## Journal of Differential Geometry

### Asymptotic Behaviour of Tame Nilpotent Harmonic Bundles with Trivial Parabolic Structure

Takuro Mochizuki

#### Abstract

Let E be a holomorphic vector bundle. Let θ be a Higgs field, that is a holomorphic section of End (E) ⊗ Ω1,0X satisfying θ2 = 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 XD. 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 solution of a conjecture of Simpson.

#### Article information

Source
J. Differential Geom., Volume 62, Number 3 (2002), 351-559.

Dates
First available in Project Euclid: 21 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1090426286

Digital Object Identifier
doi:10.4310/jdg/1090426286

Mathematical Reviews number (MathSciNet)
MR2005295

Zentralblatt MATH identifier
1069.32010

#### Citation

Mochizuki, Takuro. Asymptotic Behaviour of Tame Nilpotent Harmonic Bundles with Trivial Parabolic Structure. J. Differential Geom. 62 (2002), no. 3, 351--559. doi:10.4310/jdg/1090426286. https://projecteuclid.org/euclid.jdg/1090426286