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In mathematics, L2 cohomology is a cohomology theory for smooth non-compact manifolds ''M'' with Riemannian metric. It is defined in the same way as de Rham cohomology except that one uses square-integrable differential forms. The notion of square-integrability makes sense because the metric on ''M'' gives rise to a norm on differential forms and a volume form. L2 cohomology, which grew in part out of L2 d-bar estimates from the 1960s, was studied cohomologically, independently by Steven Zucker (1978) and Jeff Cheeger (1979). It is closely related to intersection cohomology; indeed, the results in the preceding cited works can be expressed in terms of intersection cohomology. Another such result is the Zucker conjecture, which states that for a Hermitian locally symmetric variety the L2 cohomology is isomorphic to the intersection cohomology (with the middle perversity) of its Baily–Borel compactification (Zucker 1982). This was proved in different ways by Looijenga (1988) and by Saper and Stern (1990). ==References== * * *Cheeger, Jeff ''Spectral geometry of singular Riemannian spaces.'' J. Differential Geom. 18 (1983), no. 4, 575–657 (1984). *Cheeger, Jeff ''On the Hodge theory of Riemannian pseudomanifolds.'' Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), pp. 91–146, Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980. *Cheeger, Jeff ''On the spectral geometry of spaces with cone-like singularities.'' Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 5, 2103–2106. *J. Cheeger, M. Goresky, R. MacPherson, ''L2 cohomology and intersection homology for singular algebraic varieties'', Seminar on differential geometry, vol. 102 of Annals of Mathematics Studies, pages 303–340. *M. Goresky (L2 cohomology is intersection cohomology ) *Frances Kirwan, Jonathan Woolf ''An Introduction to Intersection Homology Theory,'', chapter 6 ISBN 1-58488-184-4 *Looijenga, Eduard ''L2-cohomology of locally symmetric varieties.'' Compositio Mathematica 67 (1988), no. 1, 3–20. * *Saper, Leslie; Stern, Mark ''L2-cohomology of arithmetic varieties.'' Ann. of Math. (2) 132 (1990), no. 1, 1–69. *Zucker, Steven, ''Théorie de Hodge à coefficients dégénérescents.'' Comptes Rendus Acad. Sci. 286 (1978), 1137–1140. *Zucker, Steven, ''Hodge theory with degenerating coefficients: L2-cohomology in the Poincaré metric.'' Annals of Math. 109 (1979), 415–476. *Zucker, Steven, ''L2-cohomology of warped products and arithmetic groups.'' Inventiones Math. 70 (1982), 169–218. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「L² cohomology」の詳細全文を読む スポンサード リンク
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