12 C. TAUBE S with V o to defin e a covariant derivativ e (whic h will also be denote d b y V0) o n section s o f T*X ® End(Vr). Then , se t (2.5) d Vo a = p(V Q a), where p : T*X ® End(F) ® T*X A 2 TX ® End(F) i s th e obviou s extension o f the anti-symmetrizatio n homomorphism . Remark tha t th e sam e definitio n (mor e o r less ) define s th e exterio r covariant derivativ e dVo o n sections of APT*X® V7 when V i s any vector bundle ove r X wit h a covariant derivativ e Vo - Thi s dy 0 take s a sectio n of A P T*X ® V an d spew s out a section o f AP+1T*X ® V. Thi s exterio r covariant derivativ e i s characterized b y th e fac t tha t (2.6) d Vo {v Aw) = dvAw + ( - l )V A d Vo w for an y pai r o f (rea l valued ) g-for m v an d sectio n w o f A m T*X ® V 7 . By th e way , not e tha t {d^ Q )2w = i? Vo A IU, where th e latte r expressio n involves both exterio r multiplication o f forms an d the action of End(F' ) on V' . Anyway, with (2.4 ) understood, th e anti-self duality equations for th e u(V)-valued 1-for m a ar e obtaine d b y applyin g P + t o (2.4 ) an d settin g the resul t equa l t o zero . b) Lin e bundle s The simplest case to consider has V X a complex line bundle. Fi x a hermitia n covarian t derivativ e V o o n V an d th e anti - sel f dua l equa - tions becom e a syste m o f linear , inhomogeneous , first orde r equation s for a real 1-for m a on X. Tha t is , the covariant derivativ e V = V o + i a has anti-sel f dua l curvatur e i f and onl y if (2.7) P + da - i P+R v ° = 0 . (Here, i = (-1) 1 /2.) This equatio n can' t alway s b e solved . A necessar y an d sufficien t condition i s tha t (2.8) / v A P+RVo = 0 for ever y self-dual , close d 2-for m v. (Integratio n b y parts demonstrate s the necessit y o f (2.8) . Th e Fredhol m alternativ e give s th e sufficiency. ) Now, th e se t o f self-dual close d 2-form s o n X for m a finite dimensiona l vector space whose dimension can be computed in terms of the homology data o f X. Thi s number , b^iX), i s equa l t o on e hal f th e su m o f th e second Bett i numbe r of X wit h th e signature of X. (Alternately , b^iX) is the numbe r o f positiv e eigenvalue s o f th e symmetri c bilinea r pairin g
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