arXiv research

Analyzes structure of log smooth pairs when equality holds in Bogomolov-Gieseker inequality.

Gibbs sampler mixes quickly for certain smooth distributions.

We provide examples of families of (log) smooth canonically polarized varieties, including smooth weighted pointed curves and smooth hypersurfaces in $P^3$ with large degree such that the Chow semistable limits under distinct pluricanonical embeddings do not stabilize.

New sampling algorithm for non-log-concave distributions requires many queries.

Study shows solutions to certain equations form smooth manifolds.

Generalizes complex manifolds to manifolds with corners and generalized corners.

Guarantees convergence for black-box variational inference without modifications.

Motivated by the study of Fano type varieties we define a new class of log pairs that we call asymptotically log Fano varieties and strongly asymptotically log Fano varieties. We study their properties in dimension two under an additional assumption of log smoothness, and give a complete classification of two dimension…

LMC achieves sqrt(d) dependence in sampling error, improving previous bounds.

In this paper, we prove that the zero-locus of any global holomorphic log-one-form on a projective log-smooth pair $\left(X,D\right)$ of log-general type must be non-empty. Applying this result, we give an answer to the algebraic hyperbolicity part of Shafarevich's conjecture, with the generic fiber being Kawamata-log-…

New lower bounds for sampling from log-concave distributions in higher dimensions.

Let (X,D) be a klt pair. Assuming either K_X+D big or -(K_X+D) ample, and that the coefficients of D are greater than 1/2, we show that the Kähler-Einstein metric attached to (X,D) -whenever it exists- has cone singularities along D on the log-smooth locus of the pair intersected with the ample locus of K_X+D (in the n…

We prove the Yau-Tian-Donaldson's conjecture for any $\mathbb{Q}$-Fano variety that has a log smooth resolution of singularities such that the discrepancies of all exceptional divisors are non-positive. In other words, if such a Fano variety is K-polystable, then it admits a Kähler-Einstein metric. This extends the pre…

Formally constructs metrics near timelike geodesics in vacuum spacetimes.

We consider the family of constant curvature fiber metrics for a Lefschetz fibration with regular fibers of genus greater than one. A result of Obitsu and Wolpert is refined by showing that on an appropriate resolution of the total space, constructed by iterated blow-up, this family is log-smooth, i.e. polyhomogeneous …

McDiarmid's inequality under dependence via approximate tensorization of entropy

We develop a general theory of log spaces, in which one can make sense of the basic notions of logarithmic geometry, in the sense of Fontaine-Illusie-Kato. Many of our general constructions with log spaces are new, even in the algebraic setting. In the differentiable setting, our theory yields a framework for treating …

The paper studies Kähler-Einstein metrics with singularities and their limits.

MALA mixes efficiently under smoothness and isoperimetry assumptions.

Study geometric properties of log Calabi-Yau manifolds, focusing on Fano manifolds with smooth or two proportional components.

Study improves sampling from non-log-concave distributions using Fisher information.

BBVI converges nearly dimensionally independent for log-concave targets.

Study of 0-instantons on hyperbolic manifolds, proving invariants and energy formulas.

MALA mixes optimally in κ√d steps for log-concave sampling.

The paper studies fundamental groups of compact Kahler varieties with nef anti-canonical bundles.

Let $S$ be a smooth projective variety and $Δ$ a simple normal crossing $\mathbb{Q}$-divisor with coefficients in $(0,1]$. For any ample $\mathbb{Q}$-line bundle $L$ over $S$, we denote by $\mathscr{E}(L)$ the extension sheaf of the orbifold tangent sheaf $T_S(-\log(Δ))$ by the structure sheaf $\mathcal{O}_S$ with the …

New method improves sampling from logconcave distributions truncated on polytopes.

Improved log-concave sampling to $O(d^{1/2})$ with warm starts.