Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization
Author:
Carlos T. Simpson
Journal:
J. Amer. Math. Soc. 1 (1988), 867-918
MSC:
Primary 58E15; Secondary 32L15, 53C25, 53C55
DOI:
https://doi.org/10.1090/S0894-0347-1988-0944577-9
MathSciNet review:
944577
Full-text PDF
References | Similar Articles | Additional Information
- [1] Lars V. Ahlfors, An extension of Schwarz’s lemma, Trans. Amer. Math. Soc. 43 (1938), no. 3, 359–364. MR 1501949, https://doi.org/10.1090/S0002-9947-1938-1501949-6
- [2] Thierry Aubin, Sur la fonction exponentielle, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1514–A1516 (French). MR 0271870
- [3] Kevin Corlette, Flat 𝐺-bundles with canonical metrics, J. Differential Geom. 28 (1988), no. 3, 361–382. MR 965220
- [4] Maurizio Cornalba and Phillip Griffiths, Analytic cycles and vector bundles on non-compact algebraic varieties, Invent. Math. 28 (1975), 1–106. MR 0367263, https://doi.org/10.1007/BF01389905
- [5] 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, https://doi.org/10.1007/978-1-4899-6664-3_1
- [6] P. Deligne and J. Milne, Tannakian categories, Lecture Notes in Math., no. 900, Springer, New York, 1982, 101-228.
- [7] 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, https://doi.org/10.1112/plms/s3-50.1.1
- [8] S. K. Donaldson, Infinite determinants, stable bundles and curvature, Duke Math. J. 54 (1987), no. 1, 231–247. MR 885784, https://doi.org/10.1215/S0012-7094-87-05414-7
- [9] S. K. Donaldson, Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc. (3) 55 (1987), no. 1, 127–131. MR 887285, https://doi.org/10.1112/plms/s3-55.1.127
- [10] P. Griffiths, Periods of integrals on algebraic manifolds I, II, Amer. J. Math. 90 (1968); III, Inst. Hautes Études Sci. Publ. Math. 38 (1970).
- [11] Phillip Griffiths (ed.), Topics in transcendental algebraic geometry, Annals of Mathematics Studies, vol. 106, Princeton University Press, Princeton, NJ, 1984. MR 756842
- [12] Phillip Griffiths and Wilfried Schmid, Locally homogeneous complex manifolds, Acta Math. 123 (1969), 253–302. MR 0259958, https://doi.org/10.1007/BF02392390
- [13] Richard S. Hamilton, Harmonic maps of manifolds with boundary, Lecture Notes in Mathematics, Vol. 471, Springer-Verlag, Berlin-New York, 1975. MR 0482822
- [14] N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126. MR 887284, https://doi.org/10.1112/plms/s3-55.1.59
- [15] 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, https://doi.org/10.1007/BFb0099962
- [16] 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
- [17] Ryoichi Kobayashi, Einstein-Kaehler metrics on open algebraic surfaces of general type, Tohoku Math. J. (2) 37 (1985), no. 1, 43–77. MR 778371, https://doi.org/10.2748/tmj/1178228722
- [18] Shoshichi Kobayashi and Takushiro Ochiai, Holomorphic structures modeled after hyperquadrics, Tôhoku Math. J. (2) 34 (1982), no. 4, 587–629. MR 685426, https://doi.org/10.2748/tmj/1178229159
- [19] Martin Lübke, Stability of Einstein-Hermitian vector bundles, Manuscripta Math. 42 (1983), no. 2-3, 245–257. MR 701206, https://doi.org/10.1007/BF01169586
- [20] 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
- [21] Yoichi Miyaoka, On the Chern numbers of surfaces of general type, Invent. Math. 42 (1977), 225–237. MR 0460343, https://doi.org/10.1007/BF01389789
- [22] Yoichi Miyaoka, The maximal number of quotient singularities on surfaces with given numerical invariants, Math. Ann. 268 (1984), no. 2, 159–171. MR 744605, https://doi.org/10.1007/BF01456083
- [23] Calvin C. Moore, Compactifications of symmetric spaces. II. The Cartan domains, Amer. J. Math. 86 (1964), 358–378. MR 0161943, https://doi.org/10.2307/2373170
- [24] 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
- [25] 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 0184252, https://doi.org/10.2307/1970710
- [26] Richard S. Palais, Foundations of global non-linear analysis, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0248880
- [27] Wilfried Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211–319. MR 0382272, https://doi.org/10.1007/BF01389674
- [28] Bernard Shiffman, Complete characterization of holomorphic chains of codimension one, Math. Ann. 274 (1986), no. 2, 233–256. MR 838467, https://doi.org/10.1007/BF01457072
- [29] Carlos T. Simpson, Yang-Mills theory and uniformization, Lett. Math. Phys. 14 (1987), no. 4, 371–377. MR 922832, https://doi.org/10.1007/BF00402147
- [30] 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, https://doi.org/10.1002/cpa.3160390714
- [31] 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 0451180
- [32] A. Weil, Introduction à l'étude des variétés kähleriennes, Hermann, Paris, 1952.
- [33] Steven Zucker, Hodge theory with degenerating coefficients. 𝐿₂ cohomology in the Poincaré metric, Ann. of Math. (2) 109 (1979), no. 3, 415–476. MR 534758, https://doi.org/10.2307/1971221
- [34] Shoshichi Kobayashi, Curvature and stability of vector bundles, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), no. 4, 158–162. MR 664562
- [35] Martin Lübke, Chernklassen von Hermite-Einstein-Vektorbündeln, Math. Ann. 260 (1982), no. 1, 133–141 (German). MR 664372, https://doi.org/10.1007/BF01475761
- [36] 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, https://doi.org/10.1007/BF01389140
- [37] Fumio Takemoto, Stable vector bundles on algebraic surfaces, Nagoya Math. J. 47 (1972), 29–48. MR 0337966
Retrieve articles in Journal of the American Mathematical Society with MSC: 58E15, 32L15, 53C25, 53C55
Retrieve articles in all journals with MSC: 58E15, 32L15, 53C25, 53C55
Additional Information
DOI:
https://doi.org/10.1090/S0894-0347-1988-0944577-9
Article copyright:
© Copyright 1988
American Mathematical Society


