According to the Hopf conjecture, the Euler characteristic \(\chi(M)\) of a negatively curved \(2n\)-manifold \(M\) should have sign \((-1)^n\). For \(n>1\) this is not implied by the Chern-Gauss-Bonnet Theorem. For negatively curved Kähler manifolds and more generally for so-called Kähler hyperbolic manifolds the conjecture was proved by M. Gromov, [J. Differ. Geom. 33, No. 1, 263–292 (1991; Zbl 0719.53042)]. There, a manifold is called Kähler hyperbolic, if the pull-back of the Kähler form to its universal covering space is the differential of a bounded form. Gromov proved that under this assumption one obtains a lower bound for the spectrum of the Laplacian \(\Delta_d\) acting on \(k\)-forms for \(k\not=n\). In particular, there are no harmonic \(L^2\)-forms in degrees \(k\not=n\) and thus the \(L^2\)-Betti numbers \(b_k^{(2)}\) vanish in these degrees. Since the Euler characteristics is the alternating sum of the \(L^2\)-Betti numbers, one obtains \(\chi(M)=(-1)^nb_n^{(2)}(M)\), which implies the Hopf conjecture.
The paper under review generalizes Gromov’s result to so-called special symplectic hyperbolic closed manifolds. These are almost Kähler manifolds, where the pull-back of the Kähler form is the differential of a bounded form \(\theta\) with the additional condition \(\Vert\theta\Vert^{-2}_\infty\ge \frac{C}{4}\Vert N_J\Vert^2_\infty\) for the Nijenhuis tensor \(N_J\). (For the Kähler hyperbolic case, this is trivially true because \(N_J=0\). For a closed almost Kähler manifold with sectional curvature bounded above by a negative constant \(-K\), one has \(\Vert\theta\Vert_\infty\le\sqrt{n}K^{-\frac{1}{2}}\), from which the inequality follows for large enough \(K\), that is, after scaling.)
Gromov’s proof in the Kähler case built on the identity \(\left[L,\Delta_d\right]=0\), derived from the Kähler identities for the Lefschetz operator \(L\), its adjoint \(\Lambda\), and the Dolbeault \((1,0)\)- and \((0,1)\)-operators \(\partial, \overline{\partial}\). For almost Kähler manifolds one has a decomposition \(d=\partial+\overline{\partial}+A_J+\overline{A_J}\), where the last two summands come from the Nijenhuis tensor and vanish in the Kähler case, and one has almost Kähler identities due to J. Cirici and S. O. Wilson, [Sel. Math., New Ser. 26, No. 3, Paper No. 35, 27 p. (2020; Zbl 1442.32034)].
The identity \(\left[L,\Delta_d\right]=0\) does actually hold if and only if the almost Kähler manifold is Kähler. However, denoting \(d^\prime=\partial+\overline{A_J}\), the author considers the Laplacian \(\Delta_{d^\prime}\), for which he proves \(\left[L,\Delta_{d^\prime}\right]=0\) and hence the Hard Lefschetz Condition, that is the existence of an \(\mathfrak{sl}_2\)-structure on \(d^\prime\)-harmonic forms. From this, the author obtains (for every almost Kähler manifold, where the pull-back of the Kähler is the differential of a bounded form) a lower bound for the spectrum and hence nonexistence of \(d^\prime\)-harmonic forms in degree \(k\not=n\). In the case of special symplectic hyperbolic manifolds, the author can improve this to show that for \(n\) odd resp.even the kernel of \(d^\prime+d^{\prime *}\) consists only of \(L^2\)-forms in odd resp.even degrees. Because a suitable \(L^2\)-index of this operator computes the Euler characteristics, this implies the Hopf conjecture in this case.