In the Fall 2018 Semester the CMSA will be hosting a seminar on Hodge and Noether-Lefschetz loci, with lectures given by Hossein Movasati (IMPA). The seminar will occur weekly on Wednesday at 1:30 in room G10 of the CMSA.
The schedule below will be updated as talks are confirmed.
|11/7/2018||Title: Hodge and Noether-Lefschetz loci
Abstract: Hodge cycles are topological cycles which are conjecturally (the millennium Hodge conjecture) supported in algebraic cycles of a given smooth projective complex manifold. Their study in families leads to the notion of Hodge locus, which is also known as Noether-Lefschetz locus in the case of surfaces. The main aim of this mini course is to introduce a computational approach to the study of Hodge loci for hypersurfaces and near the Fermat hypersurface. This will ultimately lead to the verification of the variational Hodge conjecture for explicit examples of algebraic cycles inside hypersurfaces and also the verification of integral Hodge conjecture for examples of Fermat hypersurfaces. Both applications highly depend on computer calculations of rank of huge matrices. We also aim to review some classical results on this topic, such as Cattani-Deligne-Kaplan theorem on the algebraicity of the components of the hodge loci, Deligne’s absolute Hodge cycle theorem for abelian varieties etc.
In the theoretical side another aim is to use the available tools in algebraic geometry and construct the moduli space of projective varieties enhanced with elements in their algebraic de Rham cohomology ring. These kind of moduli spaces have been useful in mathematical physics in order to describe the generating function of higher genus Gromov-Witten invariants, and it turns out that the Hodge loci in such moduli spaces are well-behaved, for instance, they are algebraic leaves of certain holomorphic foliations. Such foliations are constructed from the underlying Gauss-Manin connection. This lectures series involves many reading activities on related topics, and contributions by participants are most welcome.
|11/14/2018||Title: Integral Hodge conjecture for Fermat varieties
Abstract: We describe an algorithm which verifies whether linear algebraic cycles of the Fermat variety generate the lattice of Hodge cycles. A computer implementation of this confirms the integral Hodge conjecture for quartic and quintic Fermat fourfolds. Our algorithm is based on computation of the list of elementary divisors of both the lattice of linear algebraic cycles, and the lattice of Hodge cycles written in terms of vanishing cycles, and observing that these two lists are the same. This is a joint work with E. Aljovin and R. Villaflor.
|11/21/2018||Title: Periods of algebraic cycles
Abstract: The tangent space of the Hodge locus at a point can be described by the so called infinitesimal variation of Hodge structures and the cohomology class of Hodge cycles. For hypersurfaces of dimension $n$ and degree $d$ it turns out that one can describe it without any knowledge of cohomology theories and in a fashion which E. Picard in 1900’s wanted to study integrals/periods. The data of cohomology class is replaced with periods of Hodge cycles, and explicit computations of these periods, will give us a computer implementable description of the tangent space. As an application of this we show that for examples of $n$ and $d$, the locus of hypersurfaces containing two linear cycles whose intersection is of low dimension, is a reduced component of the Hodge locus in the underlying parameter space.
|11/28/2018||Title: Periods of Complete Intersection Algebraic Cycles
Speaker: Roberto Villaflor
Abstract: In order to compute periods of algebraic cycles inside even dimensional smooth degree d hypersurfaces of the projective space, we restrict ourselves to cycles supported in a complete intersection subvariety. When the description of the complete intersection is explicit, we can compute its periods, and furthermore its cohomological class. As an application, we can use this data to describe the Zariski tangent space of the corresponding Hodge locus, as the degree d part of some Artinian Gorenstein ideal of the homogeneous coordinate ring of the projective space. Using this description, we can show that for d>5, the locus of hypersurfaces containing two linear cycles, is a reduced component of the Hodge locus in the underlying parameter space.
|Title: Some explicit Hodge cycles
Abstract: Explicit examples of Hodge cycles are due to D. Mumford and A. Weil in the case of CM abelian varieties. In this talk, I will describe few other examples for the Fermat variety. Effective verification of the Hodge conjecture for these cycles is not known.
|12/12/2018||Title: A conjectural Hodge locus for cubic tenfold
Abstract: In this talk we will consider the difference of two linear algebraic cycles of dimension 5 inside a smooth cubic tenfold and such that the dimension of their intersection is 3. We will show some computer assisted evidences to the fact that the corresponding Hodge locus is bigger than the expected locus of algebraic deformations of the cubic tenfold together with its linear cycles. A similar discussion will be also presented for cubic six and eightfold, for which we will prove that the corresponding second and third order infinitesimal Hodge loci are smooth. The main ingredient is a computer implementation of power series of periods of hypersurfaces.
|1/16/2019||Title: Algebraic BCOV anomaly equation
Abstract: We introduce the moduli space T of non-rigid compact Calabi-Yau threefolds enhanced with differential forms and a Lie algebra of vector fields in T. This will be used in order to give a purely algebraic interpretation of topological string partition functions and the Bershadsky-Cecotti-Ooguri-Vafa holomorphic anomaly equation (joint work with M. Alim, E. Scheidegger, S.-T. Yau). We will also define similar moduli spaces for even dimensional Calabi-Yau varieties, where we have the notion of Hodge locus.
|1/23/2019||Title: A new model for modular curves
Abstract: One of the non-trivial examples of a Hodge locus is the modular curve X_0(N), which is due to isogeny of elliptic curves (a Hodge/algebraic cycle in the product of two elliptic curves). After introducing the notion of enhanced moduli of elliptic curves, I will describe a new model for X_0(N) in the weighted projective space of dimension 4 and with weights (2,3,2,3,1). I will also introduce some elements in the defining ideal of such a model.
The talk is based on the article arXiv:1808.01689.