UNIVERSITY OF CAMBRIDGE MATHEMATICAL TRIPOS Part III Thursday, 4 June, 2015 1:30 pm to 4:30 pm PAPER 18 COMPLEX MANIFOLDS Attempt no more than FOUR questions. There are FIVE questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS Cover sheet Treasury Tag Script paper SPECIAL REQUIREMENTS None You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
UNIVERSITY OF CAMBRIDGE 2 1 1. Let X be a real manifold with dim X = 2n. Define what is meant by a holomorphic atlas on X and also what is meant by an almost complex structure on X. Prove that any holomorphic atlas on X induces a natural almost complex structure. 2. Now suppose that J is an arbitary almost complex structure on X. Explain how J defines the vector bundle TX0,1 and TX0,1. If 21, ... , Zn are local holomorphic 0 coordinates on X define the quantities azj and o and prove that the sets 0 az1 · , ? 0 and 2 az1 ., -0 azn give local frames for TX1,0 and TX0,1 respectively. 3. For smooth vector fields a, 6 on X define N (a, B) = 2([Ja, JB] - [a, B] - J[a, JB] - J[Ja, B]) where [a, B] = aß - Ba denotes the Lie bracket on vector fields. Prove that if J is the almost complex structure induced by a complex structure then N = 0. 4. Now write the decomposition of a smooth vector field a as a = a + a where a' I CR(X,TX1,0) and a" I CR(X,TX0,1). Prove that N(a,B)" = - 8[a', ß']" and show that N = 0 if and only if both TX1,0 and TX0,1 are involutive under the Lie bracket (i.e. if and only if whenever a and B are smooth sections of T1,0 X then so is [a, B] and similarly for T0,1 X). Part III, Paper 18
UNIVERSITY OF CAMBRIDGE 3 2 (a). Let X be a complex manifold. Define the sheaf Ak of k forms and the sheaf AP,9 of (p, q)-forms on X and show that Ak(X)= p+q=k M AP,q(X). Finally define the Dolbeaut cohomology groups HP,q (X). (b). The Bott-Chern cohomology group of X is defined to be BC( HPg(X) = 80Ap-1,9-1(X) {al AP,9(X) : da = 0} . Explain how this is well defined, and prove that complex conjugation gives an identification HB% (X) = HBc(X). (c). Prove that if B is a polydisk then HP;9 BC(B) = 0 for all p, q > 1. Does this statement still hold if B is replaced by an arbitary complex manifold? Justify your answer. [You may use the Poincare Lemmas which states that if B is a polydisk then the de-Rham cohomology HIR(B) = 0 for all k > 1 and HP,q (B) = 0 for all q > 1. ] (d). Consider next the map : HP%(X) - HP,ª(X) which takes the class of a I AP,q(X) in HP(X)