Some Morse-type inequalities for symplectic manifolds

Morse-type inequalities are given for the symplectic versions of the Bott-Chern and Aeppli cohomology groups defined by Tseng and Yau. The symplectic versions of the Bott-Chern and Aeppli cohomology groups, defined by Tseng and Yau [28], are finite-dimensional cohomology groups given in terms of differential forms on a closed symplectic manifold (M,ω) of dimension 2n. They are denoted H d+dΛ(M) and H ddΛ(M), 1 ≤ k ≤ 2n, and satisfy the following inequalities [3] relating their dimensions to the Betti numbers of M , bk, (1) dimH d+dΛ(M) = dimH ddΛ(M) ≥ bk. The purpose of this paper is to prove a set of Morse-type inequalities for these groups. Theorem 1. Let (M,ω) be an 2n-dimensional closed symplectic manifold and f a Morse function with mi critical points of index i, then for k ≤ n dimH d+dΛ(M) ≤∑ i=0 mk−2i, (2) dimH ddΛ(M) ≤∑ i=0 m2n−k+2i. (3) Remark 1. The dimensions of the d+dΛ and dd groups can vary under homotopy of the symplectic form [26]. (M,ω) satisfying the hard Lefschetz condition is equivalent to the inequality (1) being saturated for all k [28], and in this case Theorem 1 is strictly weaker than the weak Morse inequalities if n ≥ 2 (and equivalent if n = 1). If (M,ω) does not satisfy the hard Lefschetz condition, Theorem 1 can give constraints on the mi additional to those obtained from the weak Morse inequalities. In particular, a short calculation gives the following. Corollary 1. The total number of critical points of a Morse function on a closed symplectic manifold (M,ω) of dimension 2n satisfies the inequality h + 2h + h ≤ 2n ∑ i=0 mi, where h = dimH d+dΛ(M). The quantity h + 2h +hn in Corollary 1 can be larger than the sum of the Betti numbers, as the following example shows.