arXiv subjects

Takuya Murata

Publications and source records attributed to Takuya Murata.

4 recordsLinked to original sources

arXiv2026

Weak Hodge Theorem on Piecewise-Algebraic Spaces

We prove a weak version of the classical Hodge theorem on piecewise-algebraic spaces, a class of spaces introduced by Kontsevich and Soibelman in [KS00]. Precisely, we first prove the Poincare lemma that computes singular cohomology as a variant of de Rham cohomology. Then, as a weak Hodge theorem, we naturally embed the singular cohomology into the space of harmonic forms, instead of establishing an isomorphism (which does not hold for those spaces). Our approach in the latter is classical: Sobolev space theory. In addition, we give more detailed proofs for the claims in the appendix to [KS00]. This work is part of a program of extending arithmetic intersection theory to singular spaces. In particular, a type of currents in this singular setup is introduced.

math.AG↗

arXiv2022

Maps to toric varieties, toric degenerations and integrable systems \`a la Harada--Kaveh

Given a toric degeneration (a degeneration to a toric variety), over the complex numbers, we construct a surjective continuous map from a general fiber to the special fiber of the degeneration in the classical topology. The construction is a variant of one due to Goresky and MacPherson based on the Thom--Mather theory of stratified spaces. As an application, we recover and extend the construction of integrable systems \`a la Harada--Kaveh in "Integrable systems, toric degenerations and okounkov bodies." Compared to their result, our map is constructed more explicitly and we also construct the integrable systems on the boundary strata. This paper is a part of the authors' research on maps to toric degenerations; we refer the readers to "Toric degenerations and projections," arxiv and "Notes on multi-proj and maps to not-necessarily-normal toric varieties," researchgate for more algebraic approaches.

math.AG↗

arXiv2021

Generic tropical initial ideals of Cohen-Macaulay algebras

We study the generic tropical initial ideals of a positively graded Cohen-Macaulay algebra $R$ over an algebraically closed field $\mathbf{k}$. Building on work of Römer and Schmitz, we give a formula for each initial ideal, and we express the associated quasivaluations in terms of certain $I$-adic filtrations. As a corollary, we show that in the case that $R$ is a domain, every initial ideal coming from the codimension-$1$ skeleton of the tropical variety is prime, so "generic presentations of Cohen-Macaulay domains are well-poised in codimension-$1$."

math.AG↗

arXiv2020

On degenerations of projective varieties to complexity-one T-varieties

Let $R$ be a positively graded finitely generated $\textbf{k}$-domain with Krull dimension $d+1$. We show that there is a homogeneous valuation $\mathfrak{v}: R \setminus \{0\} \to \mathbb{Z}^d$ of rank $d$ such that the associated graded $\text{gr}_\mathfrak{v}(R)$ is finitely generated. This then implies that any polarized $d$-dimensional projective variety $X$ has a flat deformation over $\mathbb{A}^1$, with reduced and irreducible fibers, to a polarized projective complexity-one $T$-variety (i.e. a variety with a faithful action of a $(d-1)$-dimensional torus $T$). As an application we conclude that any $d$-dimensional complex smooth projective variety $X$ equipped with an integral Kähler form has a proper $(d-1)$-dimensional Hamiltonian torus action on an open dense subset that extends continuously to all of $X$.

math.AG↗