Related papers: Singularities on Fano fibrations and beyond

We give a treatment of relative Calabi--Yau structures on functors between $R$-linear stable $\infty$-categories, with $R$ any $\mathbb{E}_\infty$-ring spectrum, generalizing previous treatments in the setting of dg-categories. Using their…

In this paper, we study equivariant Hurewicz fibrations, obtain their internal characteristics, and prove theorems on relationship between equivariant fibrations and fibrations generated by them. Local and global properties of equivariant…

We give some bounds on the anticanonical degrees of Fano varieties with Picard number 1 and mild singularities, extending results of Koll\'ar et al. from the early 90's and improving them even in the smooth case. The proof is based on a…

We introduce and study a new notion of stability for varieties fibered over curves, motivated by Koll\'ar's stability for homogeneous polynomials with integral coefficients. We develop tools to study geometric properties of stable…

We propose a combinatorical duality for lattice polyhedra which conjecturally gives rise to the pairs of mirror symmetric families of Calabi-Yau complete intersections in toric Fano varieties with Gorenstein singularities. Our construction…

The principal aim of this note is to give an elementary proof of the fact that any two fiber functors of a Tannakian category are locally isomorphic. This builds on an idea of Deligne concerning scalar extensions of Tannakian categories and…

Toric Landau--Ginzburg models of Givental's type for Fano complete intersections are known to have Calabi--Yau compactifications. We give an alternative proof of this fact. As an output of our proof we get a description of fibers over…

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…

In this paper, we propose a connection between the field theory local model (Katz-Vafa field theory) and the type of singular fibre in flat crepant resolutions of elliptic Calabi-Yau fourfolds, a class of fourfolds considered by Esole and…

Let $(X,\Delta)$ be a projective klt pair, and $f:X\to Y$ a fibration to a smooth projective variety $Y$ with strictly nef relative anti-log canonical divisor $-(K_{X/Y}+\Delta)$. We prove that $f$ is a locally constant fibration with…

A known conjecture of Grinenko in birational geometry asserts that a Mori fibre space with the structure of del Pezzo fibration of low degree is birationally rigid if and only if its anticanonical class is an interior point in the cone of…

We prove birational superrigidity of every hypersurface of degree N in P^N with singular locus of dimension s, under the assumption that N is at least 2s+8 and it has only quadratic singularities of rank at least N-s. Combined with the…

Assuming the Morrison-Kawamata cone conjecture for the generic fiber of a Calabi-Yau fibration and the abundance conjecture, we show (1) the finiteness of minimal models, (2) the existence of a weak rational polyhedral fundamental domain…

Let $f\colon S\to B$ a complex fibred surface with fibres of genus $g\geq 2$. Let $u_f$ be its unitary rank, i.e., the rank of the maximal unitary summand of the Hodge bundle $f_*\omega_f$. We prove many new slope inequalities involving…

We resolve a mathematically precise SYZ conjecture for $A_n$ singularity by building a quantum-corrected T-duality between two singular torus fibrations related to the Kahler geometry of the $A_n$-smoothing and the Berkovich geometry of the…

We study the quasimap invariants of elliptic and K3 fibrations. Oberdieck and Pixton conjectured that the Gromov-Witten potentials of elliptic fibrations are quasi-modular forms. Analogously, we propose similar conjecture for the quasimap…

We prove a structure theorem for the Albanese maps of varieties with Q-linearly trivial log canonical divisors. Our start point is the action of a nonlinear algebraic group on a projective variety.

In the context of string dualities, fibration structures of Calabi-Yau manifolds play a prominent role. In particular, elliptic and K3 fibered Calabi-Yau fourfolds are important for dualities between string compactifications with four flat…

We show boundedness of polarized Calabi--Yau fibrations over curves only with fixed volumes of general fibers and Iitaka volumes. As its application, we construct a separated coarse moduli space of K-stable Calabi-Yau fibrations over curves…

This is the first of two papers studying both the geometric structure of Fano fibrations and the application to K\"ahler-Ricci flows developing a singularity in finite time. Given a Fano fibration which is generated by Kawamata's theorem…