L² cohomology

Template:Downsize In mathematics, L² cohomology is a cohomology theory for smooth non-compact manifolds M with Riemannian metric. It 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 volume form.

L² cohomology, which grew in part out of L^{2 d-bar estimates from c. 1970, 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 work 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. This was proved by Looijenga (1988) and Saper and Stern (1990).}

References

• B. Brent Gordon (2001) [1994], "Baily-Borel compactification", Encyclopedia of Mathematics, EMS Press
• Cheeger, Jeff Spectral geometry of singular Riemannian spaces. J. Differential Geom. 18 (1983), no. 4, 575--657 (1984).MR0730920
• 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. MR0573430
• 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. MR0530173
• J. Cheeger, M. Goresky, R. MacPherson, L² cohomology and intersection homology for singular algebraic varieties, Seminar on differential geometry, vol. 102 of Annals of mathamtical studies, pages 303-340.MR0645745
• M. Goresky L² cohomology is intersection cohomology
• Frances Kirwan, Jonathan Woolf An Introduction to Intersection Homology Theory,, chapter 6 ISBN 1584881844
• Looijenga, Eduard L²-cohomology of locally symmetric varieties. Compositio Math. 67 (1988), no. 1, 3-20. MR0949269
• Saper, Leslie; Stern, Mark L₂-cohomology of arithmetic varieties. Ann. of Math. (2) 132 (1990), no. 1, 1-69. MR1059935
• 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.