By Greg Friedman, Eugénie Hunsicker, Anatoly Libgober, Laurentiu Maxim, Editors

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Then we consider the complex d d l 1 l    ! X / ! X / !    l 1 =wl ;wl  l =wl C1 ;wl C1  When the cohomology of this complex can be computed from a local computation (that is, when there is a Poincar´e lemma characterizing the cohomology of this complex), then the L2 cohomology of X can be obtained from the degree k cohomology space of this complex. This method has been used successfully by T. Hausel, E. Hunsicker and R. Mazzeo in [14] to obtain a topological interpretation of the L2D1 cohomology of manifolds with fibered cusp ends or with fibered boundary ends.

0; / D 0, the above expression only makes sense for ai ¤ 1. Y I z; /. X; D/ ! X; D/ are equal to 1. X; D/ > 1. Xj ; Dj / ¤ 1. X2 ; D2 I z; /, we must connect these two resolutions by a sequence of blow-ups and blow-downs, applying functoriality of the elliptic genus of pairs at each stage. X1 ; D1 / are greater than 1, and others less than 1, then after blowing up X1 , we may acquire discrepancy coefficients equal to 1. X2 ; D2 /. Xj ; Dj / > 1. This constraint is quite familiar to minimal model theorists; singular varieties Y possessing this property are said to have log-terminal singularities.

X / if k > n=2: There are different proofs of this result. The first one uses the scattering calculus developed by Melrose; see [21, Theorem 4] and [14, Theorem 1A]. 11]. 5]. 2. The case of 3 points A vanishing result outside degree 4. C 2 / constructed by D. Joyce coincides with the one of H. 5 According to N. Hitchin, the L2 cohomology of a hyperk¨ahler reduction of a Euclidean quaternionic space is trivial except perhaps for the degree equal to the middle (real) dimension [15]. C 2 / in degree 4.

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