Memoirs of the American Mathematical Society 2007; 240 pp; softcover Volume: 185 ISBN10: 0821839438 ISBN13: 9780821839430 List Price: US$84 Individual Members: US$50.40 Institutional Members: US$67.20 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. Table of Contents 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 sesquilinear 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 Pluriharmonic 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 pluriharmonic metric
 Bibliography
 Index
