Related papers: Shokurov's Rational Connectedness Conjecture
Conjectures on the existence of zero-cycles on arbitrary smooth projective varieties over number fields were proposed by Colliot-Th\'el\`ene, Sansuc, Kato and Saito in the 1980's. We prove that these conjectures are compatible with…
Given a family of varieties over the projective line, we study the density of fibres that are everywhere locally soluble in the case that components of higher multiplicity are allowed. We use log geometry to formulate a new sparsity…
We associate at each link a connectivity space which describes its splittability properties. Then, the notion of order for finite connectivity spaces results in the definition of a new numerical invariant for links, their connectivity…
We state the analogs of Kontsevich's formality conjecture for Hochschild and cyclic chains, as well as their
We study rationality properties of irreducible characters of finite groups. We show that the continuity of $2$-rationality is a phenomenon that can be detected in the principal $2$-block, thus refining a recent result of N. N. Hung. We also…
In this paper, we proved two results regarding the arithmetics of separably $\mathbb{A}^1$-connected varieties of rank one. First we proved over a large field, there is an $\mathbb{A}^1$-curve through any rational point of the boundary, if…
Let T_n denote the set of log canonical thresholds of pairs (X,Y), with X a nonsingular variety of dimension n, and Y a nonempty closed subscheme of X. Using non-standard methods, we show that every limit of a decreasing sequence in T_n…
We show that normalized Schur polynomials are strongly log-concave. As a consequence, we obtain Okounkov's log-concavity conjecture for Littlewood-Richardson coefficients in the special case of Kostka numbers.
This paper presents three results on F-singularities. First, we give a new proof of Eisenstein's restriction theorem for adjoint ideal sheaves, using the theory of F-singularities. Second, we show that a conjecture of Musta\c{t}\u{a} and…
We consider surjective endomorphisms f of degree > 1 on projective manifolds X of Picard number one and their f^{-1}-stable hypersurfaces V, and show that V is rationally chain connected. Also given is an optimal upper bound for the number…
In this paper, we study rational sections of the relative Picard variety of a linear system on a smooth projective variety. Specifically, we prove that if the linear system is basepoint-free and the locus of non-integral divisors has…
The main result of this note is an effective uniform bound for the number of deformation types of certain nonisotrivial families of canonically polarized manifolds. It extends the author's earlier such bound for the classical Shafarevich…
We prove that a one-parameter family of rationally connected varieties (over an algebraically closed field of characteristic 0) always has a section.
We study the connectedness of the real locus of smooth geometrically rational Fano threefolds and prove a sufficient criterion of $\mathbb{R}$-rationality.
The notion of 'slope rational connectedness' is introduced in the context of smooth orbifold pairs. The main result parallels the characterization of the rational connectedness of projective manifolds in terms of either the non-existence of…
We resolve Schinzel's Hypothesis (H) for $100\%$ of polynomials of arbitrary degree. We deduce that a positive proportion of diagonal conic bundles over $\mathbb{Q}$ with any given number of degenerate fibres have a rational point, and…
A theorem of Graber, Harris, and Starr states that a rationally connected fibration over a curve has a section. We study an analogous question in symplectic geometry. Namely, given a rationally connected fibration over a curve, can one find…
We determine the relationship between the contact structure induced by a fibered knot, K, in the three-sphere and the contact structures induced by its various cables. Understanding this relationship allows us to classify fibered cable…
We prove the ACC conjecture for local volumes. Moreover, when the local volume is bounded away from zero, we prove Shokurov's ACC conjecture for minimal log discrepancies.
A recurring difficulty in the Minimal Model Program is that while log terminal singularities are quite well behaved (for instance, they are rational), log canonical singularities are much more complicated; they need not even be…