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Asymptotic Behaviour of Tame Harmonic Bundles and an Application to Pure Twistor $$D$$-Modules, Part 2
Takuro Mochizuki, Kyoto University, Japan
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Memoirs of the American Mathematical Society
2007; 240 pp; softcover
Volume: 185
ISBN-10: 0-8218-3943-8
ISBN-13: 978-0-8218-3943-0
List Price: US$80 Individual Members: US$48
Institutional Members: US\$64
Order Code: MEMO/185/870

The author studies the asymptotic behaviour of tame harmonic bundles. First he proves a local freeness of the prolongment of deformed holomorphic bundle by an increasing order. Then he obtains the polarized mixed twistor structure from the data on the divisors. As one of the applications, he obtains the norm estimate of holomorphic or flat sections by weight filtrations of the monodromies.

As another application, the author establishes the correspondence of semisimple regular holonomic $$D$$-modules and polarizable pure imaginary pure twistor $$D$$-modules through tame pure imaginary harmonic bundles, which is a conjecture of C. Sabbah. Then the regular holonomic version of M. Kashiwara's conjecture follows from the results of Sabbah and the author.

Part 4. An Application to the theory of Pure Twistor $$D$$-modules
• Pure twistor $$D$$-module
• Prolongation of $$\mathcal R$$-module $$\mathcal E$$
• The filtrations of $$\mathfrak{E} [\eth_t]$$
• The weight filtration on $$\psi_{t,u}\mathfrak{E}$$ and the induced $$\mathcal{R}$$-triple
• The sesqui-linear pairings
• Polarized pure twistor $$D$$-module and tame harmonic bundles
• The pure twistor $$D$$-modules on a smooth curve (Appendix)
Part 5. Characterization of Semisimplicity by Tame Pure Imaginary Pluri-harmonic Metric
• Preliminary
• Tame pure imaginary harmonic bundle
• The Dirichlet problem in the punctured disc case
• Control of the energy of twisted maps on a Kahler surface
• The existence of tame pure imaginary pluri-harmonic metric
• Bibliography
• Index