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Wild Harmonic Bundles and Wild Pure Twistor \(D\)-Modules
Takuro Mochizuki, Kyoto University, Japan
A publication of the Société Mathématique de France.
cover
Astérisque
2011; 607 pp; softcover
Number: 340
ISBN-10: 2-85629-332-8
ISBN-13: 978-2-85629-332-4
List Price: US$135
Member Price: US$108
Order Code: AST/340
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The author studies (i) the asymptotic behaviour of wild harmonic bundles, (ii) the relation between semisimple meromorphic flat connections and wild harmonic bundles, (iii) the relation between wild harmonic bundles and polarized wild pure twistor \(D\)-modules. As an application, he shows the hard Lefschetz theorem for algebraic semisimple holonomic \(D\)-modules, conjectured by M. Kashiwara. He also studies resolution of turning points for algebraic meromorphic flat bundles.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

Readership

Graduate students and research mathematicians interested in pure mathematics, analysis, algebra, and algebraic geometry.

Table of Contents

  • Introduction
Part I. Good meromorphic \(\varrho\)-flat bundles
  • Good formal property of a meromorphic \(\varrho\)-flat bundle
  • Stokes structure of a good \(\varrho\)-meromorphic flat bundle
  • Full Stokes data and Riemann-Hilbert-Birkhoff correspondence
  • \(L^2\)-cohomology of filtered \(\lambda\)-flat bundle on curves
  • Meromorphic variation of twistor structure
Part II. Prolongation of wild harmonic bundle
  • Prolongments \(\mathcal{PE}^\lambda\) for unramifiedly good wild harmonic bundles
  • Some basic results in the curve case
  • Associated family of meromorphic \(\lambda\)-flat bundles
  • Smooth divisor case
  • Prolongation and reduction of variations of polarized pure twistor structures
  • Prolongation as \(\mathcal{R}\)-triple
Part III. Kobayashi-Hitchin correspondence
  • Preliminaries
  • Construction of an initial metric and preliminary correspondence
  • Preliminaries for the resolution of turning points
  • Kobayashi-Hitchin correspondence and some applications
Part IV. Application to wild pure twistor \(D\)-modules
  • Wild pure twistor \(D\)-modules
  • The Hard Lefschetz Theorem
  • Correspondences
Part V. Appendix
  • Preliminaries from analysis on multi-sectors
  • Acceptable bundles
  • Review on \(\mathcal{R}\)-modules, \(\mathcal{R}\)-triples and variants
  • Bibliography
  • Index
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