Harmonic cohomology groups on compact symplectic nilmanifolds
The following results on Brylinski's symplectic harmonic \(k\)-cohomology group \(H^k_{hr}(M)\) [\textit{J. L. Brylinski}, J. Differ. Geom. 28, 93-114 (1988; Zbl 0634.58029)] are proved. 1. Let \(H^q_{\omega-hr}(M)\) be the symplectic harmonic cohomology with respect to a symplectic form \(\omega\) on \(M\). Then \(H^q_{\omega-hr}(M) = H^q_{\omega'-hr}(M)\) if \(\omega' =\omega + d\gamma\). 2. If \(M = G/\Gamma\) is a compact symplectic nilmanifold, \(\omega_0\) a left \(G\)-invariant 2-form and \(\omega = \omega_0+ d\gamma\), then \(H^q_{\omega-h}(M)\) is isomorphic to the Lie algebra cohomology group \(H^q_{\omega_0-hr}({\mathfrak g})\). 3. If \(M^{2m} = G/\Gamma\) is a compact \((r+1)\)-step nilmanifold, then \(\dim H^1_{hr}(M)-\dim H^{2m-1}_{hr}(M)\geq \dim {\mathfrak g}^{(r)}\). In particular, if \(M\) is a 2-step nilmanifold, then \(\dim H^1_{hr}(M)- \dim H^{2m-1}_{hr}(M)=\dim [{\mathfrak g},{\mathfrak g}]\). 4. There exists a 6-dimensional nilmanifold which has a family \(\omega_t\) of symplectic forms such that the dimension \(H^4_{\omega-hr}(M)\) varies. There are confusions on numbering these results. In the Introduction, 1 and 2 are stated as Prop. 1 and 2, 3 is stated as Theorem 3. 4 is obtained to apply precise computation of symplectic harmonic cohomology of six dimensional 2-step nilmanifold which is stated as Theorem 4. While in the body of the paper, Theorem 3 is referred to as Theorem 1 and Theorem 4 is refered to as Theorem 3, Theorem 2 does not exist. It is known that \(H^k_{DR}(M)=H^k_{hr}(M)\), \(k=1,2\) [\textit{O. Mathieu}, Harmonic cohomology classes of symplectic manifolds, Comment. Math. Helv. 70, 1-9 (1995; Zbl 0831.58004)]. There exists a 4-manifold which has a family \(\omega_t\) of symplectic forms such that the dimension of \(H^3_{\omega_t-hr}(M)\) varies. But the dimension of \(H^3_{\omega_t-hr}(M)\) is independent of the symplectic form for compact 4-dimensional nilmanifolds [\textit{D. Yan}, Adv. Math. 120, 143-154 (1996; Zbl 0872.58002)]. So 4 is an optimal example. The outline of the paper is as follows: In section 2, Brylinski's symplectic \(*\)-operator and symplectic harmonic cohomology group are defined, and their basic properties are reviewed. In section 3, regarding the space \(\Omega^*(M)\) of differential forms on \(M\) as a \(\mathfrak{sl}(2)\)-module, by using \(*\)-operation, dualities of the space of symplectic harmonic forms and space of left \(G\)-invariant harmonic forms in the case \(M = G/\Gamma\) are given (Prop. 3.4 and 3.5). In section 4, by using Yan's results, result 1, (Prop. 1 in the Introduction) is proved (Prop. 4.4). Then showing \(H^q_{hr}(M) = H^q_{hr}({\mathfrak g})\), when \(M\) is a nilmanifold and \(\omega\) is a left invariant 2-form (Prop. 4.5), result 2 (Prop. 2 in the Introduction) is shown. By using these results and precise calculus on Lie algebra structure of \(\mathfrak g\), 3 (Theorem 3 in the Introduction, called Theorem 1 in other parts of the paper) is proved. This section also contains precise computation of \(\dim H^1_{hr}(M)-\dim H^{2m-1}_{hr}(M)\), when \(M\) is a compact symplectic \((r+1)\)-step nil manifold (Prop. 5.3). Section 6 is devoted to the computation of examples such as the generalized Heisenberg group. Then in section 7, the last section, symplectic harmonic cohomologies of 6 dimensional 2-step nilmanifolds are studied precisely (Theorem 4 in the introduction, called Theorem 3 in this section) and the example stated in 4 is given.
scientific article; zbMATH DE number 1799579