometry.
method Yamamoto's tropical contractions and Li's Fermat degeneration, with toric plurisubharmonic metrics.
result Constant potential along fibers of retraction under discrete symmetry assumption.
Study thresholds and K-stability using valuations in complex projective varieties.
problem Understanding thresholds and K-stability in complex projective varieties.
method Using valuations to study log canonical and stability thresholds, proving infima are attained for ample L, and obtaining simple expressions in toric cases.
result The thresholds can be written as infima of functionals on the space of valuations, and are attained for ample L.
Let M be a projective toric manifold. We prove two results concerning respectively Kaehler-Einstein submanifolds of M and symplectic embeddings of the standard euclidean ball in M. Both results use the well-known fact that M contains an open dense subset biholomorphic to C^n.
The paper studies compactifications of SL(2,C) character varieties for punctured surfaces.
problem Compactifying character varieties of punctured surfaces.
method Projective compactifications using ideal triangulations and Komyo's method.
result Boundary divisors are toric varieties and the boundary complex is a sphere.
The paper proposes a noncommutative deformation of toric varieties.
problem Deforming complex structures into generalized complex structures.
method Using holomorphic Poisson tensors and generalized complex branes.
result Noncommutative deformation of homogeneous coordinate rings.
Let (L,h)→(X,ω) denote a polarized toric Kähler manifold. Fix a toric submanifold Y and denote by ρ^tk:X→R the partial density function corresponding to the partial Bergman kernel projecting smooth sections of Lk onto holomorphic sections of Lk that vanish to order at least tk along…
Paper describes holomorphic polyvector fields on toric varieties.
problem No specific problem stated; general description of fields.
method Explicit description of holomorphic polyvector fields on smooth compact toric varieties.
result Generalizes Demazure's result of holomorphic vector 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…
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…
Quantum Kirwan maps between K-theories of G-varieties and GIT quotients.
problem Constructing maps between K-theories of G-varieties and their GIT quotients.
method Formal construction of maps in quantum K-theory, using equivariant and non-equivariant quantum K-theory.
result Presentation of quantum K-theory for smooth proper toric DM stacks.
Sharp bounds on Fano varieties' heights proven for specific cases.
problem Determining the maximal height of arithmetic Fano varieties.
method Logarithmic extension of a conjecture, applied to specific Fano varieties.
result The conjecture settled for specific cases, including hypersurfaces and toric varieties.
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.
problem Characterizing and understanding co-Higgs sheaves on toric varieties.
method Characterization and explicit computation of examples.
result Explicit examples of co-Higgs sheaves on toric varieties computed.
Proves Witten genus vanishes for certain string intersections.
problem Proving the Witten genus vanishes for specific string intersections.
method Using equivariant localization formula of toric varieties.
result Witten genus vanishes for some string complete intersections.
A theorem of Delzant states that any symplectic manifold $(M,\om)$ of dimension 2n, equipped with an effective Hamiltonian action of the standard n-torus $\T^n = \R^{n}/2π\Z^n$, is a smooth projective toric variety completely determined (as a Hamiltonian $\T^n$-space) by the image of the moment map φ:M→Rn, a …
Uniformly K-stable toric varieties are asymptotically Chow stable if their Futaki-Ono invariant vanishes.
problem Determining asymptotic Chow stability of uniformly K-stable toric varieties.
method Detailed study of triangulations of moment polytope neighborhoods and analysis of Futaki-Ono invariant.
result Every uniformly K-stable polarized smooth toric variety with vanishing Futaki-Ono invariant is asymptotically Chow polystable.
Study BV operators on holomorphic polyvector fields on toric varieties.
problem Existence of BV operators in Gerstenhaber algebras.
method Analyzing BV operators on holomorphic polyvector fields on smooth compact toric varieties.
result Necessary and sufficient condition for BV operators existence.
Survey various symmetry notions for toric varieties.
problem Understanding different types of symmetries in toric varieties.
method Exploring algebraic, complex, representation, combinatorial, convex, and geometric stability perspectives.
result Establishes relationships between different symmetry notions.
Proves weight polytope matches with energy vectors in toric varieties.
problem Understanding the relationship between weight polytopes and energy functionals in toric varieties.
method Combines two slope formulas of K-energy in the toric setting.
result Weight polytope of Hurwitz form matches with convex hull of characteristic vectors.
The paper generalizes K-stability results to singular and weighted settings.
problem Generalizing K-stability to singular and weighted settings.
method Generalization of results in \cite{Li22a} to singular and weighted settings.
result The \(\mathbb{G}\)-uniform weighted K-stability for models implies \(\mathbb{G}\)-coercivity of the weighted Mabuchi functional.
We show that the complex projective space has maximal degree (volume) among all n-dimensional Kahler-Einstein Fano manifolds admitting a holomorphic C^*-action with a finite number of fixed points. The toric version of this result, translated to the realm of convex geometry, thus confirms Ehrhart's volume conjecture fo…
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.
The paper proves a new stability condition for certain Fano varieties.
problem Stability conditions for Fano varieties with Gorenstein singularities.
method Using toric test configurations and combinatorial criteria.
result Asymptotic Chow semistability implies Ding polystability for Gorenstein toric Fano varieties.
Moment polytope of toric exponential families is a projection of a simplex.
problem Understanding the geometry of exponential families in finite sample spaces.
method Toric torification and projection of higher-dimensional simplices.
result Moment polytope is a projection of a higher-dimensional simplex.
Logarithmic connections on principal bundles over normal varieties are studied.
problem Existence and properties of logarithmic connections on principal bundles over normal varieties.
method Introducing logarithmic connections, showing equivalence to covariant derivatives, and proving existence conditions.
result Existence of logarithmic connections on principal bundles over normal varieties is equivalent to certain conditions on the associated vector bundles and adjoint bundles.
The paper solves a geometric P=W conjecture for SL(2,C) using Thurston's compactification.
problem Addressing the geometric P=W conjecture for SL(2,C) in projective compactifications of character varieties of closed surfaces.
method Using Thurston's compactification of Teichmüller space and new results, the paper constructs a projective compactification of the SL(2,C)-character variety of any closed surface of genus g>1.
result The boundary divisors are toric varieties and the dual intersection complex is a sphere.
Kähler-Einstein metrics on toric submanifolds cannot be induced by projective immersions.
problem Characterizing Kähler-Einstein metrics induced by projective immersions.
method Analyzing four families of symmetric and non-symmetric toric Fano manifolds.
result Kähler-Einstein metrics on toric submanifolds cannot be induced by projective immersions.
Contact manifolds' momentum polytopes are convex.
problem Understanding the structure of contact manifolds.
method Using isomorphism to toric varieties.
result Momentum polytopes of contact manifolds are convex.
We compute the Chen-Ruan orbifold cohomology ring of the Batyrev mirror orbifold of a smooth quintic hypersurface in 4-dimensional projective space. We identify the obstruction bundle for this example by using the Riemann bilinear relations for periods. We outline a general method of computing the Chen-Ruan ring for Ca…
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 CP2. The fundamental groups related to th…
Study on non-archimedean μ-entropy for toric varieties, proving existence and uniqueness.
problem Exploring non-archimedean μ-entropy for toric varieties and its thermodynamical structure.
method Established a Rellich type compactness result for convex functions on simple polytope, proving existence and uniqueness of optimizer.
result Existence and uniqueness of optimizer for toric non-archimedean μ^λ-entropy for λ ≤ 0.
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…