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We study M-Theory solutions with $G$-flux on the Fermat sextic Calabi-Yau fourfold, focussing on the relationship between the number of stabilized complex structure moduli and the tadpole contribution of the flux. We use two alternative approaches to define the fluxes: algebraic cycles and (appropriately quantized) Griffiths residues. In both cases, we collect evidence for the non-existence of solutions which stabilize all moduli and stay within the tadpole bound, Comment: v2: typos corrected and references added

We determine the cycle class of join algebraic cycles inside smooth hypersurfaces by means of their periods. We show that being a join algebraic cycle is equivalent to have its associated Artin Gorenstein algebra isomorphic to the tensor product of the Artin Gorenstein algebras of each generating cycle. This decomposition allows us to relate the quadratic fundamental form associated to the Hodge loci of each algebraic cycle with the one associated to their join. As a first application we study at a second order approximation several Hodge loci of join algebraic cycles given as combinations of two linear cycles (previously studied by Movasati, Dan, Kloosterman and the second author), showing the non-smoothness in some new cases. Our main application is to show the existence of fake linear cycles inside hypersurfaces of any dimension and degree, given as join of $0$-dimensional fake linear cycles inside hypersurfaces of $\mathbb{P}^1$ with all their closed points defined over $\mathbb{Q}$. This shows that also for high degree hypersurfaces, there are infinitely many (non smooth) Hodge loci whose Zariski tangent space has the same codimension as the codimension of the component associated to a linear cycle.

Let $n$ be an even natural number. We compute the periods of any $\frac{n}{2}$-dimensional complete intersection algebraic cycle inside an $n$-dimensional non-degenerated intersection of a projective simplicial toric variety. Using this information we determine the cycle class of such algebraic cycles. As part of the proof we develop a toric generalization of a classical theorem of Macaulay about complete intersection Artin Gorenstein rings, and we generalize an algebraic cup formula for residue forms due to Carlson and Griffiths to the toric setting.

Using a refinement of the differential method introduced by Oguiso and Yu, we provide effective conditions under which the automorphisms of a smooth degree $d$ hypersurface of $\mathbf{P}^{n+1}$ are given by generalized triangular matrices. Applying this criterion we compute all the remaining automorphism groups of Klein hypersurfaces of dimension $n\geq 1$ and degree $d\geq 3$ with $(n,d)\neq (2,4)$. We introduce the concept of extremal polarized Hodge structures, which are structures that admit an automorphism of large prime order. Using this notion, we compute the automorphism group of the polarized Hodge structure of certain Klein hypersurfaces that we call of Wagstaff type, which are characterized by the existence of an automorphism of large prime order. For cubic hypersurfaces and some other values of $(n,d)$, we show that both groups coincide (up to involution) as predicted by the Torelli Principle., Comment: 23 pages

We introduce a new class of Hodge cycles with non-reduced associated Hodge loci, we call them fake linear cycles. We characterize them for all Fermat varieties and show that they exist only for degrees $d=3,4,6$, where there are infinitely many in the space of Hodge cycles. These cycles are pathological in the sense that the Zariski tangent space of their associated Hodge locus is of maximal dimension, contrary to a conjecture of Movasati. Moreover, they provide examples of algebraic cycles not generated by their periods in the sense of Movasati-Sert\"oz. To study them we compute their Galois action in cohomology and their second-order invariant of the IVHS. We conclude that for any degree $d\ge 2+\frac{6}{n}$, the minimal codimension component of the Hodge locus passing through the Fermat variety is the one parametrizing hypersurfaces containing linear subvarieties of dimension $\frac{n}{2}$, extending results of Green, Voisin, Otwinowska and the second author., Comment: Final version to appear in Algebra & Number Theory

We consider the moduli space of abelian varieties with two marked points and a frame of the relative de Rham cohomolgy with boundary at these points compatible with its mixed Hodge structure. Such a moduli space gives a natural algebro-geometric framework for higher genus quasi Jacobi forms of index zero and their differential equations which are given as vector fields. In the case of elliptic curves we compute explicitly the Gauss-Manin connection and such vector fields.

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., Comment: Final version, to appear in CCM

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 hypersurfaces containing two linear cycles whose intersection is of dimension less than $\frac{n}{2}-\frac{d}{d-2}$, corresponds to the Hodge locus of any integral combination of such linear cycles., Comment: Final version

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., Comment: 7 pages

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 homology class of such algebraic cycles. This is known as the variational Hodge conjecture., Comment: 10 pages