MATHEMATICAL TRIPOS Part III PAPER 118 COMPLEX MANIFOLDS
MATHEMATICAL TRIPOS Part III Monday, 11 June, 2018 1:30 pm to 4:30 pm PAPER 118 COMPLEX MANIFOLDS Attempt no more than FOUR questions. There are FIVE questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS Coversheet None Treasury Tag Script paper You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator. 2 1 (a) Let M be a complex manifold. Define Ap,q(M), the space of global forms of type (p,q). Define the operators ∂ : Ap,q(M) →Ap+1,q(M) and ∂¯ : Ap,q(M) →Ap,q+1(M). (b) Let Pn be an n-dimensional polydisc. Consider α ∈Ap,q(Pn), p > 1, such that ∂α = 0. Show that there exists β ∈Ap−1,q(Pn) with ∂β = α. [You may use the ∂¯-Poincar´elemma which states that if Pn is a polydisc then the p,q n > Dolbeault cohomology H∂¯ (P ) = 0 for all q 1.] (c) The Bott-Chern cohomology group of M is defined to be {α ∈Ap,q(M), dα = 0} Hp,q (M) := . BC ∂∂¯Ap−1,q−1(M) n p,q Prove that when M = P , HBC (M) = 0 when p,q > 1. [You may use the ∂¯-Poincar´elemma as well as the smooth Poincar´elemma, which k n states that the de-Rham cohomology HdR(P ) = 0 for all k > 1.] (d) Let M be a compact K¨ahler manifold. Construct an isomorphism p,q ∼ p,q φ : HBC (M) = H∂¯ (M). Part III, Paper 118 3 2 (a) Let X be a topological space and F a sheaf of abelian groups.
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