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Unformatted text preview: 18.966 Homework 3 Solutions. 1. Given a point p C (a two-dimensional oriented submanifold), let ( e, f ) be an oriented basis of T p C , orthonormal with respect to the metric g induced by and J . Then ( 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 | C dvol g | C at every point of C ; integrating, we deduce that [ ] [ C ] = C vol g ( C ). In the case of C (an almost-complex 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 ) = ( e, e ) = 0). Then ( e, Je ) = g ( Je, Je ) = 1 = dvol g | C ( e, Je ), so | C = dvol g | C , and [ ] [ C ] = C = vol g ( C ). In conclusion, vol g ( C ) = [ ] [ C ] = [ ] [ 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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