Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSpecial subvarieties of A_g
119
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 to the moduli stack of Abelian varieties. In particular we show that Y is a Hilber modular surfaces if and only if the dergee of the Hodge bundle satisfies the Arakelov equality. In the revised version, we corrected some minor mistakes, pointed out by the referee, and we tried to improve the presentation of the text.
rate research
Read More
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 consider Kobayashi geodesics in the moduli space of abelian varieties A_g that is, algebraic curves that are totally geodesic submanifolds for the Kobayashi metric. We show that Kobayashi geodesics can be characterized as those curves whose logarithmic tangent bundle splits as a subbundle of the logarithmic tangent bundle of A_g. Both Shimura curves and Teichmueller curves are examples of Kobayashi geodesics, but there are other examples. We show moreover that non-compact Kobayashi geodesics always map to the locus of real multiplication and that the Q-irreducibility of the induced variation of Hodge structures implies that they are defined over a number field.
We classify the subvarieties of infinite dimensional affine space that are stable under the infinite symmetric group. We determine the defining equations and point sets of these varieties as well as the containments between them.
Fix integers $ageq 1$, $b$ and $c$. We prove that for certain projective varieties $Vsubset{bold P}^r$ (e.g. certain possibly singular complete intersections), there are only finitely many components of the Hilbert scheme parametrizing irreducible, smooth, projective, low codimensional subvarieties $X$ of $V$ such that $$ h^0(X,Cal O_X(aK_X-bH_X)) leq lambda d^{epsilon_1}+c(sum_{1leq h < epsilon_2}p_g(X^{(h)})), $$ where $d$, $K_X$ and $H_X$ denote the degree, the canonical divisor and the general hyperplane section of $X$, $p_g(X^{(h)})$ denotes the geometric genus of the general linear section of $X$ of dimension $h$, and where $lambda$, $epsilon_1$ and $epsilon_2$ are suitable positive real numbers depending only on the dimension of $X$, on $a$ and on the ambient variety $V$. In particular, except for finitely many families of varieties, the canonical map of any irreducible, smooth, projective, low codimensional subvariety $X$ of $V$, is birational.
We establish a Grothendieck--Lefschetz theorem for smooth ample subvarieties of smooth projective varieties over an algebraically closed field of characteristic zero and, more generally, for smooth subvarieties whose complement has small cohomological dimension. A weaker statement is also proved in a more general context and in all characteristics. Several applications are included.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
