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

## TL;DR

This paper establishes a lower bound for the L^2-norm of the Hermitian scalar curvature on pre-quantized symplectic manifolds, linking the symplectic Futaki invariant to curvature obstructions.

## Contribution

It introduces the asymptotic nature of the symplectic Futaki invariant and extends Donaldson's L^2-norm bounds from K"ahler to almost-K"ahler settings.

## Key findings

- Lower bound for L^2-norm of Hermitian scalar curvature
- Symplectic Futaki invariant as an asymptotic invariant
- Extension of Donaldson's results to almost-K"ahler manifolds

## Abstract

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-K\"ahler 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 \cite{Don} in the K\"ahler case.