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

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**Example text**

It is interesting to point out that, although dS ˝ q is defined on any oriented manifold, it is only rigid for spin manifolds. Heuristically this makes sense when we view dS ˝ q as the signature operator on the loop space of X . For if X is oriented, the signature operator dS is easily seen to be rigid. But the the loop space is oriented precisely when X is spin. 34 ROBERT WAELDER Dirac operators on the loop space provide concrete examples of elliptic genera. These are homomorphisms ' W ˝ SO !

X be the blow-up along a smooth subvariety which P has normal crossings with respect to the components z D ai D z i C mE be the sum of the proper transforms of Di of D. KX D/. To avoid getting bogged down in technical details, assume f W Xz ! E/. ponents of D. 0/ 2 i #. 2 i 2 i #. 2 i x zi D #. 2j i /#. z/ #. 2 T 0X iD1 i /#. ai C 1/z/ E #. 0/ 2 i 2 E #. 2 i /#. 0/ 2 i #. 2 i 2 i #. 2 i xj Di #. 2 i /#. z/ X 0 #. 2 i /#. ai C 1/z/ iD1 T X Z Y Here, for ease of exposition, we have omitted the dependence of # on .

C/ ' C 2 =f˙ Idg is asymptotic to the Euclidean metric. Such a metric is called asymptotically locally euclidean (ALE in short). 2. The case of 3 points. C 2 /. C 2 /30 W qi D qj g; where i < j: This is illustrated in Figure 2, left. These three spheres are interchanged by the action of S3 ; hence the singular set of S7=S3 is a sphere S3 and the geometry of THE ALMOST CLOSED RANGE CONDITION 23 Figure 2. C 2 /30 . Right: S7=S3 . S7=S3 near the singular set is the one of B2 =f˙ Idg S3 , where B2 is the unit ball in C 2 .