22 C. TAUBE S higher derivative s an d w e ca n quot e tha t Arzoli-Ascol i theore m (se e any introductory rea l analysis text) t o conclude that th e sequence has a convergent subsequence in said ball. I f we can cover X b y a fixed system of balls where uniform bound s on {a a } exist , then w e can conclude tha t the sequenc e {V a } ha s a convergen t subsequence . With th e preceedin g understood , on e mus t as k whethe r unifor m bounds o n a ca n b e foun d whic h allo w on e to star t th e bootstrapping ? At first sight , ther e ar e n o aprior i reason s wh y suc h bound s shoul d ex - ist. Afte r all , the only constraint o n [V ] is that it s curvature be anti-sel f dual. However , thi s anti-sel f dualit y constrain t i s not totall y worthless , for, a s w e sa w above , i t actuall y constrain s th e siz e o f R v i n term s o f the secon d Cher n numbe r o f the bundl e V \ Now, Uhlenbeck' s grea t discover y wa s t o find a universal , positiv e constant n wit h th e followin g property : I f th e integra l ove r som e suffi - ciently small ball of |i? v | 2 i s less than K , then [V ] differs fro m th e trivia l covariant derivativ e [V 0 = d] over said bal l by an su(2)-valued 1-for m a which i s smal l enoug h t o star t th e jus t describe d bootstrapping . (Th e value o f K is independent o f the metri c o n X an d als o of X.) The preceedin g ha s a direc t implicatio n fo r a sequence, {[V a ]} of co- variant derivative s on a fixed bundle V, all with anti-self dual curvature this bein g th e first assertio n of Proposition 2.3 . / / {[V a ]} is a sequence of covariant derivatives on a fixed bundle V, all with anti-self dual curvature, then there exists an infinite subsequence and a finite set Q C X such that the subsequence converges in the C°°-topology on the compliment of this set Q. Further- more, the limiting covariant derivative extends over ft as a covariant derivative with anti-self dual curvature on a bundle V —* X with sec- ond Chern class less than V 's. (The furthermor e assertio n abov e i s a har d "removabl e singularity " theorem o f Uhlenbeck [48]. ) The preceedin g proposition ca n b e strengthened t o give a descriptio n of a compac t ification o f M.{V) a s a variet y wher e th e complimen t o f M(V) consist s o f union s o f product s o f X wit h th e M(V) fo r bun - dles V X wit h secon d Cher n clas s smalle r tha t V' s (and , apriori , non-negative). (See , e.g. , [22 ] an d [20]. ) Thi s i s th e "geometri c com - pactification" t o whic h (2.9 ) alludes . h) Algebrai c construction s Suppose tha t X i s a comple x submanifol d o f some large dimensiona l complex projectiv e space , wit h th e induce d Riemannia n metric . Here ,
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