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Tropical geometry and algebraic cycles

Mikami, Ryota 京都大学 DOI:10.14989/doctor.k22976

2021.03.23

概要

In this thesis, we study tropical geometry and its relation with algebraic cycles. Tropical geometry, a relatively new area of mathematics, is a combinatorial shadow of algebraic geometry. Several kinds of problems of algebraic cycles are proved by reducing them to tropical analogs, e.g., calculations of Gromov-Witten invariants. In the first part of this thesis, we give a tropical characterization of closed algebraic subvarieties of toric varieties over non-archimedean fields among all Zariski closed analytic subvarieties. This is a converse of the celebrated result of Bieri-Groves. The second and third parts are devoted to a very challenging program. We propose a tropical approach to some of the most important problems on algebraic cycles, i.e., some problems on algebraic classes of cohomology groups, such as the Hodge conjecture. As a part of this approach, we prove a tropical analog of the Hodge conjecture for smooth algebraic varieties over trivially valued fields. To prove this, in the second part of this paper, inspired by Liu’s construction of tropical cycle class maps by Milnor K-groups, we introduce tropical analogs of Milnor K-groups and show the exactness of the Gersten complexes for the Zariski sheaf cohomology of the sheaves of them on smooth algebraic varieties over trivially valued fields. Then the tropical analog of the Hodge conjecture follows from that the tropical cohomology groups for smooth algebraic varieties over trivially valued fields are isomorphic to the Zariski sheaf cohomology groups of the sheaves of tropical analogs of Milnor K-groups. In the third part of this paper, we prove these isomorphisms of cohomology groups by a theorem for general “cohomology theories”, developed by many mathematicians, e.g., Quillen, and explicit calculations of tropical cohomology of the trivial line bundle by non-archimedean geometry. Usually, non-archimedean geometry of height 1 valuations is used to study tropical geometry. To study tropical analogs of Milnor K-groups and tropical cohomology, we introduce tropicalizations of valuations of higher heights.

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