arXiv subjects

Takuro Mochizuki

Publications and source records attributed to Takuro Mochizuki.

40 records · Page 3Linked to original sources

arXiv2004

A characterization of semisimple local system by tame pure imaginary pluri-harmonic metric

Let $L$ be a local system on a smooth quasi projective variety over $\cnum$. We see that $L$ is semisimple if and only if there exists a tame pure imaginary pluri-harmonic metric on $L$. Although it is a rather minor refinement of a result of Jost and Zuo, it is significant for the study of harmonic bundles and pure twistor $D$-modules. As one of the application, we show that the semisimplicity of local systems are preserved by the pull back via a morphism of quasi projective varieties. This paper will be included as an appendix to our previous paper, ``Asymptotic behaviour of tame harmonic bundles and an application to pure twistor $D$-modules'', math.DG/0312230. Keywords: Higgs fields, harmonic bundle, semisimple local system.

math.DG↗

arXiv2003

The structure of the cohomology ring of the filt schemes

Let $C$ be a smooth projective curve over the complex number field $\cnum$. We investigate the structure of the cohomology ring of the quot schemes $\Quot(r,n)$, i.e., the moduli scheme of the quotient sheaves of $\nbigo_C^{\oplus r}$ with length $n$. We obtain a filtration on $H^{\ast}(\Quot(r,n))$, whose associated graded ring has a quite simple structure. As a corollary, we obtain a small generator of the ring. We also obtain a precise combinatorial description of $H^{\ast}(\Quot(r,n))$ itself. For that purpose, we consider the complete filt schemes and we use a `splitting principle'. A complete filt scheme has an easy geometric description: a sequence of bundles of projective spaces. Thus the cohomology ring of the complete filt schemes are easily determined. In the cohomology ring of complete filt schemes, we can do some calculations for the cohomology ring of quot schemes. Keywords: Quot scheme, cohomology ring, torus action, splitting principle, symmetric function.

math.AG↗

arXiv2002

Asymptotic behaviour of tame nilpotent harmonic bundles with trivial parabolic structure

Let $E$ be a holomorphic vector bundle. Let $θ$ be a Higgs field, that is a holomorphic section of $End(E)\otimesΩ^{1,0}_X$ satisfying $θ^2=0$. Let $h$ be a pluriharmonic metric of the Higgs bundle $(E,θ)$. The tuple $(E,θ,h)$ is called a harmonic bundle. Let $X$ be a complex manifold, and $D$ be a normal crossing divisor of $X$. In this paper, we study the harmonic bundle $(E,θ,h)$ over $X-D$. We regard $D$ as the singularity of $(E,θ,h)$, and we are particularly interested in the asymptotic behaviour of the harmonic bundle around $D$. We will see that it is similar to the asymptotic behaviour of complex variation of polarized Hodge structures, when the harmonic bundle is tame and nilpotent with the trivial parabolic structure. For example, we prove constantness of general monodromy weight filtrations, compatibility of the filtrations, norm estimates, and the purity theorem. For that purpose, we will obtain a limiting mixed twistor structure from a tame nilpotent harmonic bundle with trivial parabolic structure, on a punctured disc. It is a partial solution of a conjecture of Simpson.

math.DG↗

arXiv2002

The geometry of the parabolic Hilbert schemes

We discuss the parabolic Hilbert scheme of points on smooth surface, which is an interesting generalization of Hilbert scheme of points on surface. Some of the studies by Ellingsrud-Stromme, Goettsche, Cheah, Nakajima and Grojnowski can be generalized in our parabolic case.

math.AG↗