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Unformatted text preview: + 18.966 Homework 3 due Thursday April 19, 2007. 1. Let ( M, ) be a symplectic manifold, J a compatible almost-complex structure, and g the corresponding Riemannian metric. Show that two-dimensional almost-complex sub- manifolds of M are absolutely volume minimizing in their homology class, i.e.: let C , C be two-dimensional compact closed oriented submanifolds of M , representing the same ho- mology class [ C ] = [ C ] H 2 ( M, Z ). Assume that J ( TC ) = TC (and the orientation of C agrees with that induced by J ). Then vol g ( C ) vol g ( C ). Hint: compare | C and the area form induced by g . 2. We will admit the fact that the cohomology ring of CP n (the set of complex lines through 0 in C n +1 ) is H ( CP n , Z ) = Z [ h ] /h n +1 , where h H 2 ( CP n , Z ) is Poincar e dual to the homology class represented by a linear CP n 1 CP n ....
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This note was uploaded on 03/08/2010 for the course MATHEMATIC 570 taught by Professor Denisauroux during the Spring '10 term at Caltech.
- Spring '10