Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization

Simpson, Carlos T.

doi:10.1090/S0894-0347-1988-0944577-9

Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization
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by Carlos T. Simpson

J. Amer. Math. Soc. 1 (1988), 867-918

DOI: https://doi.org/10.1090/S0894-0347-1988-0944577-9

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References

Lars V. Ahlfors, An extension of Schwarz’s lemma, Trans. Amer. Math. Soc. 43 (1938), no. 3, 359–364. MR 1501949, DOI 10.1090/S0002-9947-1938-1501949-6
Thierry Aubin, Sur la fonction exponentielle, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1514–A1516 (French). MR 271870
Kevin Corlette, Flat $G$-bundles with canonical metrics, J. Differential Geom. 28 (1988), no. 3, 361–382. MR 965220
Maurizio Cornalba and Phillip Griffiths, Analytic cycles and vector bundles on non-compact algebraic varieties, Invent. Math. 28 (1975), 1–106. MR 367263, DOI 10.1007/BF01389905
P. Deligne, Un théorème de finitude pour la monodromie, Discrete groups in geometry and analysis (New Haven, Conn., 1984) Progr. Math., vol. 67, Birkhäuser Boston, Boston, MA, 1987, pp. 1–19 (French). MR 900821, DOI 10.1007/978-1-4899-6664-3_{1}

Tannakian categories

S. K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. (3) 50 (1985), no. 1, 1–26. MR 765366, DOI 10.1112/plms/s3-50.1.1
S. K. Donaldson, Infinite determinants, stable bundles and curvature, Duke Math. J. 54 (1987), no. 1, 231–247. MR 885784, DOI 10.1215/S0012-7094-87-05414-7
S. K. Donaldson, Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc. (3) 55 (1987), no. 1, 127–131. MR 887285, DOI 10.1112/plms/s3-55.1.127

Periods of integrals on algebraic manifolds

90

38

Phillip Griffiths (ed.), Topics in transcendental algebraic geometry, Annals of Mathematics Studies, vol. 106, Princeton University Press, Princeton, NJ, 1984. MR 756842, DOI 10.1515/9781400881659
Phillip Griffiths and Wilfried Schmid, Locally homogeneous complex manifolds, Acta Math. 123 (1969), 253–302. MR 259958, DOI 10.1007/BF02392390
Richard S. Hamilton, Harmonic maps of manifolds with boundary, Lecture Notes in Mathematics, Vol. 471, Springer-Verlag, Berlin-New York, 1975. MR 0482822
N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126. MR 887284, DOI 10.1112/plms/s3-55.1.59
M. Kashiwara, Vanishing cycle sheaves and holonomic systems of differential equations, Algebraic geometry (Tokyo/Kyoto, 1982) Lecture Notes in Math., vol. 1016, Springer, Berlin, 1983, pp. 134–142. MR 726425, DOI 10.1007/BFb0099962
D. A. Kajdan, On arithmetic varieties, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) Halsted, New York, 1975, pp. 151–217. MR 0486316
Ryoichi Kobayashi, Einstein-Kaehler metrics on open algebraic surfaces of general type, Tohoku Math. J. (2) 37 (1985), no. 1, 43–77. MR 778371, DOI 10.2748/tmj/1178228722
Shoshichi Kobayashi and Takushiro Ochiai, Holomorphic structures modeled after hyperquadrics, Tohoku Math. J. (2) 34 (1982), no. 4, 587–629. MR 685426, DOI 10.2748/tmj/1178229159
Martin Lübke, Stability of Einstein-Hermitian vector bundles, Manuscripta Math. 42 (1983), no. 2-3, 245–257. MR 701206, DOI 10.1007/BF01169586
B. Malgrange, Polynômes de Bernstein-Sato et cohomologie évanescente, Analysis and topology on singular spaces, II, III (Luminy, 1981) Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp. 243–267 (French). MR 737934
Yoichi Miyaoka, On the Chern numbers of surfaces of general type, Invent. Math. 42 (1977), 225–237. MR 460343, DOI 10.1007/BF01389789
Yoichi Miyaoka, The maximal number of quotient singularities on surfaces with given numerical invariants, Math. Ann. 268 (1984), no. 2, 159–171. MR 744605, DOI 10.1007/BF01456083
Calvin C. Moore, Compactifications of symmetric spaces. II. The Cartan domains, Amer. J. Math. 86 (1964), 358–378. MR 161943, DOI 10.2307/2373170
David Mumford and John Fogarty, Geometric invariant theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 34, Springer-Verlag, Berlin, 1982. MR 719371, DOI 10.1007/978-3-642-96676-7
M. S. Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) 82 (1965), 540–567. MR 184252, DOI 10.2307/1970710
Richard S. Palais, Foundations of global non-linear analysis, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0248880
Wilfried Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211–319. MR 382272, DOI 10.1007/BF01389674
Bernard Shiffman, Complete characterization of holomorphic chains of codimension one, Math. Ann. 274 (1986), no. 2, 233–256. MR 838467, DOI 10.1007/BF01457072
Carlos T. Simpson, Yang-Mills theory and uniformization, Lett. Math. Phys. 14 (1987), no. 4, 371–377. MR 922832, DOI 10.1007/BF00402147
K. Uhlenbeck and S.-T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math. 39 (1986), no. S, suppl., S257–S293. Frontiers of the mathematical sciences: 1985 (New York, 1985). MR 861491, DOI 10.1002/cpa.3160390714
Shing Tung Yau, Calabi’s conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 5, 1798–1799. MR 451180, DOI 10.1073/pnas.74.5.1798

Introduction à l’étude des variétés kähleriennes

Steven Zucker, Hodge theory with degenerating coefficients. $L_{2}$ cohomology in the Poincaré metric, Ann. of Math. (2) 109 (1979), no. 3, 415–476. MR 534758, DOI 10.2307/1971221
Shoshichi Kobayashi, Curvature and stability of vector bundles, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), no. 4, 158–162. MR 664562
Martin Lübke, Chernklassen von Hermite-Einstein-Vektorbündeln, Math. Ann. 260 (1982), no. 1, 133–141 (German). MR 664372, DOI 10.1007/BF01475761
V. B. Mehta and A. Ramanathan, Restriction of stable sheaves and representations of the fundamental group, Invent. Math. 77 (1984), no. 1, 163–172. MR 751136, DOI 10.1007/BF01389140
Fumio Takemoto, Stable vector bundles on algebraic surfaces, Nagoya Math. J. 47 (1972), 29–48. MR 337966