Symplectic domination. (English) Zbl 1472.53087

Fine, Joel; Panov, Dmitri

Bull. Lond. Math. Soc. 53, No. 1, 100-103 (2021).

In this short note the authors prove that every compact oriented symplectic manifold \(M\) of even dimension is dominated by a symplectic manifold \(S\) of the same dimension, that is to say that there exists a map of positive degree \(f\,:\,S \rightarrow M\).
The sketch of the proof deserves to be briefly recalled. The first remark is that the twistor bundle \(Z\rightarrow M\) of an oriented even-dimensional Riemannian manifold has model fiber \(F:=\operatorname{SO}(2n)/\operatorname{U}(n)\) which has the structure of an homogeneous integral symplectic manifold. Thus, for each model fiber \(F\) we have the circle bundle of the quantization \(P_F\rightarrow F\) which induces a circle bundle \(P\rightarrow Z\); by using the Levi-Civita connection for defining the horizontal distribution transverse to the fibers of \(Z\rightarrow N\) we have a connection for the circle bundle \(P\rightarrow Z\) whose curvature determines a closed integral two form \(\omega\), which is not symplectic in general. By a result of A. G. Reznikov [Ann. Global Anal. Geom. 11, No. 2, 109–118 (1993, Zbl 0810.53056)] \(\omega\) is symplectic when the metric on \(M\) is a sufficiently pinched negatively curved metric; this is not restrictive since, by a result of P. Ontaneda [Publ. Math., Inst. Hautes Étud. Sci. 131, 1–72 (2020, Zbl 1442.53026)], there exists a manifold \(N\) of the same dimension of \(M\) and degree one map \(f\,:\,N\rightarrow M\) such that the sectional curvature of \(N\) is in \([-1-\epsilon,\,-1]\). So, up to replacing \(M\) by \(N\) we have a map \(f\,:\,Z \rightarrow N\), the last part of the proof consists in considering a Donaldson hypersurface in \(Z\) and studying the degree of the resulting map [S. K. Donaldson, J. Differ. Geom. 44, No. 4, 666–705 (1996, Zbl 0883.53032)].

Reviewer: Andrea Galasso (Taipei)

MSC:

53D05

Symplectic manifolds (general theory)

Keywords:

symplectic manifold; symplectic domination; twistor bundle; quantization; Donaldson hypersurface

Citations:

Zbl 0810.53056; Zbl 1442.53026; Zbl 0883.53032

References:

[1]	R. M.Charney and M. W.Davis, ‘Strict hyperbolization’, Topology34 (1995) 329-350. · Zbl 0826.53040
[2]	S. K.Donaldson, ‘Symplectic submanifolds and almost‐complex geometry’, J. Differential Geom.44 (1996) 666-705. · Zbl 0883.53032
[3]	Y.Eliashberg, ‘Recent advances in symplectic flexibility’, Bull. Amer. Math. Soc.52 (2015) 1-26. · Zbl 1310.53001
[4]	J.Fine and D.Panov, ‘Symplectic Calabi-Yau manifolds, minimal surfaces and the hyperbolic geometry of the conifold’, J. Differential Geom.82 (2009) 155-205. · Zbl 1177.32014
[5]	J.Fine and D.Panov, ‘Hyperbolic geometry and non‐Kähler manifolds with trivial canonical bundle’, Geom. Topol.14 (2010) 1723-1763. · Zbl 1214.53058
[6]	M.Gromov, ‘Hyperbolic groups’, Essays in group theory, MSRI Publications (ed. S. M.Gersten (ed.); Springer, New York, 1987) 75-284.
[7]	A. A.Kirillov, Lectures on the orbit method, Graduate Studies in Mathematics 64 (American Mathematical Society, Providence, RI, 2004). · Zbl 1229.22003
[8]	P.Ontaneda, ‘Pinched smooth hyperbolization’, Preprint, 2011, arXiv:1110.6374.
[9]	P.Ontaneda, ‘Riemannian hyperbolization’, Publ. math. IHES131 (2020) 1-72, https://doi.org/10.1007/s10240-020-00113-1. · Zbl 1442.53026
[10]	P.Ontaneda, ‘Normal smoothings for smooth cube manifolds’, Asian J. Math.20 (2016) 709-724. · Zbl 1357.57055
[11]	P.Ontaneda, ‘On the Farrell and Jones warping deformation’, J. Lond. Math. Soc. (2)92 (2015) 566-582. · Zbl 1332.53051
[12]	P.Ontaneda, ‘Deforming an \(\varepsilon \)‐close to hyperbolic medtric to a warp metric’, Preprint, Michigan Math. J.65 (2016) 693-701. · Zbl 1360.53038
[13]	P.Ontaneda, ‘Cut limits on hyperbolic extensions’, Michigan Math. J.65 (2016) 703-714. · Zbl 1381.58004
[14]	P.Ontaneda, ‘Normal smoothings for Charney-Davis strict hyperbolizations’, J. Topol. Anal.9 (2017) 127-165. · Zbl 1358.57024
[15]	P.Ontaneda, ‘Deforming an \(\varepsilon \)‐close to hyperbolic metric to a hyperbolic metric’, Proc. Roy. Soc. Edinburgh148 (2018) 629-641. · Zbl 1404.53048
[16]	A. G.Reznikov, ‘Symplectic twistor spaces’, Ann. Global Anal. Geom.11 (1993) 109-118. · Zbl 0810.53056
[17]	A.Weinstein, ‘Fat bundles and symplectic manifolds’, Adv. Math.37 (1980) 239-250. · Zbl 0449.53035