+�18.966–Homework3–dueThursdayApril19,2007.1.Let (M, ω) be a symplectic manifold, J a compatible almost-complexstructure,and gthe corresponding Riemannianmetric. Show that two-dimensional almost-complexsub-′ manifolds ofMare absolutelyvolume minimizing intheir homologyclass, i.e.: let C,C be two-dimensionalcompactclosedorientedsubmanifoldsof M, representing the sameho-mologyclass [C] = [C′ ]∈H2(M,Z). Assume that J(TC) =TC(andthe orientation ofC agrees withthat inducedbyJ).Thenvolg(C)≤volg(C′). Hint:compare ω|C′andthe area form inducedbyg. 2.We willadmitthe factthat the cohomologyring of CPn(the set of complex lines through 0 inCn+1) is H∗(CPn,Z) =Z[h]/hn+1, where h∈ H2(CPn,Z) isPoincar´e dualtothe homologyclass representedbya linear CPn−1⊂CPn. The tautologicalline bundle L → CPnisthe subbundle of the trivial bundle Cn+1×CPnwhose fiber ata pointofCPnis the corresponding line inCn+1. The homogeneous
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