Constructs Kahler-Einstein metrics near isolated log canonical singularities.
problem Constructing metrics near singularities in complex geometry.
method Constructs Kahler-Einstein metrics with negative scalar curvature near isolated log canonical singularities.
result Metrics are complete near the singularity if the underlying space has complex dimension 2 or if the singularity is smoothable.
Classifies degenerations of complex projective plane with rational singularities.
problem Classifying singularities of complex projective plane.
method Assuming Wahl's conjecture, classifies degenerations using rational homology disk smoothing.
result Classifies surfaces with rational singularities, including new degenerations with non-log canonical singularities.
Compact Kahler-Einstein manifolds converge to semi-log canonical models.
problem Compactness of Kahler-Einstein manifolds of negative scalar curvature.
method Gromov-Hausdorff convergence and Weil-Petersson metric extension.
result Convergence to a finite union of complete Kahler-Einstein metric spaces.
Study improves sampling from non-log-concave distributions using Fisher information.
problem Sampling from non-log-concave distributions with high Fisher information guarantees.
method Proximal sampler with RGO implementation, leveraging log-concave sampling results.
result Improved complexity guarantee in relative Fisher information for non-log-concave sampling.
Study Langevin Monte Carlo for sampling non-log-concave distributions.
problem Sampling from non-log-concave distributions, especially Gaussian mixtures.
method Discretizations of overdamped Langevin diffusions.
result Numerical simulations compare Langevin Monte Carlo algorithms' performance.
Let π:X∗→B∗ be an algebraic family of compact Kähler manifolds of complex dimension n with negative first Chern class over a punctured disc B∗∈C. Let gt be the unique Kähler-Einstein metric on Xt=π−1(t). We show that as t→0, $(\mathcal{X}_t, g…
New algorithms improve convergence rates for non-log-concave sampling and log-partition estimation.
problem Efficiently sampling from non-log-concave distributions and estimating their log-partition function.
method Analysis of information-based complexity, study of polynomial-time sampling algorithms.
result Optimal rates for sampling and log-partition estimation sometimes exceed those for optimization.
Improved convergence for non-log-concave sampling.
problem Sampling from non-log-concave distributions.
method Novel conductance analysis of SGLD with auxiliary Markov Chain.
result SGLD achieves ε-sampling error with fewer evaluations.
The paper extends deformation theory to Calabi-Yau varieties with isolated log canonical singularities.
problem Deformation theory of Calabi-Yau varieties with log canonical singularities.
method Study of higher Du Bois and rational singularities, focusing on 0-liminal singularities.
result Existence of first order smoothings for isolated 0-liminal hypersurface singularities.
The paper sets lower bounds for sampling non-log-concave distributions using Fisher information.
problem Understanding the complexity of sampling non-log-concave distributions.
method Proves two lower bounds using Fisher information in the context of sampling.
result Lower bounds on the complexity of sampling non-log-concave distributions, ruling out high-accuracy algorithms.
We study global log canonical thresholds of cubic surfaces with canonical singularities, and we prove the existence of a Kahler-Einstein metric on two singular cubic surfaces.
New insights into symplectic singularities via canonical torus actions.
problem Understanding symplectic singularities and their actions.
method Use of (C∗)r actions and Donaldson-Sun theory. result Symplectic singularities admit canonical (C∗)r actions. New sampling algorithm for non-log-concave distributions requires many queries.
problem Sampling from non-log-concave distributions with good accuracy.
method Lower bound on query complexity and algorithm for sampling.
result Tight query complexity characterization for sampling from non-log-concave distributions.
We establish the existence of the K"ahler-Ricci flow on projective varieties with log canonical singularities. This generalizes some of the existence results of Song-Tian \cite{ST3} in case of projective varieties with klt singularities. We also prove that the normalized K"ahler-Ricci flow will converge to the \ka-Eins…
Study improves sampling from complex distributions using annealed Langevin Monte Carlo.
problem Sampling from non-log-concave and multimodal distributions.
method Annealed Langevin Monte Carlo algorithm with theoretical guarantees.
result Oracle complexity of O(dβ²A²/ε⁶) for achieving ε² accuracy in Kullback-Leibler divergence.
We identify the canonical contact structure on the link of a simple elliptic or cusp singularity by drawing a Legendrian handlebody diagram of one of its Stein fillings. We also show that the canonical contact structure on the link of a numerically Gorenstein surface singularity is trivial considered as a real plane bu…
Introduces new stability concept for Fano fibrations.
problem Stability of Fano fibrations with singularities.
method Introduces f-stability and shows its implications. result Fibered semi log canonical singularities are restricted.
Uniformizes varieties with log-canonical singularities using ball quotients.
problem Uniformizing complex projective varieties with log-canonical singularities.
method Criteria based on Miyaoka-Yau inequality and log-resolutions of singularities.
result Criteria for isomorphism to Baily-Borel-Mok compactifications.
Proves orbifold singularities for Ricci-flat metrics on certain Kähler varieties.
problem Regularity of singular Ricci-flat Kähler metrics on Kähler varieties with log terminal singularities.
method Analyzes orbifold singularities of metrics restricted to the orbifold locus.
result Singular Ricci-flat Kähler metrics on Kähler varieties with log terminal singularities have orbifold singularities.
We construct a compact symplectic manifold with a Hamiltonian circle action for which the Duistermaat-Heckman function is not log-concave.
Optimizes bounds for threefold singularity volumes.
problem Bounding local volumes of threefold singularities.
method Analyzes Gorenstein canonical non-hypersurface threefold singularities.
result Establishes optimal upper bound for local volumes.
The note proves positive currents induced by VKE with mixed singularities.
problem Variation of Kahler-Einstein metrics with mixed singularities.
method Fiberation between compact Kahler manifolds with generic smooth log canonical pairs.
result Current induced by VKE with mixed cone and Poincare singularities is positive.
We study one parameter degenerations of complex projective manifolds by introducing certain type of Hodge metrics coming from the pluricanonical forms. We show that degenerations with at most canonical singularities are all in the finite distance boundary of moduli spaces. We also propose the converse to be true in the…
Entropy corrections improve GBM's predictive accuracy for non-log-normal distributions.
problem Log-normal distribution limitations in GBM predictions.
method Entropy corrections to geometric Brownian motion (GBM).
result Improved predictive accuracy for non-log-normal distributions.
Study the relationship between canonical polynomials and elliptic sequences for elliptic singularities.
problem Understanding the relationship between canonical polynomials and elliptic sequences for elliptic singularities.
method An inductive setup of elliptic germs and comparison of their canonical polynomials.
result The exponents of the canonical polynomial determine the elliptic sequence and vice versa under certain conditions.
Holomorphic vector fields and anti-canonical divisors on complex manifolds are studied.
problem Existence of non-trivial holomorphic vector fields on compact complex manifolds.
method Vanishing result for measure preserving holomorphic vector fields, Gibbs stability, and log terminal singularities.
result No non-trivial holomorphic vector fields on compact complex manifolds with big anti-canonical line bundle.
Classifies normal stable Horikawa surfaces with smoothable singularities.
problem Characterizing surfaces with specific singularities and smoothability criteria.
method Classification and smoothability criterion based on log canonical singularities.
result Provides a criterion for global Q-Gorenstein smoothability of Horikawa surfaces. Paper tackles sampling from non-log-concave distributions using denoising diffusion.
problem Sampling from non-log-concave distributions efficiently.
method DDMC framework, Zeroth-Order Diffusion Monte Carlo (ZOD-MC) algorithm.
result ZOD-MC achieves inverse polynomial dependence on sampling accuracy, efficient for low dimensions.
Canonical bundle formula due to Kawamata and others has played fundamental roles in algebraic geometry. We show that the canonical bundle formula has analytic characterization in terms of fiberwise integration, which confirms a folklore conjecture. The proof uses L2 metrics and the valuative equivalence of plurisubh…
The Miyaoka-Yau inequality is proven for certain singular varieties with big canonical or anticanonical divisors.
problem Establishing the Miyaoka-Yau inequality for singular varieties with specific divisors.
method Defining the non-pluripolar product and establishing the Bogomolov-Gieseker type inequality for Higgs sheaves; investigating second Chern class inequalities.
result Proven the Miyaoka-Yau inequality for projective klt varieties with big canonical or anticanonical divisors.
Classifies surfaces with T-singularities and ample canonical class.
problem Classifying surfaces with T-singularities and specific properties.
method Using KSBA moduli space and techniques for surfaces with T-singularities.
result Identifies surfaces with only T-singularities in the KSBA space and proves non-smoothability conditions.
Smooth flows with surgery approximate weak mean curvature flows with spherical and neck-pinch singularities.
problem Approximating weak mean curvature flows with singularities using smooth flows.
method Combining Choi-Haslhofer-Hershkovits and Choi-Haslhofer-Hershkovits-White work on canonical neighbourhoods and barriers to flows with surgery.
result Smooth flows with surgery can approximate weak mean curvature flows with spherical and neck-pinch singularities.
The classical Beauville-Bogomolov Decomposition Theorem asserts that any compact Kähler manifold with numerically trivial canonical bundle admits an étale cover that decomposes into a product of a torus, and irreducible, simply-connected Calabi-Yau-- and holomorphic-symplectic manifolds. The decomposition of the simply…
Given a Lagrangian submanifold L of the affine symplectic 2n-space, one can canonically and uniquely define a center-chord and a special improper affine sphere of dimension 2n, both of whose sets of singularities contain L. Although these improper affine spheres (IAS) always present other singularities away fro…
Injectivity proven for measure homology of certain wild spaces.
problem Injectivity of measure homology for mildly wild spaces.
method Proving injectivity of the canonical map from singular to measure homology.
result Injectivity of measure homology for certain mildly wild spaces.
It is proved by Kawamata that the canonical bundle of a projective manifold is semi-ample if it is big and nef. We give an analytic proof using the Ricci flow, degeneration of Riemannian manifolds and L2-theory. Combined with our earlier results, we construct unique singular Kahler-Einstein metrics with a global Rie…
We study variuos homological structures associated with Poisson algebra, the canonical differential complex for singular Poisson structure and the analogue of the star operator for such manifolds. Give the interpretation of the classical Koszul differential of exterior forms, as the supercommutator with some second ord…
New sampling method guarantees approximate first-order stationary points for non-convex functions.
problem Sampling from non-log-concave densities with non-convex potential functions.
method Averaged Langevin Monte Carlo with complexity analysis.
result Langevin Monte Carlo outputs a sample with ε-relative Fisher information after O(L²d²/ε²) iterations.
Study shows polystability of tangent and canonical sheaves on Kähler-Einstein log Fano pairs.
problem Stability of tangent and canonical sheaves on Kähler-Einstein log Fano pairs.
method Analysis of adapted tangent and canonical sheaves under singular Kähler-Einstein metrics.
result Adapted tangent and canonical sheaves are polystable.
Higher-dimensional Ricci flows are shown to have unique and stable solutions.
problem Stability and uniqueness of Ricci flows in higher dimensions.
method Generalization of Bamler-Kleiner's proof to higher dimensions, use of Brendle's classification of κ-solutions, and maximum principle for linearized Ricci-DeTurck flow.
result Canonical evolution through singularities for manifolds with positive isotropic curvature.
Solves complex Hessian equations in unstable cases, proving unique canonical solutions with singularities.
problem Existence of smooth solutions to complex Hessian equations in unstable cases.
method Parabolic flows and moment-map energy functionals, focusing on J-equation and deformed Hermitian Yang-Mills equation.
result Proves existence of unique canonical solutions with singularities on Kahler surfaces.
In this paper, we study the convergence of Calabi-Yau manifolds under Kähler degeneration to orbifold singularities and complex degeneration to canonical singularities (including the conifold singularities), and the collapsing of a family of Calabi-Yau manifolds.
We describe Milnor open books and Legendrian surgery diagrams for canonical contact structures of links of some rational surface singularities. We also describe an infinite family of Milnor fillable contact 3-manifolds so that the Milnor genus (resp. Milnor norm) is strictly greater than the support genus (resp. suppor…
New 1-parameter family of ovals identified in 4d Ricci flow classification.
problem Classifying κ-solutions in 4d Ricci flow. method Introducing conjectures and constructing new examples.
result Established canonical neighborhood theorem for 4d Ricci flow.
This paper studies a specific blow-up algorithm for sop polynomials and their RLCT.
problem Determining the RLCT of sum-of-products polynomials through blow-up.
method Investigates a specific blow-up algorithm for sop polynomials to resolve their singularities.
result It is possible to resolve the singularities of sop polynomials using a specific blow-up algorithm.
Let X be a canonically polarized variety, i.e. a complex projective variety such that its canonical class KX defines an ample $\Q-$line bundle, and satisfying the conditions G1 and S2. Our main result says that X admits a Kähler-Einstein metric iff X has semi-log canonical singularities i.e. iff X is…
Evaluation of the marginal likelihood plays an important role in model selection problems. The widely applicable Bayesian information criterion (WBIC) and singular Bayesian information criterion (sBIC) give approximations to the log marginal likelihood, which can be applied to both regular and singular models. When the…
An isolated complex surface singularity induces a canonical contact structure on its link. In this paper, we initiate the study of the existence problem of Stein cobordisms between these contact structures depending on the properties of singularities. As a first step we construct an explicit Stein cobordism from any co…