By Bruce Bennett, Stephen S. -T. Yau (auth.), Ronald J. Knill, Morris Kalka, Howard C. J. Sealey (eds.)
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Extra 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
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. 1) = 0 (resp. surjective if depthp(~) _> 2 . we see that the map is inJective (resp. surjective) ~(V,~) = O) .
Of c o u r s e , of case, p Is our form there example 55 is v e r y special. F o l l o w i n g Calabi, one can c a l c u a l t e terms of the degrees of function f~P) the e n e r g y of f , f(q)_ , or in terms of the g e n e r a t i n g g , by the c l a s s i c a l P l u c k e r formulas. E e l l s and W o o d have e s t i m a t e d the index of the n o n - h o l o m o r p h i c h a r m o n i c ~pl + ~pn in , again u s i n g c o m p l e x curve theory. s h o u l d refer to in  for details, f : The r e a d e r as well as further r e f e r e n c e s to the r e c e n t l i t e r a t u r e on the problem.
The g while keeping on S I x M' The Ricci Derdzinski M for any metrics are author among be t o r s i o n free res s a t i s f y position general into : ([i]) The D metrics these metrics with constant sense when one scalar one uncouples connection D the on the t a n g e n t is a Y a n g - M i l l s metric connection on p r o d u c t positive scalar (hence the metrics is of s p e c i a l interest. The flat. on a c o m p a c t relies heavily identity. orientable curvature on t h e has Since D is).