Related papers: A note on Fano manifolds whose second Chern charac…

This article settles the question of existence of smooth weak Fano threefolds of Picard number two with small anti-canonical map and previously classified numerical invariants obtained by blowing up certain curves on smooth Fano threefolds…

In this note, we classify smooth equivariant compactifications of $\mathbb{G}_a^n$ which are Fano manifolds with index $\geq n-2$.

We prove that smooth Fano 5-folds with nef tangent bundles and Picard numbers greater than one are rational homogeneous manifolds.

Fano varieties are subvarieties of the Grassmannian whose points parametrize linear subspaces contained in a given projective variety. These expository notes give an account of results on Fano varieties of complete intersections, with a…

We show that there exists a toric Fano manifold of dimension $10$ that does not admit an extremal K\"ahler metric in the first Chern class, answering a question of Mabuchi. By taking a product with a suitable toric Fano manifold, one can…

The complete local classification and geometric description of n-dimensional submanifolds F with recurrent nonparallel second fundamental form in the spaces of constant curvature M(c) are obtained in this article.

For any positive integer $k$ and any integer $n$ large enough, we construct a Fano variety $X$ with Picard number $k$ and dimension $n$ such that $((-K_X)^n)^{1/n}$ grows like $n^k/(\log n)^{k-1}$.

We address the problem of classification of contact Fano manifolds. It is conjectured that every such manifold is necessarily homogeneous. We prove that the Killing form, the Lie algebra grading and parts of the Lie bracket can be read from…

We find all K-stable smooth Fano threefolds in the family No. 2.22.

By using the classification of Fano 3-folds we prove: Let $X$ be Fano 3-fold. Assume that the tangent bundle $T_X$ of $X$ is not stable (i.e. semi-stable or unstable). Then $b_2\geq 2$ and a relative tangent sheaf $T_{X/Y}$ of a contraction…

We present a unifying point of view which allows to understand spectral features reported in recent experiments with two-dimensional arrays of subwavelength holes in metal films. We develop a Fano analysis of the related scattering problem…

This paper aims to develop a theory of projective and affine structures on higher-dimensional varieties in positive characteristic. This theory deals with Frobenius-projective and Frobenius-affine structures, which have been previously…

Fano manifolds are basic building blocks in geometry - they are, in a precise sense, atomic pieces of shapes. The classification of Fano manifolds is therefore an important problem in geometry, which has been open since the 1930s. One can…

We completely classify toric weakened Fano 3-folds, that is, smooth toric weak Fano 3-folds which are not Fano but are deformed to smooth Fano 3-folds. There exist exactly 15 toric weakened Fano 3-folds up to isomorphisms.

We classify compact K\"ahler manifolds with semi-positive holomorphic bisectional and big tangent bundles. We also classify compact complex surfaces with semi-positive tangent bundles and compact complex $3$-folds of the form $P(T^*X)$…

In this note we collect some results on the deformation theory of toric Fano varieties.

In this paper, we prove a general principle of lifting an automorphism from positive characteristic to zero characteristic. We based on the principle to prove the automorphism group of Fano variety of cubic threefold (fourfold) acts on its…

This is a survey on the Fano schemes of linear spaces, conics, rational curves, and curves of higher genera in smooth projective hypersurfaces, complete intersections, Fano threefolds, etc.

We describe the closed cone of moving curves of smooth Fano three- and fourfolds by giving finitely many equations that cut out the cone. The equations are induced by the exceptional divisors of divisorial contractions and by nef divisors…

Fano surfaces parametrize the lines of smooth cubic threefolds. In this paper, we study their quotients by some of their automorphism sub-groups. We obtain in that way some interesting surfaces of general type.