METRICS, CONNECTIONS AND GLUING THEOREMS 23 X i s given a s th e zer o se t o f a syste m o f complex , algebrai c equations . And, a remarkabl e theore m o f Donaldson assert s tha t th e spac e M(V) is also the zero set o f some system o f complex, algebraic equations [17]. The poin t her e i s tha t wit h enoug h knowledg e o f algebrai c geometry , you can , i n principle , answe r an y questio n aske d abou t M(V). And , surprisingly enough , ther e ar e peopl e aroun d wh o ar e expert s i n thi s sort o f algebrai c geometr y (see , e.g. , [4] , [11] , [22]) . Th e fac t i s tha t there are quite a few examples where the whole of M(V) ca n be writte n down i n close d form . I won' t g o int o detail s her e becaus e I profes s a rhyolitic ignoranc e o f the algebrai c side o f the game . However, I will say that th e algebrai c geometry example s do serve as building block s fo r a surprisingl y successfu l cu t an d past e approac h t o understanding M(V) i n the general case. Here , I refer th e reader to th e recent remarkabl e result s i n [31 ] and [23] . i) Application s Here i s a summar y o f th e ramification s o f (2.9) : Suppos e tha t non - empty, M ca n b e assume d t o b e a smooth , oriente d submanifol d o f a n infinite dimensiona l manifol d B = B°/SU(2). (Thi s i s th e cas e whe n the numbe r 62"(X ) i s positive. ) Thi s Ai(V) i s not , b y itsel f anythin g canonical becaus e its construction depend s o n the choic e of a metric o n X. However , i f 6j(X ) 2 , then a pai r o f such Ms correspondin g t o a pair o f metrics o n X ar e cobordan t i n [0,1 ] x B vi a a smooth , oriente d cobordism. Se e the followin g diagram . \ 1 / / 1 / / (2.21)
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