Effective estimate of positivity loss in current regularizations (Q1426548)

Let \((X,\omega)\) be a compact complex Hermitian manifold, \(\gamma\) a real continuous \((1,1)\) form and \(T \geq \gamma\) be a \(d\)-closed \((1,1)\) almost positive current on \(X\). Then \(T = \alpha + i \partial \bar{\partial} \varphi\), \(\alpha\) a differentiable \((1,1)\) form and \(\varphi\) an almost plurisubharmonic function. A result of Demailly states the existence of a series of almost plurisubharmonic functions \(\varphi_m\) with analytic singularities satisfying several nice properties, including \(\alpha + i \partial \bar{\partial}\varphi_m \geq \gamma - \epsilon_m \omega\) with \(\epsilon_m >0\) tending to zero. In this paper the author gives, in case \(\gamma\) is \(C^1\), the bound \(C \root 4\of{ m}\) for \(\epsilon_m\), where \(C\) a constant not depending on \(m\). The estimate is improved to \(C/m\) in case \(\gamma\) is \(C^\infty\) and closed.