arXiv subjects
Publications and source records attributed to Ivan Cheltsov.
We study the rationality problem for nodal quartic double solids. In particular, we prove that nodal quartic double solids with at most six singular points are irrational, and nodal quartic double solids with at least eleven singular points are rational.
We show that the only finite quasi-simple non-abelian groups that can faithfully act on rationally connected threefolds are the following groups: $\mathfrak{A}_5$, $\operatorname{PSL}_2(\mathbf{F}_7)$, $\mathfrak{A}_6$, $\operatorname{SL}_2(\mathbf{F}_8)$, $\mathfrak{A}_7$, $\operatorname{PSp}_4(\mathbf{F}_3)$, $\operatorname{SL}_2(\mathbf{F}_{7})$, $2.\mathfrak{A}_5$, $2.\mathfrak{A}_6$, $3.\mathfrak{A}_6$ or $6.\mathfrak{A}_6$. All of these groups with a possible exception of $2.\mathfrak{A}_6$ and $6.\mathfrak{A}_6$ indeed act on some rationally connected threefolds.
We verify Katzarkov-Kontsevich-Pantev conjecture for Landau-Ginzburg models of smooth Fano threefolds.
We give a simple sufficient condition for K-stability of polarized del Pezzo surfaces and for the existence of a constant scalar curvature Kahler metric in the Kahler class corresponding to the polarization.
We prove that the sum of the $α$-invariants of two different Kollár components of a Kawamata log terminal singularity is less than $1$.
We study a wide class of affine varieties, which we call affine Fano varieties. By analogy with birationally super-rigid Fano varieties, we define super-rigidity for affine Fano varieties, and provide many examples and non-examples of super-rigid affine Fano varieties.
We study two rational Fano threefolds with an action of the icosahedral group $\mathfrak{A}_5$. The first one is the famous Burkhardt quartic threefold, and the second one is the double cover of the projective space branched in the Barth sextic surface. We prove that both of them are $\mathfrak{A}_5$-Fano varieties that are $\mathfrak{A}_5$-birationally superrigid. This gives two new embeddings of the group $\mathfrak{A}_5$ into the space Cremona group.
We study Tian's $α$-invariant in comparison with the $α_1$-invariant for pairs $(S_d,H)$ consisting of a smooth surface $S_d$ of degree $d$ in the projective three-dimensional space and a hyperplane section $H$. A conjecture of Tian asserts that $α(S_d,H)=α_1(S_d,H)$. We show that this is indeed true for $d=4$ (the result is well known for $d\leqslant 3$), and we show that $α(S_d,H)<α_1(S_d,H)$ for $d\geqslant 8$ provided that $S_d$ is general enough. We also construct examples of $S_d$, for $d=6$ and $d=7$, for which Tian's conjecture fails. We provide a candidate counterexample for $S_5$.
We give an explicit counter-example to a conjecture of Kyusik Hong and Joonyeong Won about $α$-invariants of polarized smooth del Pezzo surfaces of degree one.
We prove that $\frac{2}{d}$, $\frac{2d-3}{(d-1)^2}$, $\frac{2d-1}{d(d-1)}$, $\frac{2d-5}{d^2-3d+1}$ and $\frac{2d-3}{d(d-2)}$ are the smallest log canonical thresholds of reduced plane curves of degree $d\geqslant 3$, and we describe reduced plane curves of degree $d$ whose log canonical thresholds are these numbers. As an application, we prove that $\frac{2}{d}$, $\frac{2d-3}{(d-1)^2}$, $\frac{2d-1}{d(d-1)}$, $\frac{2d-5}{d^2-3d+1}$ and $\frac{2d-3}{d(d-2)}$ are the smallest values of the $α$-invariant of Tian of smooth surfaces in $\mathbb{P}^3$ of degree $d\geqslant 3$. We also prove that every reduced plane curve of degree $d\geqslant 4$ whose log canonical threshold is smaller than $\frac{5}{2d}$ is GIT-unstable for the action of the group $\mathrm{PGL}_3(\mathbb{C})$, and we describe GIT-semistable reduced plane curves with log canonical thresholds $\frac{5}{2d}$.
We prove that the quartic threefolds defined by $$ \sum_{i=0}^{5}x_i=\sum_{i=0}^{5}x_i^4-t\left(\sum_{i=0}^{5}x_i^2\right)^2=0 $$ in $\mathbb{P}^5$ are rational for $t=\frac{1}{6}$ and $t=\frac{7}{10}$.
On del Pezzo surfaces, we study effective ample $\mathbb{R}$-divisors such that the complements of their supports are isomorphic to $\mathbb{A}^1$-bundles over smooth affine curves.
We study quartic double solids admitting icosahedral symmetry.
For each del Pezzo surface $S$ with du Val singularities, we determine whether it admits a $(-K_S)$-polar cylinder or not. If it allows one, then we present an effective $\mathbb{Q}$-divisor $D$ that is $\mathbb{Q}$-linearly equivalent to $-K_S$ and such that the open set $S\setminus\mathrm{Supp}(D)$ is a cylinder. As a corollary, we classify all the del Pezzo surfaces with du Val singularities that admit nontrivial $\mathbb{G}_a$-actions on their affine cones defined by their anticanonical divisors.
We show that affine cones over smooth cubic surfaces do not admit non-trivial $\mathbb{G}_a$ -actions.
We study log canonical thresholds on quartic threefolds, quintic fourfolds, and double spaces. As an application, we show that they have a Kaehler-Einstein metric if they are general.
We prove that every quasi-smooth hypersurface in the 95 families of weighted Fano threefold hypersurfaces is birationally rigid.
For every smooth del Pezzo surface $S$, smooth curve $C\in|-K_{S}|$ and $β\in(0,1]$, we compute the $α$-invariant of Tian $α(S,(1-β)C)$ and prove the existence of Kähler--Einstein metrics on $S$ with edge singularities along $C$ of angle $2πβ$ for $β$ in certain interval. In particular we give lower bounds for the invariant $R(S,C)$, introduced by Donaldson as the supremum of all $β\in(0,1]$ for which such a metric exists.