Tropical Normal Functions -- Higher Abel-Jacobi Invariants of Tropical cycles

• Published in: arXiv.org 2020-12
• Main Author: Rahmati, Mohammad Reza
• Format: Article
• Language: English
• Subjects: Construction; Invariants; Jacobians; Rings (mathematics)
• ISSN: 2331-8422

We consider the variation of tropical Hodge structure (TVHS) associated to families of tropical varieties. The family of the tropical intermediate Jacobians of the associated tropical Hodge structure defines a bundle of tropical Jacobians, whose sections we call the tropical normal functions. We define formal sequential derivatives of these functions on the base with respect to the natural Gauss-Manin connection as the Hodge theoretic invariants detecting tropical cycles in the fibers. The associated invariants which are defined inductively are the higher Abel-Jacobi invariants in the tropical category. They naturally identify the tropical Bloch-Beilinson filtration on the tropical Chow group. We examine this construction on the moduli of tropical curves with marked points, in order to study the tropical tautological classes in the tautological ring of \(\mathcal{M}_{g,n}^{\text{trop}}\). The expectation is the nontriviality of these cycles could be examined with less complexity in the tropical category. The construction is compatible with the tropicalization functor on the category of schemes, and the aforementioned procedure will also provide an alternative way to examine the relations in the tautological ring of \(\mathcal{M}_{g,n}\) in the schemes category.