Einstein almost cokähler manifolds (Q2825990)

The Goldberg conjecture is the statement whereby a compact Einstein \(2n\)-manifold \((M,g)\) should not admit strict almost Kähler (\(g\)-compatible, symplectic but not Kähler) forms \(\omega\). It famously holds when the scalar curvature is non-negative [\textit{K. Sekigawa}, J. Math. Soc. Japan 39, 677--684 (1987; Zbl 0637.53053)]. The negative curvature case is still open, and the conventional wisdom is that it should be true always. Removing compactness leads to counterexamples [\textit{V. Apostolov} et al., Int. J. Math. 12, No. 7, 769--789 (2001; Zbl 1111.53303)].NEWLINENEWLINEThere is a similar conjecture adapted to odd dimensions, with mild variations depending on the point of view [\textit{C. P. Boyer} and \textit{K. Galicki}, Proc. Am. Math. Soc. 129, No. 8, 2419--2430 (2001; Zbl 0981.53027); \textit{V. Apostolov} et al., Math. Nachr. 279, No. 9--10, 948--952 (2006; Zbl 1172.53321); \textit{B. Cappelletti Montano} and \textit{A. M. Pastore}, J. Adv. Math. Stud. 3, No. 2, 27--40 (2010; Zbl 1210.53038)]. Let \((M,\alpha, \omega, g)\) be a compact, Einstein, almost co-Kähler manifold, meaning \((M,g)\) is a smooth compact Einstein manifold of real dimension \((2n+1)\), \(\alpha\) is a closed \(1\)-form and \(\omega\) a symplectic form on the complement \(\alpha^\perp\). The central question is to understand when such a structure is an honest co-Kähler structure, i.e., when \(\alpha\) and \(\omega\) are \(g\)-parallel. Both the geometry and topology of (almost) co-Kähler manifolds have been much studied, see the references.NEWLINENEWLINEThe authors dutifully prove that if the odd-dimensional conjecture (compact, Einstein, almost co-Kähler \(\Rightarrow\) Kähler) were false so would the Goldberg conjecture, since the `square' of an Einstein, strictly almost co-Kähler manifold is Einstein and strictly almost Kähler. Now take a compact, Einstein, almost co-Kähler manifold \((M^{2n+1},\alpha, \omega, g)\) with volume \(V\), and call the normalised scalar curvatures \(\tau\) and \(\tau^*\). The main result of this paper states thatNEWLINENEWLINE-- either this structure is co-Kähler, and \(\tau=0=\tau^*\), orNEWLINENEWLINE-- it is not co-Kähler, and NEWLINE\[NEWLINE\frac1{2n}\leq \frac1V\int \frac{\tau-\tau^*}{\tau} \leq \frac{4n-1+\sqrt{16n^2-8n-14}}{10n}.NEWLINE\]NEWLINENEWLINENEWLINEConsequently if \(\tau^*\geq 0\) the conjecture holds, and in the non-integrable case the authors establish bounds for \(\tau^*\).NEWLINENEWLINEWhat is more, exploiting \textit{J. Lauret}'s landmark results [Ann. Math. (2) 172, No. 3, 1859--1877 (2010; Zbl 1220.53061)] they show that the odd-dimensional conjecture, too, requires compactness, for they exhibit examples of non-compact but complete, Einstein, strictly almost co-Kähler manifolds.NEWLINENEWLINEThe arguments rely mainly on the quintessential Weitzenböck formula, used to prove that 1) Einstein co-Kähler implies Ricci-flat, and 2) Einstein almost co-Kähler forces \(\tau\leq 0\), and a useful local formula due to Apostolov et al. [loc. cit.] involving the curvatures and the Nijenhuis tensor of an almost Kähler manifold.