Roberto Villaflor Loyola

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 d hypersurface of P n + 1 \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 ≥ 1 n\geq 1 and degree d ≥ 3 d\geq 3 with ( n , d ) ≠ ( 2 , 4 ) (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 ) (n,d) , we show that both groups coincide (up to involution) as predicted by the Torelli Principle.

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.

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

We consider the moduli space of abelian varieties with two marked points and a frame of the relative de Rham cohomology 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 introduce a new class of Hodge cycles with nonreduced 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. They provide examples of algebraic cycles not generated by their periods in the sense of Movasati and Sertöz (2021). 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 ≥ 2 + 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 n 2 , extending results of Green, Voisin, Otwinowska and Villaflor Loyola.