MAT 545: Complex GeometryFall 2008Problem Set 4Due on Tuesday, 11/4, at 2:20pm in Math P-131(or by 2pm on 11/4 in Math 3-111)Please write up concise solutions to problems worth 20 pts.Problem 1 (5 pts)Let (V, J) be a complex vector space of complex dimension m and g a Riemannian metricon V compatible with J. Show that∗: Λp,qV −→ Λm−p,m−qV ⊂ Λ2m−p−qC(V ⊗RC),where ∗ is the Hodge operator on (V, g) with the orientation induced by J and extendedC-anti-linearly to V ⊗RC.Problem 2 (5 pts)Let M and N be complex manifolds with hermitian metrics hMand hN. Show that∆M×N= ∆M⊗ 1 + 1 ⊗ ∆N,where ∆M, ∆N, and ∆M×Nare the Laplacians on M, N , and M ×N with respect to themetrics hM, hN, and hM⊗1+1⊗hN.Problem 3 (5 pts)Let E −→ M be a holomorphic vector bundle with a hermitian inner-product over a com-pact complex manifold with a hermitian metric and ∆Ethe corresponding Laplacian on E.Show that(a) all eigenvalues of ∆Eare non-negative;(b) eigenfunctions corresponding to distinct eigenvalues of ∆Eare orthogonal;(c) eigenspaces of ∆Eare finite-dimensional;(d) the set of eigenvalues of ∆Ehas no limit point.Problem 4 (10 pts)With notation as in Problem 3, show that(a) ∆Ehas a positive eigenvalue;(b) ∆Ehas infinitely many positive eigenvalues;(c) the linear span of eigenfunctions of ∆Eis L2-dense in Γ(M; Λ∗C(T∗M ⊗RC)⊗CE);(d) the linear span of eigenfunctions of ∆Eis L∞-dense in Γ(M; Λ∗C(T∗M ⊗RC)⊗CE).This problems supplies the missing ingredient for the proof of Kunneth formula in thetextbook: the existence of a complete set of eigenfunctions of a laplacian.Hint for (a): Let G be the inverse of ∆Eon H∗,∗(M, E)⊥. Show that λ1∈R+given by1/λ1= supα∈H∗,∗(M,E)⊥,kαk=1kGαkis the smallest positive eigenvalue of