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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.

Video

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.

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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.

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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.

Room G02

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.

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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.

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.

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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.

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Abstract: In this talk I will first introduce algebraic Yukawa couplings for any moduli of enhanced Calabi-Yau n-folds. Then I will list many examples in support of the following conjecture. A moduli of Calabi-Yau n-folds is a quotient of a Hermitian symmetric domain (constructed from periods) by an arithmetic group if and only if the corresponding Yukawa couplings are constants.

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Abstract: The integrality of the coefficients of the mirror map is a central problem in the arithmetic of Calabi-Yau varieties and it has been investigated  by Lian-Yau (1996, 1998), Hosono-Lian-Yau (1996), Zudilin (2002), Kontsevich-Schwarz-Vologodsky (2006) Krattenthaler-Rivoal (2010). The central tool in most of these works has been the so called Dwork method.  In this talk we use this method and classify all hypergeometric differential equations with a maximal unipotent monodromy whose mirror map has integral coefficients.

We also  give a computable condition on the parameters of a hypergeometric function which conjecturally computes all the primes which appear in the denominators of the coefficients of the mirror map. This is a joint work with Kh. Shokri.

Abstract: In this talk I will introduce a holomorphic foliation in a larger parameter space attached to families of enhanced projective varieties. Irreducible components of the Hodge locus with constant periods are algebraic leaves of such a foliation. Under the hypothesis that these are all the algebraic leaves,  we get the fact that such algebraic leaves are defined over the algebraic closure of the base field and that Hodge classes are weak absolute in the sense of C. Voisin.