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Hodge decompositions and Dolbeault complexes on normal surfaces


Authors: Jeffrey Fox and Peter Haskell
Journal: Trans. Amer. Math. Soc. 343 (1994), 765-778
MSC: Primary 58G05; Secondary 14C30, 14F32, 32S60, 58A14
DOI: https://doi.org/10.1090/S0002-9947-1994-1191611-9
MathSciNet review: 1191611
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Abstract: Give the smooth subset of a normal singular complex projective surface the metric induced from the ambient projective space. The $ {L^2}$ cohomology of this incomplete manifold is isomorphic to the surface's intersection cohomology, which has a natural Hodge decomposition. This paper identifies Dolbeault complexes whose $ \bar \partial $-closed and $ \bar \partial $-coclosed forms represent the classes of pure type in the corresponding Hodge decomposition of $ {L^2}$ cohomology.


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  • [B] A. Borel, $ {L^2}$-cohomology and intersection cohomology of certain arithmetic varieties, Emmy Noether at Bryn Mawr (J. Sally and B. Srinivasan, eds.), Springer-Verlag, Berlin, Heidelberg, and New York, 1983, pp. 119-131. MR 713796 (85e:32038)
  • [BCa] A. Borel and W. Casselman, Cohomologie d'intersection et $ {L^2}$-cohomologie de variétés arithmétiques de rang rationnel deux, C. R. Acad. Sci. Paris 301 (1985), 369-373. MR 808630 (86m:22015)
  • [BoT] R. Bott and L. Tu, Differential forms in algebraic topology, Springer-Verlag, Berlin and New York, 1982. MR 658304 (83i:57016)
  • [C1] J. Cheeger, Hodge theory of complex cones, Analyse et Topologie sur les Espaces Singulièrs, Astérisque, vol. 101-102, Soc. Math. France, 1983, pp. 118-134. MR 737929 (85j:58016)
  • [C2] -, On the Hodge theory of Riemannian pseudomanifolds, Proc. Sympos. Pure Math., vol. 36, Amer. Math. Soc., Providence, RI, 1980, pp. 91-146. MR 573430 (83a:58081)
  • [C3] -, Spectral geometry of singular Riemannian spaces, J. Differential Geom. 18 (1983), 575-657. MR 730920 (85d:58083)
  • [C4] -, On the spectral geometry of spaces with cone-like singularities, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), 2103-2106. MR 530173 (80k:58098)
  • [CGM] J. Cheeger, M. Goresky, and R. MacPherson, $ {L^2}$-cohomology and intersection homology of singular algebraic varieties, Seminar on Differential Geometry (S. T. Yau, ed.), Princeton Univ. Press, Princeton, NJ, 1982, pp. 303-340. MR 645745 (84f:58005)
  • [GM1] M. Goresky and R. MacPherson, Intersection homology theory, Topology 19 (1980), 135-162. MR 572580 (82b:57010)
  • [GM2] -, Intersection homology. II, Invent. Math. 71 (1983), 77-129. MR 696691 (84i:57012)
  • [GrHa] P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, New York, 1978. MR 507725 (80b:14001)
  • [H1] P. Haskell, Index theory on curves, Trans. Amer. Math. Soc. 288 (1985), 591-604. MR 776394 (86m:58142)
  • [H2] -, $ {L^2}$-Dolbeault complexes on singular curves and surfaces, Proc. Amer. Math. Soc. 107 (1989), 517-526. MR 975647 (90c:58170)
  • [HsP] W. C. Hsiang and V. Pati, $ {L^2}$-cohomology of normal algebraic surfaces I, Invent. Math. 81 (1985), 395-412. MR 807064 (87f:32024)
  • [L] E. Looijenga, $ {L^2}$-cohomology of locally symmetric varieties, Compositio Math. 67 (1988), 3-20. MR 949269 (90a:32044)
  • [N1] M. Nagase, On the heat operators of normal singular algebraic surfaces, J. Differential Geom. 28 (1988), 37-57. MR 950554 (89h:58202)
  • [N2] -, Pure Hodge structure of the harmonic $ {L^2}$-forms on singular algebraic surfaces, Publ. Res. Inst. Math. Sci. 24 (1988), 1005-1023. MR 1000125 (90h:32027)
  • [N3] -, Remarks on the $ {L^2}$-cohomology of singular algebraic surfaces, J. Math. Soc. Japan 41 (1989), 97-116. MR 972167 (90m:58005)
  • [Pa] W. Pardon, The $ {L^2}{\text{-}}\bar \partial $-cohomology of an algebraic surface, Topology 28 (1989), 171-195. MR 1003581 (90i:32014)
  • [S] M. Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), 849-995. MR 1000123 (90k:32038)
  • [Sa1] L. Saper, $ {L_2}$-cohomology and intersection homology of certain algebraic varieties with isolated singularities, Invent. Math. 82 (1985), 207-255. MR 809713 (87h:32029)
  • [Sa2] -, $ {L_2}$-cohomology of Kähler varieties with isolated singularities, J. Differential Geom. 36 (1992), 89-161. MR 1168983 (93e:32038)
  • [SaSt1] L. Saper and M. Stern, $ {L_2}$-cohomology of arithmetic varieties, Proc. Nat. Acad. Sci. U.S.A. 84 (1987), 5516-5519. MR 903789 (89g:32052)
  • [SaSt2] -, $ {L_2}$-cohomology of arithmetic varieties, Ann. of Math. (2) 132 (1990), 1-69. MR 1059935 (91m:14027)
  • [Z1] S. Zucker, The Hodge structures on the intersection homology of varieties with isolated singularities, Duke Math. J. 55 (1987), 603-616. MR 904943 (88k:32039)
  • [Z2] -, $ {L_2}$-cohomology and intersection homology of locally symmetric varieties, II, Compositio Math. 59 (1986), 339-398. MR 860320 (89i:32073)
  • [Z3] -, $ {L_2}$-cohomology of warped products and arithmetic groups, Invent. Math. 70 (1982), 169-218. MR 684171 (86j:32063)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1191611-9
Keywords: Normal surface, induced metric, $ {L^2}$ Dolbeault cohomology, Hodge decomposition
Article copyright: © Copyright 1994 American Mathematical Society

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