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Takuro Mochizuki

Publications and source records attributed to Takuro Mochizuki.

At least 37 records · Page 2Linked to original sources
arXiv2019

Kobayashi-Hitchin correspondence for analytically stable bundles

We prove the existence of a Hermitian-Einstein metric on holomorphic vector bundles with a Hermitian metric satisfying the analytic stability condition, under some assumption for the underlying Kähler manifolds. We also study the curvature decay of the Hermitian-Einstein metrics. It is useful for the study of the classification of instantons and monopoles on the quotient of $4$-dimensional Euclidean space by some types of closed subgroups. We also explain examples of doubly periodic monopoles corresponding to some algebraic data.

math.DG↗
arXiv2018

Curve test for enhanced ind-sheaves and holonomic D-modules

Recently, the Riemann-Hilbert correspondence was generalized in the context of general holonomic D-modules by A. D'Agnolo and M. Kashiwara. Namely, they proved that their enhanced de Rham functor gives a fully faithfully embedding of the derived category of cohomologically holonomic complexes of D-modules into the derived category of complexes of real constructible enhanced ind-sheaves. In this paper, we study a condition when a complex of real constructible enhanced ind-sheaves $K$ is induced by a cohomologically holonomic complex of D-modules. We characterize such $K$ in terms of the restriction of $K$ to holomorphic curves.

math.CV↗
arXiv2017

A twistor approach to the Kontsevich complexes

This is a note to revisit interesting results of H. Esnault, C. Sabbah, M. Saito and J.-D. Yu on the Kontsevich complexes from the viewpoint of mixed twistor D-modules. We explicitly describe the V-filtration of the mixed twistor D-modules and their relative de Rham complexes, associated to some meromorphic functions. We explain how such descriptions imply the results on the Kontsevich complexes.

math.AG↗
arXiv2017

Some characterizations of Dirac type singularity of monopoles

We study singular monopoles on open subsets in the $3$-dimensional Euclidean space. We give two characterizations of Dirac type singularities. One is given in terms of the growth order of the norms of sections which are invariant by the scattering map. The other is given in terms of the growth order of the norms of the Higgs fields.

math.DG↗
arXiv2016

Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces

Let $(E,\overline{\partial}_E,θ)$ be a stable Higgs bundle of degree $0$ on a compact connected Riemann surface. Once we fix the flat metric $h_{\det(E)}$ on the determinant of $E$, we have the harmonic metrics $h_t$ $(t>0)$ for the stable Higgs bundles $(E,\overline{\partial}_E,tθ)$ such that $\det(h_t)=h_{\det(E)}$. We study the behaviour of $h_t$ when $t$ goes to $\infty$. First, we show that the Hitchin equation is asymptotically decoupled under the assumption that the Higgs field is generically regular semisimple. We apply it to the study of the so called Hitchin WKB-problem. Second, we study the convergence of the sequence $(E,\overline{\partial}_E,θ,h_t)$ in the case where the rank of $E$ is two. We introduce a rule to determine the parabolic weights of a "limiting configuration", and we show the convergence of the sequence to the limiting configuration in an appropriate sense. The results can be appropriately generalized in the context of Higgs bundles with a Hermitian-Einstein metric on curves.

math.DG↗
arXiv2015

Twistor property of GKZ-hypergeometric systems

We study the mixed twistor D-modules associated to meromorphic functions. In particular, we describe their push-forward and specialization under some situations. We apply the results to study the twistor property of a type of GKZ-hypergeometric systems, and to study their specializations. As a result, we obtain some isomorphisms of mixed TEP-structures in the local mirror symmetry.

math.AG↗
arXiv2014

Holonomic D-modules with Betti structure

We define the notion of Betti structure for holonomic D-modules which are not necessarily regular singular. We establish the fundamental functorial properties. We also give auxiliary analysis of holomorphic functions of various types on the real blow up.

math.AG↗
arXiv2013

Mixed twistor D-modules

We introduce mixed twistor $D$-modules, and establish the fundamental functorial property. We also prove that they are described as the gluing of admissible variations of mixed twistor structure. In a sense, mixed twistor $D$-modules could be regarded as a twistor version of mixed Hodge modules due to M. Saito . In the other sense, they could be a mixed version of pure twistor $D$-modules studied by C. Sabbah and the author. A theory of mixed twistor $D$-modules is one of the ultimate goals in the study suggested by Simpson's Meta Theorem, and the author hopes that it would play a basic role in the Hodge theory for holonomic $D$-modules possibly with irregular singularity.

math.CV↗
arXiv2013

Harmonic bundles and Toda lattices with opposite sign

We study a certain type of wild harmonic bundles in relation with a Toda equation. We explain how to obtain a classification of the real valued solutions of the Toda equation in terms of their parabolic weights, from the viewpoint of the Kobayashi-Hitchin correspondence. Then, we study the associated integrable variation of twistor structure. In particular, we give a criterion for the existence of an integral structure. It follows from two results. One is the explicit computation of the Stokes factors of a certain meromorphic flat bundle. The other is an explicit description of the associated meromorphic flat bundle. We use the opposite filtration of the limit mixed twistor structure with an induced torus action.

math.DG↗
arXiv2010

Wild Harmonic Bundles and Wild Pure Twistor D-modules

We study (i) asymptotic behaviour of wild harmonic bundles, (ii) the relation between semisimple meromorphic flat connections and wild harmonic bundles, (iii) the relation between wild harmonic bundles and polarized wild pure twistor $D$-modules. As an application, we show the hard Lefschetz theorem for algebraic semisimple holonomic $D$-modules, conjectured by M. Kashiwara.

math.DG↗
arXiv2008

Asymptotic behaviour of variation of pure polarized TERP structures

The purpose of this paper is twofold. One is to give a survey of our study on the reductions of harmonic bundles, and the other is to explain a simple application in the study of TERP structure. In particular, we investigate the asymptotic behaviour of the "new supersymmetric index" for variation of pure polarized TERP structures.

math.DG↗
arXiv2008

Kobayashi-Hitchin correspondence for tame harmonic bundles II

Let $X$ be a smooth projective complex variety with an ample line bundle $L$, and let $D$ be a simple normal crossing divisor. We establish the Kobayashi-Hitchin correspondence between tame harmonic bundles on $X-D$ and $μ_L$-stable parabolic $λ$-flat bundles with trivial characteristic numbers on $(X,D)$. Especially, we obtain the quasiprojective version of the Corlette-Simpson correspondence between flat bundles and Higgs bundles.

math.DG↗
arXiv2006

Kobayashi-Hitchin correspondence for tame harmonic bundles and an application

We establish the correspondence between tame harmonic bundles and $μ_L$-stable parabolic Higgs bundles with trivial characteristic numbers. We also show the Bogomolov-Gieseker type inequality for $μ_L$-stable parabolic Higgs bundles. Then we show that any local system on a smooth quasi projective variety can be deformed to a variation of polarized Hodge structure. As a consequence, we can conclude that some kind of discrete groups cannot be a split quotient of the fundamental group of a smooth quasi projective variety.

math.DG↗
arXiv2006

A theory of the invariants obtained from the moduli stacks of stable objects on a smooth polarized surface

Let $X$ be a smooth polarized algebraic surface over the compex number field. We discuss the invariants obtained from the moduli stacks of semistable sheaves of arbitrary ranks on $X$. For that purpose, we construct the virtual fundamental classes of some moduli stacks, and we show the transition formula of the integrals over the moduli stacks of the $δ$-stable Bradlow pairs for the variation of the parameter $δ$. Then, we study the relation among the invariants. In the case $p_g>0$, we show that the invariants are independent of the choice of a polarization of $X$. We also show that the invariants can be reduced to the invariants obtained from the moduli of abelian pairs and the Hilbert schemes. In the case $p_g=0$, we obtain the weak wall crossing formula and the weak intersection rounding formula, which describes the dependence of the invariants on the polarization.

math.AG↗
arXiv2005

The 3-cocycles of the Alexander quandles F_q[T]/(T-omega)

We determine the third cohomology of Alexander quandles of the form F_q[T]/(T-omega), where F_q denotes the finite field of order q and omega is an element of F-q which is neither 0 nor 1. As a result, we obtain many concrete examples of non-trivial 3-cocycles.

math.GT↗
arXiv2004

Asymptotic behaviour of tame harmonic bundles and an application to pure twistor $D$-modules

We study the asymptotic behaviour of tame harmonic bundles. First of all, we prove a local freeness of the prolongation by an increasing order. Then we obtain the polarized mixed twistor structure. As one of the applications, we obtain the norm estimate of holomorphic or flat sections by weight filtrations of the monodromies. As other application, we establish the correspondence of semisimple regular holonomic $D$-modules and polarizable pure imaginary pure twistor $D$-modules through a tame pure imaginary harmonic bundles, which is a conjecture of Sabbah. Then the regular holonomic version of Kashiwara's conjecture follows from the results of Sabbah and us. Keywords: Higgs fields, harmonic bundle, variation of Hodge structure, mixed twistor structure, $D$-module.

math.DG↗