METRICS, CONNECTION S AN D GLUIN G THEOREM S 7 To be mor e precis e here , suppos e tha t V i s a comple x vecto r bundl e with a hermitia n metri c h. On e ca n thin k o f h a s a homomorphis m from th e bundl e V* ® V t o C with the property tha t h(v*,v) 0 unless v 0. (Here , th e comple x conjugatio n homomorphis m fro m V t o V* (which i s anti-linear ove r C) i s denoted b y (•)*• ) A covariant derivativ e is said t o b e metri c compatibl e whe n (1.9) d{h{v\w)) = h(Vv\w) + h(v\ Vw). The curvature of a metric compatible covariant derivative takes values in u(V ) ® A2T*, wher e u(V ) C End(V ) i s th e subalgebr a o f unitar y endomorphisms. A t times , i t wil l be convenien t t o restric t attentio n t o covariant derivative s V whos e curvatur e i s a trac e zer o sectio n o f th e bundle u(V) ® A2T*, that is , a section of su(V) ® A2T*. (Suc h restricte d covariant derivative s ma y no t exist. ) I n eithe r case , th e curvatur e i s said t o b e anti-sel f dua l whe n it s projectio n int o u(V ) ® A+ vanishe s identically. (Not e tha t thi s conditio n i s metric dependent) . One ca n as k fo r bundle s whic h admi t suc h connections , an d ther e are know n sufficien t conditions . However , a shor t digressio n i s firs t required t o se t th e stage . T o star t th e digression , agre e t o cal l a pai r of bundle s V an d V (o f th e sam e fibe r dimension ) isomorphi c whe n there exist s a n invertibl e sectio n o f Hom c (V, V). I f s : V V i s a n isomorphism an d i f V i s a covarian t derivativ e o n V wit h anti-sel f dua l curvature, the n s o V o s - 1 i s a covariant derivativ e o n V wit h anti-sel f dual curvature. Indeed , the curvature of this last covarian t derivativ e is equal t o s o Rv o s' 1 . It follow s tha t on e nee d onl y lis t th e isomorphis m classe s o f vec - tor bundles having covariant derivative s with anti-self dual connections . With th e preceding understood, remar k tha t th e isomorphism classe s of complex vecto r bundle s (wit h fixed fiber dimension ) ove r a 4-manifol d X are labeled by a pair e H 2 {X\ Z ) an d c 2 G H4(X Z) . A the bundl e admits a covariant derivativ e with curvature i n su(V) ® A2T* if and onl y if Ci = 0 as a rational class . Then , c 2 ca n hav e an y value . ( I am tacitl y assuming tha t X i s connected s o that H*(X] Z ) « Z for connecte d X.) Here i s the fundamenta l existenc e theorem : Theorem 3 . [45] : Let X be a compact, connected, oriented l^-mantfold with Riemannian metric. There exists N 0 such that if V X is a complex vector bundle with C\ = 0 and with c 2 N, then V admits a co- variant derivative V with anti-self dual curvature that is, the curvature Rv is a section of su(V) ® A". Since this theorem is easier to prove than Theore m 1, 1 will spend th e first hav e o f m y lecture s reviewin g it s proof thi s wil l serv e a s a war m
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