6 C. TAUBE S 0 Q-Q£ (1.8) A second vie w of this connec t su m wil l be give n i n a subsequent lec - ture. Th e notation X# n Y i s shorthand fo r th e iterate d connec t su m of X wit h som e n copie s of Y. Tha t is , X#nY = ( • • • (X#Y)#Y) • • • #Y) with n copie s o f Y appearing . d) Th e meanin g o f W + = 0 There i s a Principl e o f Universa l Equivalenc e whic h asserts , i n part , that al l differential equation s ar e the Cauchy-Rieman n equations . Wit h this Principle understood, one might ask: Whic h Cauchy-Riemann equa - tions correspond t o the condition W+ = 0 ? A "Twistor correspondence " gives the answe r (se e [5]) : Theorem 2 . (R . Penrose) Let X be an oriented 4-manifold with a Rie- mannian metric g which has W+ = 0 . The unit 2-sphere bundle Z C A + has the structure of a complex manifold (a complex 3-fold) for which the fiber S 2 's for the projection to X are holomorphic. Together, thes e first tw o theorems construct a whole raft o f new com- plex 3-folds . An d the y lea d t o th e followin g mystery : Question: Wil l the study of complex 3-folds provid e insights abou t real 4-manifolds ? I shall retur n t o Theorem s 1 and 2 in late r lectures . e) Anti-sel f dualit y fo r vecto r bundle s The equation W + = 0 has a simpler (thoug h not reall y simple) analo g in th e followin g se t up : Suppos e tha t a metri c g i s fixed onc e an d for al l o n th e oriente d 4-manifol d X. Now , on e consider s covarian t derivatives o n vector bundle s whic h hav e curvature s whic h ar e anti-sel f dual i n th e sens e tha t a s section s o f End(V r ) ® A2T*, thei r projectio n into End(F) ® A+ i s everywhere zero .

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