We characterize the smallest codimension components of the Hodge locus of smooth degree $d$ hypersurfaces of the projective space $\mathbb{P}^{n+1}$ of even dimension $n$, passing through the Fermat variety (with $d\neq 3,4,6$). They correspond to the locus of hypersurfaces containing a linear algebraic cycle of dimension $\frac{n}{2}$. Furthermore, we prove that among all the local Hodge loci associated to a non-linear cycle passing through Fermat, the ones associated to a complete intersection cycle of type $(1,1,\ldots,1,2)$ attain the minimal possible codimension of their Zariski tangent spaces. This answers a conjecture of Movasati, and generalizes a result of Voisin about the first gap between the codimension of the components of the Noether-Lefschetz locus to arbitrary dimension, provided that they contain the Fermat variety.
In this paper we study properties of the locus of second type lines of a general cubic threefold and fourfold. By analysing the geometry of the Fano scheme of lines of the Fermat cubic fourfold and in particular giving an explicit description of the locus of second type lines, we deduce that the Voisin map is birational over the second type locus. For a general cubic threefold, by studying properties of the second type locus again, we compute that various natural geometric monodromy groups are the full symmetric group.
We prove that the integral polarized Hodge structure on the transcendental lattice of a sextic Fermat surface is decomposable. This disproves a conjecture of Kulikov related to a Hodge theoretic approach to proving the irrationality of the very general cubic fourfold.
We study an analogue of the Noether-Lefschetz theorem in the context of Beilinson's Hodge conjecture. We estimate the dimension of the Noether-Lefschetz locus by the infinitesimal methods in Hodge theory.
Given a variation of Hodge structures on a quasi-projective base $S$, whose generic Mumford-Tate group is non-product, we prove that the (countable) union of positive components of the Hodge locus is either an algebraic subvariety of $S$, or is Zariski-dense in $S$.
For a fixed integer $d$, we study here the locus of degree $d$ hypersurfaces $X$ in $\mathbb{P}^{2n+1}$ such that $H^{2n}(X,\mathbb{Q}) \cap H^{n,n}(X,\mathbb{C}) \not= \mathbb{Q}$. We call this locus \textit{the Noether-Lefschetz locus}. Any irreducible component of this locus is locally a Hodge locus. So, we see that the study of this locus is very closely related to the variational Hodge conjecture. In this article we show that the Hodge locus corresponding to the cohomology class of a complete intersection subscheme in $\mathbb{P}^{2n+1}$ of codimension $n+1$ and degree less than $d$ satis…
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.
The main subject of study in this paper is the Noether-Lefschetz locus for Beilinson-Hodge cycles on the complement of the union of hyperplane sections in a surface in $\Bbb P^3$. We give an analogue of the result by M.Green and C.Voisin on the maximal components in the Noether-Lefschetz locus.
We introduce an "extended locus of Hodge classes" that also takes into account integral classes that become Hodge classes "in the limit". More precisely, given a polarized variation of integral Hodge structure of weight zero on a Zariski-open subset of a complex manifold, we construct a canonical analytic space that parametrizes limits of integral classes; the extended locus of Hodge classes is an analytic subspace that contains the usual locus of Hodge classes, but is finite and proper over the base manifold. The construction uses Saito's theory of mixed Hodge modules and a small generalizati…
We study the stable hyperelliptic locus, i.e. the closure, in the Deligne- Mumford moduli space of stable curves, of the locus of smooth hyperelliptic curves. Working on a suitable blowup of the relative Hilbert scheme (of degree 2) associated to a family of stable curves, we construct a bundle map ('degree-2 Brill-Noether') from a modification of the Hodge bundle to a tautological bundle, whose degeneracy locus is the natural lift of the stable hyperelliptic locus plus a simple residual scheme. Using intersection theory on Hilbert schemes and Fulton-Macpherson residual intersection theory, th…
We show that an irreducible component of the Hodge locus of a polarizable variation of Hodge structure of weight 0 on a smooth complex variety X is defined over an algebraically closed subfield k of finite transcendence degree if X is defined over k and the component contains a k-rational point. We also prove a similar assertion for the Hodge locus inside the Hodge bundle if the Hodge bundle together with the connection is defined over k. This is closely related with the theory of absolute Hodge classes. The proof uses the spread of the Hodge locus, and is quite similar to the case of the zero…
We compute the class of the locus in M_g of curves having a pencil with two unspecified triple ramification points. This is the first example of a geometric divisor on M_g which is not the pull-back of a divisor on the moduli space of pseudo-stable curves. This space, in which elliptic tails are replaced by cusps, appears as a result of the first divisorial contraction in the minimal model program for M_g. In particular, we show that our divisor picks-up the locus of Fermat cubic tails when restricted to the boundary divisor of elliptic tails. We also give various enumerative applications conc…
Given a family $π:\mc{X} \rightarrow B$ of smooth projective varieties, a closed fiber $\mc{X}_o$ and an invertible sheaf $\mc{L}$ on $\mc{X}_o$, we compare the Hodge locus in $B$ corresponding to the Hodge class $c_1(\mc{L})$ with the locus of points $b\,\in\, B$ such that $\mc{L}$ deforms to an invertible sheaf $\mc{L}_b$ on $\mc{X}_b$ with at least $h^0(\mc{L})$--dimensional space of global sections (it is a Brill-Noether type locus associated to $\mc{L}$). We finally give an application by comparing the Brill-Noether locus to a family of curves on a surface passing through a fixed set of p…
We provide a closed form expression for linear Hodge integrals on the hyperelliptic locus. Specifically, we find a succinct combinatorial formula for all intersection numbers on the hyperelliptic locus with one $λ$-class, and powers of a $ψ$-class pulled back along the branch map. This is achieved by using Atiyah-Bott localization on a stack of stable maps into the orbifold $[\mathbb{P}^1/\mathbb{Z}_2]$.
Let $f: X \rightarrow S$ be a family of non singular projective varieties parametrized by a complex algebraic variety $S$. Fix $s \in S$, an integer $p$, and a class $h \in {\rm H}^{2p}(X_s,\Z)$ of Hodge type $(p,p)$. We show that the locus, on $S$, where $h$ remains of type $(p,p)$ is algebraic. This result, which in the geometric case would follow from the rational Hodge conjecture, is obtained in the setting of variations of Hodge structures.
We study the structure of string theory flux compactification for a general family of elliptic CY 3-folds. We investigate the locus of the attractor points of the flux compactification in type IIB string theory on the boundary components of period domains. Specifically we give equations describing this locus through the asymptotic of nilpotent orbits on period domains. our approach is a mixture of techniques of asymptotic Hodge theory and the numerical period vectors used in physics.
The basic admissible locus $\mathcal{F}(G, μ, b)^a$ inside the flag variety $\mathcal{F}(G, μ)$, attached to a reductive group $G$ with a minuscule cocharacter $μ$ of $G$, is a $p$-adic analogue of the complex analytic period spaces. It has an algebraic approximation $\mathcal{F}(G, μ, b)^{wa}$ inside the flag variety, called the weakly admissible locus. On the flag variety $\mathcal{F}(G, μ)$, we have the Newton stratification which has the admissible locus as its unique open stratum. In this paper, we study the relation between the Newton strata and the weakly admissible locus. We show that …
Let $Y$ be a smooth complex projective variety of dimension $N+1$, $L$ an invertible sufficiently ample sheaf, $X\in |L|$ a smooth hypersurface and $λ\in F^kH^N(X,C)$ a vanishing cohomology class, where $F^{*}$ is the Hodge filtration and $k\in\{1,...,[N/2]\}$. Assume that $L$ is sufficiently ample and that the codimension in $|L|$ of the Hodge variety associated to $λ$ (locally defined as the locus where the image of $λ$ by flat transport over $|L|$ remains in $F^k$) is sufficiently small. I show that this forces $N$ to be even and $k=[N/2]$, and that the class $λ$ is a linear combination wit…
Given a prime $\ell\geq 3$ and a positive integer $k \le \ell-2$, one can define a matrix $D_{k,\ell}$, the so-called Demjanenko matrix, whose rank is equal to the dimension of the Hodge group of the Jacobian ${\mathrm Jac}({\mathcal C}_{k,\ell})$ of a certain quotient of the Fermat curve of exponent $\ell$. For a fixed $\ell$, the existence of $k$ for which $D_{k,\ell}$ is singular (equivalently, for which the rank of the Hodge group of ${\mathrm Jac}({\mathcal C}_{k,\ell})$ is not maximal) has been extensively studied in the literature. We provide an asymptotic formula for the number of such…
The variety of all smooth hypersurfaces of given degree and dimension has the Fermat hypersurface as a natural base point. In order to study the period map for such varieties, we first determine the integral polarized Hodge structure of the primitive cohomology of a Fermat hypersurface (as a module over the automorphism group of the hypersurface). We then focus on the degree 3 case and show that the period map for cubic fourfolds as analyzed by R. Laza and the author gives complete information about the period map for cubic hypersurfaces of lower dimension dimension. In particular, we thus rec…
In this article we show how the data of integrals of algebraic differential forms over algebraic cycles can be used in order to prove that algebraic and Hodge cycle deformations of a given algebraic cycle are equivalent. As an example, we prove that most of the Hodge and algebraic cycles of the Fermat sextic fourfold cannot be deformed in the underlying parameter space. We then take a difference of two linear cycles inside the Fermat variety, and gather evidences that the Hodge locus corresponding to this is smooth and reduced. This implies the existence of new algebraic cycles in the Fermat v…
We combine Deligne's global invariant cycle theorem, and the algebraicity theorem of Cattani, Deligne and Kaplan, for the connected components of the locus of Hodge classes, to conclude that under simple assumptions these components are defined over number fields (assuming the initial family is), as expected from the Hodge conjecture. We also show that the Hodge conjecture for (weakly) absolute Hodge classes reduces to the Hodge conjecture for (weakly) absolute Hodge classes on varieties defined over number fields.
We review the Newton stratification and Ekedahl-Oort stratification on the special fiber of a smooth integral model for a Shimura variety of Hodge type at a prime of good reduction. We show that the μ-ordinary locus coincides with the generic Ekedahl-Oort stratum, and that for any two geometric points in the μ-ordinary locus there is an isomorphism of the attached Dieudonne modules with additional structure. As a consequence, we proof that the μ-ordinary locus is open and dense, thus generalizing the results which were already known in the PEL-case. To prove our results we provide a method whi…
We present a conjecture on the geometry of the Hodge locus of a (graded polarizable, admissible) variation of mixed Hodge structure over a complex smooth quasi-projective base, generalizing to this context the Zilber-Pink Conjecture for mixed Shimura varieties (in particular the André-Oort conjecture).
Teichmueller curves are geodesic discs in Teichmueller space that project to an algebraic curve in the moduli space $M_g$. We show that for all $g \geq 2$ Teichmueller curves map to the locus of real multiplication in the moduli space of abelian varieties. Remark that McMullen has shown that precisely for $g=2$ the locus of real multiplication is stable under the $\SL_2(\RR)$-action on the tautological bundle $Ω\M_g$. We also show that Teichmueller curves are defined over number fields and we provide a completely algebraic description of Teichmueller curves in terms of Higgs bundles. As a cons…
We prove the equidistribution of the Hodge locus for certain non-isotrivial, polarized variations of Hodge structure of weight $2$ with $h^{2,0}=1$ over complex, quasi-projective curves. Given some norm condition, we also give an asymptotic on the growth of the Hodge locus. In particular, this implies the equidistribution of elliptic fibrations in quasi-polarized, non-isotrivial families of $K3$ surfaces.
The Fermat type Calabi-Yau $n$-fold, denoted by $\mathscr{F}_n$, is the hypersurface of $\mathbb{P}^{n+1}$ defined by $\sum_{i=0}^{n+1}x_i^{n+2}=0$, which is the smooth fiber over the Fermat point $ψ=0$ of the Fermat pencil $$ \sum_{i=0}^{n+1} x^{n+2}_i -(n+2)\, ψ\, \prod_{i=0}^{n+1} x_i =0. $$ The nowhere vanishing holomorphic $n$-form on $\mathscr{F}_n$ defines an $n+1$ dimensional sub-Hodge structure of $(H^n(\mathscr{F}_n,\mathbb{Q}),F_p)$. In this paper, we will formulate a conjecture which says that this $n+1$ dimensional sub-Hodge structure splits completely into the direct sum of pure …
The Eckardt hypersurface in $\mathbb{P}^{19}$ parameterizes smooth cubic surfaces with an Eckardt point, which is a point common to three of the $27$ lines on a smooth cubic surface. We describe the cubic surfaces lying on the singular locus of the model of this hypersurface in $\mathbb{P}^4$, obtained via restriction to the space of cubic surfaces possessing a so-called Sylvester form. We prove that inside the moduli of cubics, the singular locus corresponds to a reducible surface with two rational irreducible components intersecting along two rational curves. The two curves intersect in two …
We give a classification of components of the Hodge locus in any parameter space of smooth projective varieties. This is done using determinantal varieties constructed from the infinitesimal variation of Hodge structures (IVHS) of the underlying family. As a corollary we prove that the minimum codimension for the components of the Hodge locus in the parameter space of $m$-dimensional hypersurfaces of degree $d$ with $d\geq 2+\frac{4}{m}$ and in a Zariski neighborhood of the point representing the Fermat variety, is obtained by the locus of hypersurfaces passing through an $\frac{m}{2}$-dimensi…
Given a family of smooth complex projective varieties, the Hodge conjecture predicts the algebraicity of the locus of Hodge classes. This was proven unconditionnally by Cattani, Deligne and Kaplan in 1995. In a similar way, conjectures on algebraic cycles have led Green and Griffiths to conjecture the algebraicity of the zero locus of normal functions. This corresponds to a mixed version of the theorem of Cattani, Deligne and Kaplan. This result has been proven recently by Brosnan-Pearlstein, Kato-Nakayama-Usui, and Schnell building on work of M. Saito. We will present some of the ideas around…
Let G be the group A_4 or Z_2xZ_2. We compute the integral of λ_g on the Hurwitz locus H_G\subset M_g of curves admitting a degree 4 cover of P^1 having monodromy group G. We compute the generating functions for these integrals and write them as a trigonometric expression summed over the positive roots of the E_6 and D_4 root systems respectively. As an application, we prove the Crepant Resolution Conjecture for the orbifolds [C^3/A_4] and [C^3/(Z_2xZ_2)].
We establish the Hodge conjecture for some subvarieties of a class of toric varieties. First we study quasi-smooth intersections in a projective simplicial toric variety, which is a suitable notion to generalize smooth complete intersection subvarieties in the toric environment, and in particular quasi-smooth hypersurfaces. We show that under appropriate conditions, the Hodge Conjecture holds for a very general quasi-smooth intersection subvariety, generalizing the work on quasi-smooth hypersurfaces of the first author and Grassi in [3]. We also show that the Hodge Conjecture holds asymptotica…
We prove that the moduli space of Calabi-Yau 3-folds coming from eight planes of $P^3$ in general positions is not modular. In fact we show the stronger statement that the Zariski closure of the monodromy group is actually the whole $Sp(20,R)$. We construct an interesting submoduli, which we call \emph{hyperelliptic locus}, over which the weight 3 $Q$-Hodge structure is the third wedge product of the weight 1 $Q$-Hodge structure on the corresponding hyperelliptic curve. The non-extendibility of the hyperelliptic locus inside the moduli space of a genuine Shimura subvariety is proved.
We study the structure of the supersingular locus of the Rapoport--Zink integral model of the Shimura variety for $\mathrm{GU}(2,2)$ over a ramified odd prime with the special maximal parahoric level. We prove that the supersingular locus equals the disjoint union of two basic loci, one of which is contained in the flat locus, and the other is not. We also describe explicitly the structures of basic loci. More precisely, the former one is purely $2$-dimensional, and each irreducible component is birational to the Fermat surface. On the other hand, the latter one is purely $1$-dimensional, and …
For every even number $n$, and every $n$-dimensional smooth hypersurface of $\mathbb{P}^{n+1}$ of degree $d$, we compute the periods of all its $\frac{n}{2}$-dimensional complete intersection algebraic cycles. Furthermore, we determine the image of the given algebraic cycle under the cycle class map inside the De Rham cohomology group of the corresponding hypersurface in terms of its Griffiths basis and the polarization. As an application, we use this information to address variational Hodge conjecture for a non complete intersection algebraic cycle. We prove that the locus of general hypersur…
The goal of this paper is to propose a theory of mirror symmetry for varieties of general type. Using Landau-Ginzburg mirrors as motivation, we describe the mirror of a hypersurface of general type (and more generally varieties of non-negative Kodaira dimension) as the critical locus of the zero fibre of a certain Landau-Ginzburg potential. The critical locus carries a perverse sheaf of vanishing cycles. Our main results shows that one obtains the interchange of Hodge numbers expected in mirror symmetry. This exchange is between the Hodge numbers of the hypersurface and certain Hodge numbers d…
Let $\mathbb V$ be a polarized variation of Hodge structure over a smooth complex quasi-projective variety $S$. In this paper, we give a complete description of the typical Hodge locus for such variations. We prove that it is either empty or equidistributed with respect to a natural differential form, \emph{the pull-push form}. In particular, it is always analytically dense when the pull-push form does not vanish. When the weight is $2$, the Hodge numbers are $(q,p,q)$ and the dimension of $S$ is least $rq$, we prove that the typical locus where the Picard rank is at least $r$ is equidistribut…
In this article we use a theorem of Carlson and Griffiths and compute periods of linear algebraic cycles inside the Fermat variety of even dimension $n$ and degree $d$. As an application, for examples of $n$ and $d$ we prove that 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. We also check the same statement for hypersurfaces containing a complete intersection algebraic cycle. Our result confirms the Hodge conjecture for Hodge cycles obtained by the monodromy of the hom…
