Related papers: Known cases of the Hodge conjecture
Formal (mixed) Hodge structures FHS are introduced in such a way that the Hodge realization of Deligne's 1-motives extends to a realization from Laumon's 1-motives to formal Hodge structures of level 1, providing an equivalence of…
The goal of this work is to study the existence and properties of non constant entire curves f drawn in a complex irreducible n-dimensional variety X, and more specifically to show that they must satisfy certain global algebraic or…
A reformulation of the Hoop Conjecture based on the concept of trapped circle is presented. The problems of severe compactness in every spatial direction, and of how to superpose the hoops with the surface of the black hole, are resolved. A…
We give a rather informal introduction to the theory of mixed Hodge modules for young mathematicians.
Why do natural and interesting sequences often turn out to be log-concave? We give one of many possible explanations, from the viewpoint of "standard conjectures". We illustrate with several examples from combinatorics.
A construction similar to Hagge's construction for circles through the orthocentre is shown to apply for any point.
We prove a closed formula for integrals of the cotangent line classes against the top Chern class of the Hodge bundle on the moduli space of stable pointed curves. These integrals are computed via relations obtained from virtual…
The topics of Convexity and Concavity and Envelopes are central in Complex Analysis and extensively investigated. The aim of this paper is to find a possible counterpart in Algebraic Geometry. The article presents preliminary results on…
In a recent paper, M. Green and P. Griffiths used R. Thomas' works on nodal hypersurfaces to establish the equivalence of the Hodge conjecture and the existence of certain singular admissible normal functions. Inspired by their work, we…
''Positive geometries'' are a class of semi-algebraic domains which admit a unique ''canonical form'': a logarithmic form whose residues match the boundary structure of the domain. The study of such geometries is motivated by recent…
The Hamilton-Jacobi equation for a Hamiltonian section on a Lie affgebroid is introduced and some examples are discussed.
We give an introduction to the study of algebraic hypersurfaces, focusing on the problem of when two hypersurfaces are isomorphic or close to being isomorphic. Working with hypersurfaces and emphasizing examples makes it possible to discuss…
This thesis is devoted to the study of algebraic cycles in projective hyper-K\"ahler manifolds and strict Calabi-Yau manifolds. It contributes to the understanding of Beauville's and Voisin's conjectures on the Chow rings of projective…
In their study of the Yamabe problem in the presence of isometry group, Hebey and Vaugon announced a conjecture. This conjecture generalizes Aubin's conjecture, which has already been proven and is sufficient to solve the Yamabe problem. In…
We study Hodge loci as leaf schemes of foliations. The main ingredient is the Gauss-Manin connection matrix of families of projective varieties. We also aim to investigate a conjecture on the ring of definition of leaf schemes and its…
This note describes some open problems that can be examined with the purpose of gaining additional insight of how to solve the problem of finding a general classification of geodetic graphs
A central question in dynamics is whether the topology of a system determines its geometry. This is known as rigidity. Under mild topological conditions rigidity holds for many classical cases, including: Kleinian groups, circle…
We introduce and study a notion of large homomorphisms on the homotopy lie coalgebra; these homomorphisms are a variant of the large homomorphisms of Levin. As a consequence of our work, we establish new cases of a homotopy lie coalgebra…
Although it is important both in theory as well as in applications, a theory of Birkhoff interpolation with main emphasis on the shape of the set of nodes is still missing. Although we will consider various shapes (e.g. we find all the…
We obtain a system of relations between linear Hodge integrals. As an application, we show that its first non-trivial relation implies the Witten's Conjecture/Kontsevich Theorem.