Asymptotic Behaviour of Tame Nilpotent Harmonic Bundles with Trivial Parabolic Structure
By T. Mochizuki. 2002
58 citations recorded by the source corpus.
Let $E$ be a holomorphic vector bundle. Let $\theta$ be a Higgs field, that is a holomorphic section of $End(E)\otimes\Omega^{1,0}_X$ satisfying $\theta^2=0$. Let $h$ be a pluriharmonic metric of the Higgs bundle $(E,\theta)$. The tuple $(E,\theta,h)$ 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 $(E,\theta,h)$ over $X-D$. We regard $D$ as the singularity of $(E,\theta,h)$, 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.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/995bb375b282b98cc1cba6b2c8b5994ccc2a208d
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Wild Harmonic Bundles and Wild Pure Twistor D-modules (2008, methodology): “In our previous works ([66], [67], [68] and [69]), we studied tame harmonic bundles after the work of Simpson ([83], [84], [85], [87], for example.”
- Katz's middle convolution algorithm (2006, background): “It would be good to have the full middle-convolution theory for the general setup of parabolic logarithmic λ-connections [107] [108] [109] [122].”
- Asymptotic Behaviour of Tame Harmonic Bundles and an Application to Pure Twistor D-Modules (2003, methodology): “In the previous paper [37], we discussed the behaviour of tame harmonic bundle imposed the nilpotentness and the trivial parabolic structure conditions.”
- On the classification of rank-two representations of quasiprojective fundamental groups (2007, background): “Looking at harmonic mappings to trees or more generally buildings, mirrors much of the study of harmonic bundles which has been extended to the quasiprojective case [14] [27] [60] [80] [81] [82] [85] [86] [93] [116].”