ﻻ يوجد ملخص باللغة العربية
Let $f : X to S$ be a family of smooth projective algebraic varieties over a smooth connected quasi-projective base $S$, and let $mathbb{V} = R^{2k} f_{*} mathbb{Z}(k)$ be the integral variation of Hodge structure coming from degree $2k$ cohomology it induces. Associated to $mathbb{V}$ one has the so-called Hodge locus $textrm{HL}(S) subset S$, which is a countable union of special algebraic subvarieties of $S$ parametrizing those fibres of $mathbb{V}$ possessing extra Hodge tensors (and so conjecturally, those fibres of $f$ possessing extra algebraic cycles). The special subvarieties belong to a larger class of so-called weakly special subvarieties, which are subvarieties of $S$ maximal for their algebraic monodromy groups. For each positive integer $d$, we give an algorithm to compute the set of all weakly special subvarieties $Z subset S$ of degree at most $d$ (with the degree taken relative to a choice of projective compactification $S subset overline{S}$ and very ample line bundle $mathcal{L}$ on $overline{S}$). As a corollary of our algorithm we prove conjectures of Daw-Ren and Daw-Javanpeykar-Kuhne on the finiteness of sets of special and weakly special subvarieties of bounded degree.
We discuss and extend some of the results obtained in Arakelov inequalities and the uniformization of certain rigid Shimura varieties (math.AG/0503339), restricting ourselves to the two dimensional case, i.e. to surfaces Y mapping generically finite
Let (G, +) be an abelian group. A subset of G is sumfree if it contains no elements x, y, z such that x +y = z. We extend this concept by introducing the Schur degree of a subset of G, where Schur degree 1 corresponds to sumfree. The classical inequa
More than four decades ago, Eisenbud, Khimv{s}iav{s}vili, and Levine introduced an analogue in the algebro-geometric setting of the notion of local degree from differential topology. Their notion of degree, which we call the EKL-degree, can be though
In this paper we extend the construction of the canonical polarized variation of Hodge structures over tube domain considered by B. Gross in cite{G} to bounded symmetric domain and introduce a series of invariants of infinitesimal variation of Hodge
We determine the structure of the ring of Siegel modular forms of degree 2 in characteristic 3.