Related papers: Effective non-vanishing and Mukai type conjecture

We develop some ideas of Morrison and Plesser and formulate a precise mathematical conjecture which has close relations to toric mirror symmetry. Our conjecture, we call it Toric Residue Mirror Conjecture, claims that the generating…

In this article, we will prove the Generalized Nonvanishing Conjecture holds for threefolds with either $\kappa>0$ or $q>0$. As a result, we can prove the Iitaka conjecture $C_{n,m}$ holds for $n=7$ if the source space has non-negative…

We study Diophantine arithmetic properties of birational divisors in conjunction with concepts that surround $\mathrm{K}$-stability for Fano varieties. There is also an interpretation in terms of the barycentres of Newton-Okounkov bodies.…

Geometric Manin's conjecture for complex Fano varieties describes the structure of the moduli space of curves. We propose a version of this conjecture in characteristic $p$ and describe its connection to the Batyrev--Manin--Peyre--Tschinkel…

We give two proofs of a conjecture of Hori and Vafa which expresses the J-function (i.e, the generating function for 1-point descendant Gromov-Witten invariants) of a Grassmannian in terms of the J-function of a product of projective…

We construct several new families of Fano varieties of K3 type. We give a geometrical explanation of the K3 structure and we link some of them to projective families of irreducible holomorphic symplectic manifolds.

In this paper we study the algebraic ranks of foliations on $\mathbb{Q}$-factorial normal projective varieties. We start by establishing a Kobayashi-Ochiai's theorem for Fano foliations in terms of algebraic rank. We then investigate the…

We show that being a general fibre of a Mori fibre space is a rather restrictive condition for a Fano variety. More specifically, we obtain two criteria (one sufficient and one necessary) for a Q-factorial Fano variety with terminal…

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 give the first examples of flat fiber type contractions of Fano manifolds onto varieties that are not weak Fano, and we prove that these morphisms are Fano conic bundles. We also review some known results about the interaction between…

Let X be a Fano manifold of dimension n and index n-3. Kawamata proved the non vanishing of the global sections of the fundamental divisor in the case n=4. Moreover he proved that if Y is a general element of the fundamental system then Y…

The paper proposes and motivates a conjecture on the invariance of cohomological support loci under derived equivalence. It contains a proof in the case of surfaces, and explains further developments and consequences.

We show that the anti-canonical volume of an $n$-dimensional K\"ahler-Einstein $\mathbb{Q}$-Fano variety is bounded from above by certain invariants of the local singularities, namely $\mathrm{lct}^n\cdot\mathrm{mult}$ for ideals and the…

We prove a conjecture of L.Bonavero, C. Casagrande, O. Debarre and S. Druel, on the pseudo-index of smooth Fano varieties, in the special case of horospherical varieties.

We conjecture a formula for the generating function of genus one Gromov-Witten invariants of the local Calabi-Yau manifolds which are the total spaces of splitting bundles over projective spaces. We prove this conjecture in several special…

We prove the weak relative Kawamata-Morrison movable cone conjecture for K-trivial fibrations whose very general fibre is a quotient, by a finite group of automorphisms acting freely in codimension one, of a product of certain Calabi-Yau…

The Virasoro conjecture proposed by Eguchi-Hori-Xiong and S. Katz predicts that the generating function of Gromov-Witten invariants is annihilated by infinitely many differential operators which form a half branch of the Virasoro algebra.…

In this short note we show the unboundedness of the dimension of the K-moduli space of $n$-dimensional Fano varieties, and that the dimension of the stack can also be unbounded while, simultaneously, the dimension of the corresponding…

The Virasoro conjecture is a conjectured sequence of relations among the descendent Gromov-Witten invariants of a smooth projective variety in all genera; the only varieties for which it is known to hold are a point (Kontsevich) and…

We study a natural birational invariant for varieties over finite fields and show that its vanishing on projective space is equivalent to the Tate conjecture, the Beilinson conjecture, and the Grothendieck--Serre semi-simplicity conjecture…