HW206sol - Differential Geometry 2 Homework 06 1. Let M be...

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Unformatted text preview: Differential Geometry 2 Homework 06 1. Let M be a connected complex manifold (of complex dimension greater than one) on which g is a K ahler metric. Show that the only Kahler metrics on M that are conformal to g are the constant multiples of g . 2. Let V, h|i be a complex inner product space. Show explicitly that the induced fundamental two-form on complex projective space P ( V ) is closed so that P ( V ) is a Kahler manifold. Solutions (1) Let f C ( M ) be everywhere positive and let e g = fg with e = f . The Leibniz rule shows that d e = d f + f d = d f since g is Kahler; accordingly, e g is Kahler iff d f = 0. We claim (essentially on the basis of algebra) that if is a one-form then = 0 = 0. To see this, argue by contradiction: suppose 6 = 0 and choose so that ( ) = 1 (locally or at a point); with = y it follows that 0 = y ( ) = ( y ) - ( y ) or...
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HW206sol - Differential Geometry 2 Homework 06 1. Let M be...

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