14 C. TAUBE S 3. Th e solution s hav e a n algebrai c constructio n fo r a certain favore d set o f manifolds . (2.9) I will elaborate o n these three point s below . However , a digressio n i s needed t o initiat e th e discussion . e) Th e modul i spac e If there is one covariant derivativ e on V wit h anti-sel f dua l curvature , then ther e ar e many . Indeed , give n suc h a covarian t derivative , V , here i s ho w t o construc t som e more : Le t s : V V b e a vecto r bundle automorphism . (Thus , s i s a n everywher e invertibl e sectio n o f the vecto r bundl e End(V). ) Le t V = s V s"1. Th e curvatur e o f V i s Rv' = s - R v s - 1 , an d i s therefore anti-sel f dua l too . Thus, i t follow s tha t i t make s n o sens e t o "count " V' s wit h anti - self dua l curvatures . A t best , on e migh t hop e t o coun t th e equivalenc e classes of such V here, V and V ar e said t o be in the same equivalence class whe n (2.10) V ~ s - V - 5 " 1 for som e unitar y automorphis m s o f V wit h determinan t 1 . (Tha t is , the automorphis m s i s require d t o pul l th e hermitia n metri c bac k a s itself.) Th e equivalenc e class of V will be denoted b y [V] , and th e set of equivalence classes of covariant derivative s with anti-sel f dua l curvatur e will be denoted b y M = M(V). For technical reasons, it is convenient t o introduce a some what large r set, M°. Th e definitio n o f M. 0 require s th e choic e o f a fiducia l bas e point p G X. Now , defin e a s th e se t o f equivalenc e classe s o f V a s in (2.10) , but wit h th e constrain t tha t s(p) = identity . Then , th e grou p SU(2) (o f 2 x 2 unitary matrice s wit h determinan t 1 ) acts o n wit h quotient M. (Thi s i s becaus e th e grou p of unitary , determinan t 1 automorphisms, s , o f V wit h s(p) = I define s a normal subgrou p o f th e group Q of al l unitary , determinan t 1 automorphisms o f V. ) The se t i s a bit nice r tha n M becaus e o f the followin g structur e theorem: Theorem 2.1 . (Uhlenbec k [24]): There is a Baire (second category) set of smooth metrics on X for which the corresponding has a natural structure of a smooth, finite dimensional manifold with a smooth action ofSU(2). Furthermore, ifbt(X) 0, then the action of SU{2) on is everywhere free modulo the center of 377(2) (which is ±1) as long as C2(V) is positive.
Previous Page Next Page