20 C. TAUBE S g) Compactification s The modul i space s M = M°/SU(2) ar e hardl y eve r compact . I n fact, wit h X fixed, thes e space s ar e non-compac t wheneve r V s secon d Chern number is large enough. Wit h this understood, i t is an absolutel y crucial point (vi s a vis applications) tha t M ha s a canonical, geometri c compactification. Th e existence of such a compactification follow s fro m a key result of Uhlenbeck [47 , 48]. Uhlenbeck's result i s hard an d almos t pure analysis , s o i t i s usuall y overlooke d i n al l bu t th e mos t technica l discussions. Yet , i t i s Uhlenbeck' s analysi s whic h connect s th e forma l Predholm result s abov e with th e har d issue s of 4-manifold topology . The compactnes s issu e ca n b e state d loosel y a s follows : Doe s a se - quence o f point s i n M (o r M°) hav e a convergen t subsequence ? And , if not , ca n yo u discrib e th e behavio r (i n geometri c terms ) o f a non - convergent sequence . (Th e applications of Donaldson would be impossi- ble without som e sort of geometric description o f the non- compactness . As described below, Donaldson uses M. a s a homology clas s in the spac e B = B°/SU(2). T o mak e thi s notio n precise , i t i s absolutel y crucia l t o be abl e t o compensat e fo r th e fac t tha t M i s non-compact. ) I won' t giv e th e detail s o f Uhlenbeck' s arguments . Rather , I wil l describe the strategy. Th e strategy is based on two important principles . First, th e integral s ove r X o f the squar e o f the nor m o f th e curvature s of an y tw o covarian t derivativ e o n V wit h anti-sel f dua l curvatur e ar e the same . Infact , thes e integral s ar e equa l t o a universa l multipl e o f the secon d Cher n numbe r o f V. Second , th e equations for anti-sel f dua l curvature ar e elliptic in a certain wel l defined sense . (Se e below. ) The first poin t abov e i s the cru x o f the whol e compactnes s business . To elaborate on this point, one must invok e a basic fact fro m differentia l geometry: Th e secon d Cher n numbe r o f a comple x vecto r bundl e ca n be compute d usin g th e curvatur e o f an y metri c compatible , covarian t derivative on said bundle (see , e.g., [37]) . Tha t is , if V is such a covariant derivative, the n V s secon d Cher n numbe r i s equal t o (2.19) (4TT)- 2 / trace(i? v A i?v), where trac e i s the fiber trace . With i t understoo d tha t th e valu e o f (2.19 ) i s determine d b y th e topology o f V conside r th e ramification s o f th e conditio n tha t i? v = * J?v. I n thi s case , th e integran d i n (2.19 ) i s equa l t o a multipl e o f |i? v | 2 dvol. Thus , th e integra l ove r X o f the squar e o f the nor m o f th e curvature o f a covarian t derivativ e wit h anti-sel f dua l curvatur e ha s a fixed, Go d give n value .
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