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By Bruce Bennett, Stephen S. -T. Yau (auth.), Ronald J. Knill, Morris Kalka, Howard C. J. Sealey (eds.)

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Additional info for Harmonic Maps: Proceedings of the N.S.F.-C.B.M.S. Regional Conference, Held at Tulane University, New Orleans December 15–19, 1980

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Then Thus, let ~ e ker dj , ~IV' e Im(dj_llV,) Now depthp(O V) = n (because depthp(~ j-l) = n V , since is a hypersurface; n > 2 , by the surjectivity assertion ~ of HO(v,~ j-l) ; so they actually agree in ~J dj_l(~) agrees by the injectivity dJ-l(~) = w . e. , f ~' e H0(V',~ j-l) . 2) Since H0(V,~ j) . > 0 We want to see that this sequence = ~ P / ~ ( d p _ l) . extends to a section away from assertion of Adf . ,p-i . is locally free and with ~ is a global section in see remark of for WlV. --~ ~p-I ) ~P---)n~ P , ~ a free ~V resolution of by sheaves of depth n at ~ P , .

A sequence is not a zero divisor in S / ( f l , . . , f i ) ~ depthp(~ V) = dimp(~ V) at every point P . 31 vanish. 4) ~ < d - i , is supported at P . is a Cohen-Macaulay affine variety of dimension P , then (for 9 = ~p(V~v) ~ ( V ' , ~ V) = 0 for ~V ) , all of the = 0 for 1 < i < d - 2 . Exts on the right I < ~ < d - i , and we apply to get the result. Remark Every smooth variety is Cohen-Macaulay; every relative complete intersection over a Cohen-Macaulay variety is Cohen-Macaulay. (3-3) Proposition V , P , V' as above with V affine and ~ coherent.

Kahler submanifolds case of t h e fact harmonic [LZI,2]. Cauchy-Riemann I shall with to sional reason vector these cross generalize that minimal there of t h e cause the varieties maps between manifolds [WI]. This Kahler manifolds is t h a t holomorphic results in the varieties results of immersions products. some maps of is is a s p e c i a l satisfy I shall holomorphic [HLI,2] have considered are the and to m a n i f o l d s in d o i n g Lawson 3-dimensional define map a simple by H a r v e y note interest of H a r v e y interesting ~8.

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