Related papers: A characterization of Fano type varieties

We prove the termination of 4-fold canonical flips.

In this notes we classify toric Fano 4-folds having positive second Chern Character.

This note outlines some first steps in the classification of Fano manifolds for which $c_1^2-2c_2$ is positive or nef.

A Fano surface of a smooth cubic threefold X in P^4 parametrizes the lines on X. In this note, we prove that a Fano surface satisfies the Tate conjecture over a field of finite type over the prime field and characteristic not 2.

We classify Fano threefolds with only terminal singularities whose canonical class is Cartier and divisible by 2, and satisfying an additional assumption that the $G$-invariant part of the Weil divisor class group is of rank 1 with respect…

In this paper we prove that a regular foliation on a complex weak Fano manifold is algebraically integrable.

We give a brief survey of the concept of birational rigidity, from its origins in the two-dimensional birational geometry, to its current state. The main ingredients of the method of maximal singularities are discussed. The principal…

The purpose of this paper is to give basic tools for the classification of nonsingular toric Fano varieties by means of the notions of primitive collections and primitive relations due to Batyrev. By using them we can easily deal with…

We prove several extensions of the Erdos-Fuchs theorem.

We classify smooth toric Fano varieties of dimension $n\geq 3$ containing a toric divisor isomorphic to $\PP^{n-1}$. As a consequence of this classification, we show that any smooth complete toric variety $X$ of dimension $n\geq 3$ with a…

We observe a method for finding weak Landau-Ginzburg models for Fano varieties and find them for smooth Fano threefolds of genera 9, 10, and 12.

We prove a new restriction theorem for semistable sheaves on varieties in all characteristics strengthening previous results. We also prove restriction theorem for strong semistability for varieties with some non-negativity constrains on…

We prove that the variety of complete flags for any semisimple algebraic group is rigid in any smooth family of Fano manifolds.

We prove a variation of Gronwall's lemma.

We continue to study birational geometry of Fano fibrations $\pi\colon V\to {\mathbb P}^1$ the fibers of which are Fano double hypersurfaces of index 1. For a majority of families of this type, which do not satisfy the condition of…

In this paper we classify rank two Fano bundles $\cE$ on Fano manifolds satisfying $H^2(X,\Z)\cong H^4(X,\Z)\cong\Z$. The classification is obtained via the computation of the nef and pseudoeffective cones of the projectivization…

We introduce some invariants of Fano varieties and propose a Mukai-type conjecture which characterizes the product of projective spaces. Moreover, we prove that the Ambro--Kawamata effective non-vanishing conjecture implies the Mukai-type…

We develop a general approach to prove K-stability of Fano varieties. The new theory is used to (a) prove the existence of K\"ahler-Einstein metrics on all smooth Fano hypersurfaces of Fano index two, (b) to compute the stability thresholds…

We study K-stability of products of K-stable $\mathbb{Q}$-Fano varieties.

Fano mechanism is the universal explanation of asymmetric resonance appearing in different systems. We report the evidence of Fano-like resonance in selective reflection from a resonant two-level medium. We draw an analogy with the…