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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 K¨ahler 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 ω on complex projective space P ( V ) is closed so that P ( V ) is a K¨ahler manifold. Solutions (1) Let f ∈ C ∞ ( M ) be everywhere positive and let e g = fg with e ω = fω . The Leibniz rule shows that d e ω = d f ∧ ω + f d ω = d f ∧ ω since g is K¨ahler; accordingly, e g is K¨ahler iff d f ∧ ω = 0. We claim (essentially on the basis of algebra) that if α is a one-form then α ∧ ω = 0 ⇒ α = 0. To see this, argue by contradiction: suppose α 6 = 0 and choose ζ so that α ( ζ ) = 1 (locally or at a point); with β = ζ y α it follows that 0 = ζ y ( α ∧ ω ) = ( ζ y α ) ∧ ω- α ∧ ( ζ y ω ) or ω...
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This note was uploaded on 07/08/2011 for the course MTG 6256 taught by Professor Robinson during the Spring '09 term at University of Florida.
- Spring '09