arXiv subjects

Ryota Mikami

Publications and source records attributed to Ryota Mikami.

8 recordsLinked to original sources

arXiv2026

Tropical cohomology via reductions of tropical varieties

Itenberg-Katzarkov-Mikhalkin-Zharkov gave an isomorphism of tropical cohomology and cohomology of some maximally degenerate algebraic varieties. Their proof was based on tropical analogs of Steenbrink's geometric monodromy-weight spectral sequences. These were generalized to the non-realizable case by Amini-Piquerez. In this paper, we give a new construction of these tropical spectral sequences in the same way as Steenbrink's ones. For this purpose, we introduce reductions of tropical varieties. We also show that eigenwave actions are given by tropical Gauss-Manin connections.

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arXiv2024

Tropical intersection homology

Numerical equivalence of algebraic cycles is defined abstractly by intersection numbers. Classically, for smooth complex proper toric varieties, the quotients by numerical equivalence with rational coefficients can be described geometrically as singular cohomology. They are also expressed in terms of tropical geometry, tropical cohomology, introduced by Itenberg-Katzarkov-Mikhalkin-Zharkov. This paper aims to generalize this to suitable pairs of smooth proper varieties and divisors by introducing a tropical analog of intersection homology.

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arXiv2024

Differential forms and cohomology in tropical and complex geometry

Ducros, Hrushovski, and Loeser gave maps from families of archimedean diffrential forms to non-archiemedean (or tropical) ones, which are compatible with integrals on algebraic varieties. In this paper, we introduce slight modifications of their maps for complex projective varieties which give natural maps from tropical to the usual Dolbeault cohomology. We also show that our maps are compatible with integrals on generic semi-algebraic subsets and those on their weighted tropicalizations. Weighted tropicalizations induce the dual maps of the above maps of Dolbeault cohomology groups under some assumptions.

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arXiv2024

Tropical analogs of Milnor $K$-groups and tropicalizations of Zariski-Riemann spaces

As a step of a tropical approach to problems on algebraic classes of cohomology groups (such as the Hodge conjecture), in this paper, we introduce tropical analogs of (rational) Milnor $K$-groups, and prove the existence of the Gersten resolution of their Zariski sheafification. Our result will be used to prove a tropical analog of the Hodge conjecture for smooth algebraic varieties over trivially valued fields in a subsequent paper.

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arXiv2023

On tropical Hodge theory for tropical varieties

To prove log-concavity of the characteristic polynomials of matroids, Adiprasito-Huh-Katz proved the Kähler package (the hard Lefschetz theorem and the Hodge-Riemann bilinear relations) for their Chow rings. Amini-Piquerez generalized it to tropical cohomology of smooth projective tropical varieties. Their proofs were combinatorial. In this paper, we establish tropical Hodge theory except for regularity of solutions of Laplacian, and give a conditional proof of the Kähler package in the same way as compact Kähler manifolds.

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arXiv2020

On tropical cohomology of smooth algebraic varieties

In this paper, we give an explicit description of tropical cohomology of smooth algebraic varieties over trivially valued fields. We also construct ``monodromy weight'' spectral sequences for tropical cohomology of geometric strictly semi-stable reductions.

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arXiv2018

A tropical characterization of algebraic subvarieties of toric varieties over non-archimedean fields

We study the tropicalizations of analytic subvarieties of normal toric varieties over complete non-archimedean valuation fields. We show that a Zariski closed analytic subvariety of a normal toric variety is algebraic if its tropicalization is a finite union of polyhedra. Previously, the converse direction was known by the theorem of Bieri and Groves. Over the field of complex numbers, Madani, L. Nisse, and M. Nisse proved similar results for analytic subvarieties of tori.

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arXiv2017

Cyclic coverings of the projective line by Mumford curves in positive characteristic

We study the rigid analytic geometry of cyclic coverings of the projective line. We determine the defining equation of a cyclic covering of degree $p$ of the projective line by a Mumford curve over a complete discrete valuation field of positive characteristic $p$. Previously, Bradley studied that of any degree over a non-archimedean local field of characteristic zero.

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