Journal of Symplectic Geometry

Volume 18 (2020)

Number 2

On the lower bounds of the $L^2$-norm of the Hermitian scalar curvature

Pages: 537 – 558



Julien Keller (Aix Marseille Université, CNRS Centrale Marseille, Institut de Mathématiques de Marseille, France)

Mehdi Lejmi (Department of Mathematics, Bronx Community College of CUNY, Bronx, N.Y., U.S.A.)


On a pre-quantized symplectic manifold, we show that the symplectic Futaki invariant, which is an obstruction to the existence of constant Hermitian scalar curvature almost-Kähler metrics, is actually an asymptotic invariant. This allows us to deduce a lower bound for the $L^2$-norm of the Hermitian scalar curvature as obtained by S. Donaldson [15] in the Kähler case.

Received 27 April 2017

Accepted 30 April 2019

Published 8 June 2020