Related papers: Equivalence classes for smooth Fano polytopes

We propose a new method to study birational maps between Fano varieties based on multiplier ideal sheaves. Using this method, we prove equivariant birational rigidity of four Fano threefolds acted on by the group A6. As an application, we…

The Manin-Peyre conjecture is established for a class of smooth spherical Fano varieties of semisimple rank one. This includes all smooth spherical Fano threefolds of type T as well as some higher-dimensional smooth spherical Fano…

We classify closed, conformally flat Lorentzian manifolds of dimension $n \geq 3$ with unipotent holonomy in PO(2,n). They are all Kleinian and fall into four different geometric types according to the intersection of the image of the…

We classify the possible images of the action of the group of automorphisms of a smooth Fano threefold on its Picard group. We also study the first group cohomology of the Picard group for families of smooth Fano threefolds.

We classify toric Fano threefolds having at worst terminal singularities such that a rank of a $G$-invariant part of a class group equals one, where $G$ is a group acting on the variety by automorphisms.

This paper classifies toric Fano 3-folds with singular locus { 1/k(1,1,1) } for any positive integer k, building on the work of Batyrev and Watanabe-Watanabe. This is achieved by completing an equivalent problem in the language of Fano…

We study smoothings of Fano threefolds. We prove that the Picard number remains constant in the case of terminal Gorenstein singularities.

We consider unimodality and related properties of f-vectors of polytopes in various dimensions. By a result of Kalai (1988), f-vectors of 5-polytopes are unimodal. In higher dimensions much less can be said; we give an overview on current…

Let p and $\ell$ be two distinct primes, F a p-adic field and n an integer. We show that any level 0 block of the category of smooth Z $\ell$-valued representations of GL n (F) is equivalent to the unipotent block of an appropriate product…

The degree zero part of the quantum cohomology algebra of a smooth Fano toric symplectic manifold is determined by the superpotential function, W, of its moment polytope. In particular, this algebra is semisimple, i.e. splits as a product…

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

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…

Let $M$ and $N$ be smooth manifolds, with $M$ closed and connected. If the $C^r$--diffeomorphism group of $M$ is elementarily equivalent to the $C^s$--diffeomorphism group of $N$ for some $r,s\in[1,\infty)\cup\{0,\infty\}$, then $r=s$ and…

The purpose of this note is to give a generalization of the statement that the anticanonical class of a (smooth) projective toric variety is the sum of invariant prime divisors, corresponding to the rays in its fan (or facets in its…

A variety is said to satisfy Condition (A) if every finite abelian subgroup of its automorphism group has a fixed point. We show that a smooth Fano 3-fold not satisfying Condition (A) is K-polystable unless it is contained in eight…

We show that the set of families of smooth well-formed Fano weighted complete intersections admits a natural partition with respect to the variance $\mathrm{var}(X) = \mathrm{coind}(X) - \mathrm{codim}(X)$. Moreover, we obtain the…

Let $X$ be an $n$-dimensional complex Fano manifolds $(n\geq 3)$. Assume that $X$ contains a divisor $A$, which is isomorphic to a rational homogeneous space with Picard number one, such that the conormal bundle $\mathscr{N}^*_{A/X}$ is…

We give sufficient conditions for the semisimplicity of quantum cohomology of Fano varieties of Picard rank 1. We apply these techniques to prove new semisimplicity results for some Fano varieties of Picard rank 1 and large index. We also…

We study a subclass of K\"ahler-Einstein Fano polygons and how they behave under mutation. The polygons of interest are K\"ahler-Einstein Fano triangles and symmetric Fano polygons. In particular, we find an explicit bound for the number of…

Denote by $E_r$ the $r^{th}$ elementary symmetric polynomial in $\dim V$ variables for a vector space $V$ over an infinite field $\Bbbk$. We describe the rational points on the Fano scheme $F_{d-1}(Z(E_{\dim V-1}))$ of projective…