The Walker Abel–Jacobi map descends

For a complex projective manifold, Walker has defined a regular homomorphism lifting Griffiths’ Abel–Jacobi map on algebraically trivial cycle classes to a complex abelian variety, which admits a finite homomorphism to the Griffiths intermediate Jacobian. Recently Suzuki gave an alternate, Hodge-theoretic, construction of this Walker Abel–Jacobi map. We provide a third construction based on a general lifting property for surjective regular homomorphisms, and prove that the Walker Abel–Jacobi map descends canonically to any field of definition of the complex projective manifold. In addition, we determine the image of the l-adic Bloch map restricted to algebraically trivial cycle classes in terms of the coniveau filtration.

There is Science at Guadalupe

Our project seeks to disseminate science throughout the school community. 12th-grade students are building a scientific magazine and developing mini-projects, in small groups, which aim to stimulate the school community's interest in science and scientific discoveries. There are groups of students doing research and planning experiments. Other groups have establishedpartnerships with pre-school and 1st cycle classes so that, in a playful and fun way, interest in science grows more and earlier. The project started in October/November 2019 and, at this moment, we are still planning and devising strategies. In parallel, some of the mini-projects are already being implemented. In partnerships with the pre-school and 1st cycle classes, we have already started a scientific cooking project. We will also hold workshops on building terrariums, maintaining aquariums, and composting projects.

NORMAL FUNCTIONS FOR ALGEBRAICALLY TRIVIAL CYCLES ARE ALGEBRAIC FOR ARITHMETIC REASONS

For families of smooth complex projective varieties, we show that normal functions arising from algebraically trivial cycle classes are algebraic and defined over the field of definition of the family. In particular, the zero loci of those functions are algebraic and defined over such a field of definition. This proves a conjecture of Charles.

Review of Hodge theory and algebraic cycles

This chapter provides the background for the studies to be undertaken in succeeding chapters. It reviews Chow groups, correspondences and motives on the purely algebraic side, cycle classes, and (mixed) Hodge structures on the algebraic–topological side. Emphasis is placed on the notion of coniveau and the generalized Hodge conjecture which states the equality of geometric and Hodge coniveau. The chapter first follows the construction of Chow groups, the application of the localization exact sequence, the functoriality and motives of Chow groups, and cycle classes. It then turns to Hodge structures; pursuing related topics such as polarization, Hodge classes, standard conjectures, mixed Hodge structures, and Hodge coniveau.