Introduction

Hodge conjecture is one of the millennium conjectures for which the evidences are not so much: It is proved for surfaces (Lefschetz (1,1) theorem) and surface type varieties (like cubic fourfolds), and we do not know even it is true for all Fermat varieties (despite some partial result by T. Shioda and his coauthors). The main goal of the course is introduce computational methods for finding new Hodge cycles for hypersurfaces such that its verification is challenging in such cases! I will explain few results which motivate us to claim that either the verification of Hodge conjecture for such cycles must be an easy exercise in commutative algebra or they might be good candidates to be counterexamples. For this we aim to study Hodge loci for hypersurfaces. We introduce larger parameter space and holomorphic foliations on them such that Hodge loci become the leaves of such foliation. In our way we have to develop a theory of holomorphic foliations on schemes such that the leaves also enjoy scheme structures (we call them leaf schemes). We will also introduce Hasse principal or local-global conjectures for these foliations which generalize the Katz-Grothendieck conjecture for linear differential equations.

Reference

[1] N. M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of turrittin, Publications mathématiques de l’IHES 39 (1970), 175–232.

[2] Katz, Nicholas M. (1972). "Algebraic solutions of differential equations (p-curvature and the Hodge filtration)". Invent. Math. 18 (1–2): 1–118.

[3] W. Mendson, J. V. Pereira. Codimension one foliations in positive characteristic, Preprint 2023.

[4] Y. Miyaoka and T. Peternell, Geometry of higher dimensional algebraic varieties, vol. 26, Springer, 1997.

[5] H. Movasati, A Course in Hodge Theory: With Emphasis on Multiple Integrals, Somerville, MA: International Press Boston, 2021.

[6] H. Movasati, Modular and Automorphic Forms & Beyond, Monographs in Number Theory, World Scientific, 2021.

[7] H. Movasati, R. Villaflor, A Course in Hodge Theory: Periods of Algebraic cycles, 33 Colóquio Brasileiro de Matemática, IMPA, Rio de Janeiro, Brazil, 2021.

Lecturer Intro

Hossein Movasati is an Iranian-Brazilian mathematician who since 2006 has worked at the Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro. He began his mathematical career working on holomorphic foliations and differential equations on complex manifolds, and gradually moved to study Hodge theory and modular forms and the role of these in mathematical physics, and in particular mirror symmetry.