Related papers: Fano symmetric varieties with low rank

We construct an explicit semistable degeneration of a Fano eightfold of index three and deduce its Hodge numbers, in particular we show that it has Picard rank one. The Fano variety is of K3 type and it is defined as a connected component…

We give a necessary and sufficient condition for the nonsingular projective toric variety associated to the graph cubeahedron of a finite simple graph to be Fano or weak Fano in terms of the graph.

We construct exceptional Fano varieties with the smallest known minimal log discrepancies in all dimensions. These varieties are well-formed hypersurfaces in weighted projective space. Their minimal log discrepancies decay doubly…

Real Legendrian subvarieties are classical objects of differential geometry and classical mechanics and they have been studied since antiquity. However, complex Legendrian subvarieties are much more rigid and have more exceptional…

Symmetric varieties are normal equivarient open embeddings of symmetric homogeneous spaces, and they are interesting examples of spherical varieties. We prove that all smooth Fano symmetric varieties with Picard number one admit…

We obtain a combinatorial description of Gorenstein spherical Fano varieties in terms of certain polytopes, generalizing the combinatorial description of Gorenstein toric Fano varieties by reflexive polytopes and its extension to Gorenstein…

The present paper provides a geometric characterization of complete flag varieties for semisimple algebraic groups. Namely, if $X$ is a Fano manifold whose all elementary contractions are $\mathbb P^1$-fibrations then $X$ is isomorphic to…

In this article, we introduce a new approach to show the existence and smoothing of simple normal crossing varieties in a given projective space. Our approach relates the above to the existence of nowhere reduced schemes called ribbons and…

We define the nef complexity of a projective variety $X$. This invariant compares $\dim X+\rho(X)$ with the sum of the coefficients of nef partitions of $-K_X$. We prove that the nef complexity is non-negative and it is zero precisely for…

We study the Picard rank of smooth toric Fano varieties possessing families of minimal rational curves of given degree. We discuss variants of a conjecture of Chen-Fu-Hwang and prove a version of their statement that recovers the original…

We classify smooth Fano threefolds with infinite automorphism groups.

We obtain 866 isomorphism classes of five-dimensional nonsingular toric Fano varieties using a computer program and the database of four-dimensional reflexive polytopes. The algorithm is based on the existence of facets of Fano polytopes…

We prove that all smooth Fano threefolds with Picard rank 2 and degree 28 are K-polystable, except for some explicit cases which we describe. We also give a classification of the normal bundle of a rational normal quartic curve in a smooth…

In this paper, we consider terminal reflexive polytopes arising from finite directed graphs and study the problem of deciding which directed graphs yield smooth Fano polytopes. We show that any centrally symmetric or pseudo-symmetric smooth…

In this paper, we classify smooth toric Fano 5-folds of index 2. There exist exactly 10 smooth toric Fano 5-folds of index 2 up to isomorphisms.

A symplectic manifold is called symplectic rationally connected if there is a non-zero genus zero Gromov-Witten invariant with two point insertions. It is conjectured that every smooth projective rationally connected variety is symplectic…

We prove that the Satake-Baily-Borel compactification of certain Shimura varieties are Fano varieties, Calabi-Yau varieties or have ample canonical divisors with mild singularities. We also prove some variants statements, give applications…

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…

The Manin-Peyre conjecture is established for smooth spherical Fano threefolds of semisimple rank one and type N. Together with the previously solved case T and the toric cases, this covers all types of smooth spherical Fano threefolds. The…

We classify rank two vector bundles on a Fano threefold of Picard rank one whose projectivizations are weak Fano. We also prove the existence of examples for each case of the classification result. Our classification includes detailed…