Related papers: On the fibration method for zero-cycles and ration…

Let $K$ be the function field of a smooth curve over an algebraically closed field $k$. Let $X$ be a scheme, which is smooth and projective over $K$. Suppose that the cotangent bundle $\Omega_{X/K}$ is ample. Let $R:={\rm Zar}(X)(K)\cap X)$…

Kashiwara conjectured that the hard Lefshetz theorem and the semisimplicity theorem hold for any semisimple perverse sheaf M on a variety over a field of characteristic 0. He also conjectured that if you apply to such M the nearby cycle…

We conjecture that the relative Gromov-Witten potentials of elliptic fibrations are (cycle-valued) lattice quasi-Jacobi forms and satisfy a holomorphic anomaly equation. We prove the conjecture for the rational elliptic surface in all…

We develop some theory of double fibration transforms where the cycle space is a smooth manifold and apply it to complex projective space.

We study unramified sections of the fundamental group sequence of smooth projective curves of genus $\geq 2$ over $p$-adic fields together with an integral model. We are particularly interested in the induced specialized sections of the…

The main aim of this paper is to show that a cyclic cover of $\mathbb{P}^n$ branched along a very general divisor of degree $d$ is not stably rational provided that $n \ge 3$ and $d \ge n+1$. This generalizes the result of…

Let X be an algebraic curve over Q and t a non-constant Q-rational function on X such that Q(t) is a proper subfield of Q(X). For every integer n pick a point P_n on X such that t(P_n)=n. We conjecture that, for large N, among the number…

K-polystability is, on the one hand, conjecturally equivalent to the existence of certain canonical K\"ahler metrics on polarised varieties, and, on the other hand, conjecturally gives the correct notion to form moduli. We introduce a…

In [19] we studied a Fadell-Neuwirth type fibration theorem for orbifolds, and gave a short exact sequence of fundamental groups of configuration Lie groupoids of Lie groupoids corresponding to the genus zero 2-dimensional orbifolds with…

Let $X$ be a smooth projective variety over a finite field $\F$. We discuss the unramified cohomology group $H^3_\nr(X,\Q/\Z(2))$. Several conjectures put together imply that this group is finite. For certain classes of threefolds,…

A famous theorem in polytope theory states that the combinatorial type of a simplicial polytope is completely determined by its facet-ridge graph. This celebrated result was proven by Blind and Mani in 1987, via a non-constructive proof…

Inspired by their results on the Chow rings of projective K3 surfaces, Beauville and Voisin made the following conjecture: given a projective hyperkaehler manifold, for any algebraic cycle which is a polynomial with rational coefficients of…

Motivated by the Bloch-Beilinson conjectures, Voisin has made a conjecture concerning the behaviour of zero-cycles on self-products of Calabi-Yau varieties. This note contains some examples of Calabi-Yau fourfolds verifying Voisin's…

Consider a smooth, projective family of canonically polarized varieties over a smooth, quasi-projective base manifold Y, all defined over the complex numbers. It has been conjectured that the family is necessarily isotrivial if Y is special…

A field $F$ is a $\mathfrak{B}_s$-field if, for every finite extension $E'/E$ of $F$, the norm map $K_s^M(E')\to K_s^M(E)$ of the Milnor $K$-groups is surjective. In particular, finite fields ($s=1$), local fields, and certain global fields…

We show that a conjecture of Kotschick about one-forms without zeros on compact K\"ahler manifolds follows in the case of simple Albanese torus from a conjecture of Bobadilla and Koll\'ar about homologically trivial fibrations. As an…

For an arithmetical scheme X, K. Kato introduced a certain complex of Gersten-Bloch-Ogus type whose component in degree a involves Galois cohomology groups of the residue fields of all the points of X of dimension a. He stated a conjecture…

A new class of compact K\"ahler manifolds, called special, is defined, which are the ones having no surjective meromorphic map to an orbifold of general type. The special manifolds are in many respect higher-dimensional generalisations of…

We propose an intersection-theoretic method to reduce questions in genus zero logarithmic Gromov-Witten theory to questions in the Gromov-Witten theory of smooth pairs, in the presence of positivity. The method is applied to the enumerative…

We prove a new rigidity criterion for families of polarized Calabi-Yau manifolds. Motivated by known non-rigid examples, we conjecture that a family over a quasi-projective curve is rigid if it admits a smooth compactification whose…