Related papers: Weil conjectures and affine hypersurfaces

Let $X$ be a smooth projective variety defined on a finite field $\mathbb{F}_q$. On $X$ there is a special morphism $Fr_X$, which raises coordinates to exponent $q$: $t\mapsto t^q$. The two main results in this paper are: Result 1: If…

We prove a conjecture of Kleinbock which gives a clear-cut classification of all extremal affine subspaces of $\mathbb{R}^n$. We also give an essentially complete classification of all Khintchine type affine subspaces, except for some…

Consider a closed connected hypersurface in $\mathbb{R}^n$ with constant signature (k,l) of the second quadratic form, and approaching a quadratic cone at infinity. This hypersurface divides $\mathbb{R}^n$ into two pieces. We prove that one…

Motivated by the theory of Inoue-type varieties, we give a structure theorem for projective manifolds $W_0$ with the property of admitting a 1-parameter deformation where $W_t$ is a hypersurface in a projective smooth manifold $Z_t$. Their…

Let $f: X \to Y$ be a dominant morphism of smooth, proper and geometrically integral varieties over a number field $k$, with geometrically integral generic fibre. We give a necessary and sufficient geometric criterion for the induced map…

Let X be a smooth or proper variety defined over a finite field. The geometric etale fundamental group of X is a normal subgroup of the Weil group, so conjugation gives it a Weil action. We consider the pro-Q_l-algebraic completion of the…

Motivated by Emmanuel Kowalski's exponential sums over definable sets in finite fields, we generalize Ax's theorem on pseudo-finite fields to a continuous-logic setting allowing for an additive character. The role played by Weil's Riemann…

The paper contains a general construction which produces new examples of non simply-connected smooth projective surfaces. We analyze the resulting surfaces and their fundamental groups. Many of these fundamental groups are expected to be…

By relying on a new approach to Lefschetz type questions based on Beilinson's singular support and Saito's characteristic cycle, we prove an instance of the wild Lefschetz theorem envisioned by Deligne. Our main tool are new finiteness…

We give a proof of a phenomenon conjectured in our former article: "Beltrami forms, affine surfaces and the Schwarz-Christoffel formula: a worked out example of straightening". We also start an abstract discussion of the notion of limits of…

In this paper we discuss the geometry of affine Deligne Lusztig varieties with very special level structure, determining their dimension and connected and irreducible components. As application, we prove the Grothendieck conjecture for…

Let $G$ be the Weil restriction of a general linear group. By extending the method of semi-modules developed by de Jong, Oort, Viehmann and Hamacher, we obtain a stratification of the affine Deligne-Lusztig varieties for $G$ (in the affine…

With the help of hyper-ideal circle pattern theory, we have developed a discrete version of the classical uniformization theorems for surfaces represented as finite branched covers over the Riemann sphere as well as compact polyhedral…

We know that semi-regular sub-varieties satisfy the variational Hodge conjecture i.e., given a family of smooth projective varieties $\pi:\mathcal{X} \to B$, a special fiber $\mathcal{X}_o$ and a semi-regular subvariety $Z \subset…

We prove the standard conjectures for complex projective varieties that are deformations of the Hilbert scheme of points on a K3 surface. The proof involves Verbitsky's theory of hyperholomorphic sheaves and a study of the cohomology…

Let $\mathcal E$ be a torus-linearised reflexive sheaf over a smooth projective toric variety. Generalising a theorem of Perlman and Smith, we prove an explicit sufficient condition for $\mathcal E$ to be acyclic via Weil decorations.

For a reduced projective scheme over the ring of integers of a number field, the set of places over which the fibres of the scheme are not reduced is a finite set. We give an explicit upper bound for the product of the norms of places in…

For the base field of complex numbers we discuss the relationship between categories of coherent sheaves on compact Riemann surfaces and categories of coherent sheaves on weighted smooth projective curves. This is done by bringing back to…

Let $X$ be a smooth irreducible projective variety of dimension at least 2 over an algebraically closed field of characteristic 0 in the projective space ${\mathbb{P}}^n$. Bertini's Theorem states that a general hyperplane $H$ intersects…

Deligne has conjectured that certain mixed Hodge theoretic invariants of complex algebraic invariants are motivic. This conjecture specializes to an algebraic construction of the Jacobian for smooth projective curves, which was done by A.…