L² cohomology

L² cohomology

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 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

*springer|id=B/b130010|author=B. Brent Gordon|title=Baily-Borel compactification
*Cheeger, Jeff "Spectral geometry of singular Riemannian spaces." J. Differential Geom. 18 (1983), no. 4, 575--657 (1984).MathSciNet|id=0730920
*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. MathSciNet|id=0573430
*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. MathSciNet|id=0530173
*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.MathSciNet|id=0645745
*M. Goresky [http://www.math.ias.edu/~goresky/pdf/zucker.pdf 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. MathSciNet|id=0949269
*Saper, Leslie; Stern, Mark "L2-cohomology of arithmetic varieties." Ann. of Math. (2) 132 (1990), no. 1, 1-69. MathSciNet|id=1059935
*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.


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