no. 4
Harmonic metrics and connections with irregular singularities
Annales de l'Institut Fourier, Tome 49 (1999) no. 4, pp. 1265-1291.

Nous identifions le complexe de de Rham de l’extension minimale d’un fibré méromorphe à connexion sur une surface de Riemann compacte X au complexe L 2 associé à ce fibré sur la surface de Riemann privée des pôles, lorsqu’on munit celui-ci d’une métrique convenable et la surface épointée d’une métrique complète. En appliquant des résultats de C. Simpson, nous montrons l’existence d’une métrique harmonique sur ce fibré, donnant lieu au même complexe L 2 .

We identify the holomorphic de Rham complex of the minimal extension of a meromorphic vector bundle with connexion on a compact Riemann surface X with the L 2 complex relative to a suitable metric on the bundle and a complete metric on the punctured Riemann surface. Applying results of C. Simpson, we show the existence of a harmonic metric on this vector bundle, giving the same L 2 complex.

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     title = {Harmonic metrics and connections with irregular singularities},
     journal = {Annales de l'Institut Fourier},
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     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {49},
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     doi = {10.5802/aif.1717},
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Sabbah, Claude. Harmonic metrics and connections with irregular singularities. Annales de l'Institut Fourier, Tome 49 (1999) no. 4, pp. 1265-1291. doi : 10.5802/aif.1717. http://archive.numdam.org/articles/10.5802/aif.1717/

[1] O. Biquard, Fibrés de Higgs et connexions intégrables : le cas logarithmique (diviseur lisse), Ann. Scient. Éc. Norm. Sup., 4e série, 50 (1997), 41-96. | Numdam | MR | Zbl

[2] M. Cornalba, P. Griffiths, Analytic cycles and vector bundles on noncompact algebraic varieties, Invent. Math., 28 (1975), 1-106. | MR | Zbl

[3] J.-P. Demailly, "Théorie de Hodge L2 et théorèmes d'annulation", Introduction à la théorie de Hodge, Panoramas et Synthèses, vol. 3, Société Mathématique de France, 1996, 3-111.

[4] M. Kashiwara, Semisimple holonomic D-modules, in [6]. | Zbl

[5] M. Kashiwara, T. Kawai, The Poincaré lemma for variations of polarized Hodge structure, Publ. RIMS, Kyoto Univ., 23 (1987), 345-407. | MR | Zbl

[6] M. Kashiwara, K. Saito, A. Matsuo, I. Satake (eds.), Topological Field Theory, Primitive Forms and Related Topics, Progress in Math., vol. 160, Birkhäuser, Basel, Boston, 1998. | Zbl

[7] B. Malgrange, Équations différentielles à coefficients polynomiaux, Progress in Math., vol. 96, Birkhäuser, Basel, Boston, 1991. | MR | Zbl

[8] B. Opic A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics, vol. 219, Longman Scientific & Technical, Harlow, 1990. | MR | Zbl

[9] Y. Sibuya, Linear Differential Equations in the Complex Domain : Problems of Analytic Continuation, Translations of Mathematical Monographs, vol. 82, American Math. Society, Providence, RI, 1976 (japanese) and 1990. | Zbl

[10] C. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc., 1 (1988), 867-918. | MR | Zbl

[11] C. Simpson, "Harmonic bundles on noncompact curves", J. Amer. Math. Soc., 3 (1990), 713-770. | MR | Zbl

[12] W. Wasow, Asymptotic expansions for ordinary differential equations, Interscience, New York, 1965. | MR | Zbl

[13] S. Zucker, Hodge theory with degenerating coefficients : L2-cohomology in the Poincaré metric, Ann. of Math., 109 (1979), 415-476. | MR | Zbl

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