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Algebraic Geometry

By B.H. Gross, B. Mazur

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It is instructive to sketch the proof of this fact, which is due to Atiyah and Hirzebruch [3]: For simplicity, assume that T acts on X with isolated fixed points fpg, and that the action lifts to the spin bundles ˙ . p/, where 2n D dim X . D/ is a function only on the unit circle in C. D/ to a meromorphic function on S 2 , with possible poles restricted to lie on the unit circle. D/ D ak t k , all such poles on the unit circle must cancel. D/ is constant. Furthermore, by taking the limit as t !

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. C /, n  2. 26 GILLES CARRON The result in degree 4. C 2 /, the natural map from cohomology with compact support to absolute cohomology is an isomorphism in degree 4 and moreover these spaces have dimension 1. 4. r C 1//2 . C 2 / n K, the L2 range of d is almost closed in degree 4 with respect to w.

M n K//; Then the reduced L2 cohomology of M is finite-dimensional. k 1M / such that ˛ D dˇ. 8, if w is a parabolic weight the L2 range of d is almost closed in degree k with respect to w. 9. M; g/ is a complete Riemannian manifold and that w W M ! 0; C1/ is a parabolic weight. M / is finite-dimensional. the space Hw Moreover, these three properties imply the existence of a positive constant C and a compact set K  M such that the inequality (3-1) holds. The first proposition is in fact a statement about the operator d C d , whereas the second is a statement about d.

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