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Unformatted text preview: 18.966 – Homework 3 – Solutions. ′ 1. Given a point p ∈ C (a twodimensional oriented submanifold), let ( e, f ) be an oriented basis of T p C ′ , orthonormal with respect to the metric g induced by ω and J . Then ω ( e, f ) = g ( Je, f ) ≤  Je  f  =  e  f  = 1. Meanwhile, the area form dvol g  C ′ induced by g ′ on C is given by dvol g  C ′ ( e, f ) = 1. Hence ω  C ′ ≤ dvol g  C ′ at every point of C ′ ; integrating, we deduce that [ ω ] · [ C ′ ] = C ′ ω ≤ vol g ( C ′ ). In the case of C (an almostcomplex submanifold, equipped with the orientation induced by J ), an oriented orthonormal basis of T p C is given by ( e, Je ) where e is any unit length vector in T p C . (Note that  Je  =  e  = 1 and g ( Je, e ) = ω ( e, e ) = 0). Then ω ( e, Je ) = g ( Je, Je ) = 1 = dvol g  C ( e, Je ), so ω  C = dvol g  C , and [ ω ] · [ C ] = C ω = vol g ( C ). In conclusion, vol g ( C ) = [ ω ] · [ C ] = [ ω ] · [ C ′ ] ≤ vol g ( C ′ ). 2. a) The homogeneous coordinate x n is a linear form on C n +1 (namely, ( x , . . . , x n ) mapsto→ x n ) and hence, by restriction to the tautological line, a linear form on L . This section of L ∗ vanishes precisely at those points [ x : ··· : x n ] for which the last coordinate is zero, so its zero set is CP n − 1 ⊂ CP n . Moreover, it vanishes transversely, and the orientation induced on its zero set is the natural one (because all orientations agree with those induced by the complex structure). So c 1 ( L ∗ ) = e ( L ∗ ) is Poincar´ e dual to [ CP n − 1 ] ∈ H 2 n − 2 ( CP n ), i.e....
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 Spring '10
 DenisAuroux
 Linear Algebra, Geometry, Linear map, Chern, Chern class, CPN

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