Related papers: A survey on birational Rigidity of threefold Weigh…
We show that mixed-characteristic and equi-characteristic small deformations of 3-dimensional canonical (resp. terminal) singularities with perfect residue field of characteristic $p>5$ are canonical (resp. terminal). We discuss…
We prove rigidity of various types of holomorphic parabolic geometry on smooth complex projective varieties.
We give several examples of pairs of non-isomorphic cubic fourfolds whose Fano varieties of lines are birationally equivalent (and in one example isomorphic). Two of our examples, which are special families of conjecturally irrational…
Our earlier proof of mirror formulas for genus 0 Gromov -- Witten invariants of Fano and Calabi -- Yau toric complete intersections is illustrated in the example of quintic 3-folds.
In this paper, we investigate the question of triviality of the rational Chow groups of complete intersections in projective spaces and obtain improved bounds for this triviality to hold. Along the way, we study the dimension and…
We study Fano threefolds with~terminal singularities admitting a "minimal" action of a finite group. We prove that under certain additional assumptions such a variety does not contain planes. We also obtain an upper bounds of the number of…
We construct a $14$-dimensional affine variety $\Sigma^{14}_{\mathbb{A}}$ with a $\rm{GL}_3$- and a $(\mathbb{C}^*)^6$-actions. We denote by $\Sigma^{13}_{\mathbb{A}}$ the affine variety obtained from $\Sigma^{14}_{\mathbb{A}}$ by setting…
We prove birational superrigidity of direct products $V=F_1\times...\times F_K$ of primitive Fano varieties of the following two types: either $F_i\subset{\mathbb P}^M$ is a general hypersurface of degree $M$, $M\geq 6$, or…
This is an overview article, based on my 2018 Kinosaki lecture, that surveys and announces work on 3-fold flopping contractions, their affine combinatorics, stability conditions, tilting bundles and autoequivalences. Some first applications…
We prove an optimal result on the birational rigidity and K-stability of index $1$ hypersurfaces in $\mathbb{P}^{n+1}$ with ordinary singularities when $n\gg 0$ and also study the birational superrigidity and K-stability of certain weighted…
It is well-known that a nonsingular minimal cubic surface is birationally rigid; the group of its birational selfmaps is generated by biregular selfmaps and birational involutions such that all relations between the latter are implied by…
Extending some results of Crauder and Katz, and Ein and Shepherd-Barron on special Cremona transformations, we study birational transformations of the complex projective spaces onto prime Fano manifolds such that the base locus X of the…
We get sharp degree bound for generic smoothness and connectedness of the space of conics in low degree complete intersections which generalizes the old work about Fano scheme of lines on Hypersurfaces.
Here we investigate the birational geometry of projective varieties of arbitrary dimension having defective higher secant varieties. We apply the classical tool of tangential projections and we determine natural conditions for uniruledness,…
In this article, we determine all equivariant compactifications of the three-dimensional vector group $\mathbf{G}_a^3$ which are smooth Fano threefolds with Picard number greater or equal than two.
We study the physics of 2 and 3 mutually intersecting conformal defects forming wedges and corners in general dimension. For 2 defects we derive the beta function of the edge interactions for infinite and semi-infinite wedges and study them…
We list combinatorial criteria of some singularities, which appear in the Minimal Model Program or in the study of (singular) Fano varieties, for spherical varieties. Most of the results of this paper are already known or are quite easy…
We begin the study of completeness of affine connections, especially those on statistical manifolds as well as on affine hypersurfaces. We collect basic facts, prove new theorems and provide examples with remarkable properties.
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…
We consider complete Riemannian manifolds which satisfy a weighted Poincar\`e inequality and have the Ricci curvature bounded below in terms of the weight function. When the weight function has a non-zero limit at infinity, the structure of…