METRICS, CONNECTIONS AND GLUING THEOREMS 13 on H 2 (X M) whic h send s UJ an d u' t o th e evaluatio n o f u U u/ o n th e fundamental clas s of X.) A remarkable theorem of Uhlenbeck (i n [24]) asserts that whe n b^iX) is non-zero, and when the first Cher n class of V i s rationally non-trivial , then thes e necessary an d sufficien t condition s can not b e satisfied i f the metric i s chosen fro m a certai n open , dens e subse t o f metrics. (I n fact , the subse t o f metric s fo r whic h a give n comple x lin e bundl e V ha s a connection wit h anti-sel f dua l curvatur e i s a subvariety o f codimensio n 6J(X) i n th e spac e o f al l metric s o n TX.) Whe n 62"(-X" ) = 0 , ever y complex line bundle o n X ha s a covariant derivativ e with anti-sel f dua l curvature. c) SU(2) an d £7(2 ) Let V » X b e a complex 2-plan e bundle. The n V define s a natura l complex lin e bundle ove r X whic h i s denoted b y det(V) . (Thi s bundl e is th e sam e a s A 2 V.) I f V ha s a covarian t derivativ e wit h anti-sel f dual curvature , the n A 2 (V) ha s on e too . A s discusse d above , mos t o f the time , comple x lin e bundle s d o no t hav e suc h covarian t derivatives . Thus, most of the time, complex 2-plane bundles whose first Chern clas s is rationaly non-zero do not have covariant derivatives with anti-self dua l curvature. There are , still , man y non-isomorphi c 2-plan e bundle s wit h torsio n first Cher n class. A s will be seen in a subsequent lecture , these can have covariant derivative s with anti-self dua l curvature. Th e simplest t o con- sider hav e zer o first Cher n class , an d th e discussio n belo w i s restricte d to this case. (Th e general case is not s o very much harder. Th e analysi s is essentially the same though the general case requires an extra digres s or s o to clarif y certai n topologica l issues. ) d) Propertie s o f th e solutio n se t In all that follows , I am considering covariant derivative s on complex , hermitian 2-plan e bundle s wit h vanishin g first Cher n class . As I note d previously , th e solution s t o th e anti-sel f dua l equation s have see n som e remarkabl e uses , first b y Donaldso n [13]-[16 ] an d no w by man y others . (Th e reade r her e i s referre d t o [20 ] for details. ) On e can identify a small set o f features o f the anti-sel f dua l equation s whic h seem, i n hindsight , t o b e th e crucia l factors . I wil l lis t the m now , an d then digres s a t leisur e on each . 1. Th e equation s lineariz e t o Fredhol m equations . 2. Ther e i s a geometri c compactificatio n o f the solutio n set .
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