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Ivan Cheltsov

Publications and source records attributed to Ivan Cheltsov.

At least 37 records · Page 2Linked to original sources
arXiv2024

Hadamard Langevin dynamics for sampling the l1-prior

Priors with non-smooth log-densities, such as the l1-prior, are widely used in Bayesian inverse problems for their sparsity-inducing properties. Existing Langevin-based sampling methods typically rely on proximal mappings or smooth approximations, which alter the target distribution. We propose an alternative approach based on a Hadamard product parameterization of the l1-norm, leading to a smooth but nonconvex and non-globally Lipschitz potential whose marginal law exactly recovers the desired posterior. The resulting Hadamard Langevin dynamics (HLD) defines a diffusion process that is analytically distinct from proximal or mirror-type Langevin schemes. Our main contribution is a rigorous well-posedness theory for both the continuous and discrete HLD. We establish existence and uniqueness of strong solutions, geometric ergodicity of the continuous dynamics, and convergence of the discretized scheme as the step size tends to zero. These results provide the first theoretical foundation for sampling from nonconvex, nonsmooth posteriors through overparameterized Langevin dynamics.

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arXiv2024

K-stability of pointless del Pezzo surfaces and Fano 3-folds

We explore connections between existence of $\Bbbk$-rational points for Fano varieties defined over $\Bbbk$, a subfield of $\mathbb{C}$, and existence of Kähler-Einstein metrics on their geometric models. First, we show that geometric models of del Pezzo surfaces with at worst quotient singularities defined over $\Bbbk\subset\mathbb{C}$ admit (orbifold) Kähler--Einstein metrics if they do not have $\Bbbk$-rational points. Then we prove the same result for smooth Fano 3-folds with 8 exceptions. Consequently, we explicitly describe several families of pointless Fano 3-folds whose geometric models admit Kähler-Einstein metrics. In particular, we obtain new examples of prime Fano 3-folds of genus $12$ that admit Kähler--Einstein metrics. Our result can also be used to prove existence of rational points for certain Fano varieties, for example for any smooth Fano 3-fold over $\Bbbk\subset\mathbb{C}$ whose geometric model is strictly K-semistable.

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arXiv2024

K-stability and space sextic curves of genus three

We study Fano threefolds that can be obtained by blowing up the three-dimensional projective space along a smooth curve of degree six and genus three. We produce many new K-stable examples of such threefolds, and we describe all finite groups that can act faithfully on them.

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arXiv2024

On K-moduli of quartic threefolds

The family of smooth Fano 3-folds with Picard rank 1 and anticanonical volume 4 consists of quartic 3-folds and of double covers of the 3-dimensional quadric branched along an octic surface. They can all be parametrised as complete intersections of a quadric and a quartic in the weighted projective space $\mathbb{P}(1,1,1,1,1,2)$, denoted by $X_{2,4} \subset \mathbb{P}(1^5,2)$; all such smooth complete intersections are K-stable. With the aim of investigating the compactification of the moduli space of quartic 3-folds given by K-stability, we exhibit three phenomena: (i) there exist K-polystable complete intersection $X_{2,2,4} \subset \mathbb{P}(1^5,2^2)$ Fano 3-folds which deform to quartic 3-folds and are neither quartic 3-folds nor double covers of quadric 3-folds - in other words, the closure of the locus parametrising complete intersections $X_{2,4}\subset \mathbb{P}(1^5,2)$ in the K-moduli contains elements that are not of this type; (ii) any quasi-smooth $X_{2,2,4} \subset \mathbb{P}(1^5,2^2)$ is K-polystable; (iii) the closure in the K-moduli space of the locus parametrising complete intersections $X_{2,2,4} \subset \mathbb{P}(1^5,2^2)$ which are not complete intersections $X_{2,4} \subset \mathbb{P}(1^5,2)$ contains only points which correspond to complete intersections $X_{2,2,4} \subset \mathbb{P}(1^5,2^2)$.

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arXiv2024

K-stable smooth Fano threefolds of Picard rank two

We prove that all smooth Fano threefolds in the families 2.1, 2.2, 2.3, 2.4, 2.6 and 2.7 are K-stable, and we also prove that smooth Fano threefolds in the family 2.5 that satisfy one very explicit generality condition are K-stable.

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arXiv2023

K-stability of Casagrande-Druel varieties

We introduce a new subclass of Fano varieties (Casagrande-Druel varieties), that are $n$-dimensional varieties constructed from Fano double covers of dimension $n-1$. We conjecture that a Casagrande-Druel variety is K-polystable if the double cover and its base space are K-polystable. We prove this for smoothable Casagrande-Druel threefolds, and for Casagrande-Druel varieties constructed from double covers of $\mathbb{P}^{n-1}$ ramified over smooth hypersurfaces of degree $2d$ with $n>d>\frac{n}{2}>1$. As an application, we describe the connected components of the K-moduli space parametrizing smoothable K-polystable Fano threefolds in the families 3.9 and 4.2 in the Mori-Mukai classification.

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arXiv2023

Birational rigidity and alpha invariants of Fano varieties

We prove that for every $ε>0$, there is a birationally super-rigid Fano variety $X$ such that $\frac{1}{2}\leqslantα(X)\leqslant \frac{1}{2}+ε$. Also we show that for every $ε>0$, there is a Fano variety $X$ and a finite subgroup $G\subset\mathrm{Aut}(X)$ such that $X$ is $G$-birationally super-rigid, and $α_G(X)<ε$.

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arXiv2023

Fibers over infinity of Landau-Ginzburg models

We conjecture that the number of components of the fiber over infinity of Landau--Ginzburg model for a smooth Fano variety $X$ equals the dimension of the anticanonical system of $X$. We verify this conjecture for log Calabi--Yau compactifications of toric Landau--Ginzburg models for smooth Fano threefolds, complete intersections, and some toric varieties.

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arXiv2023

Toric G-solid Fano threefolds

We study toric G-solid Fano threefolds that have at most terminal singularities, where G is an algebraic subgroup of the normalizer of a maximal torus in their automorphism groups.

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