Collapsing of the line bundle mean curvature flow on Kähler surfaces (Q2225910)

Let \(X\) be a compact complex surface, \(\alpha\) a Kähler form (a closed Hermitian form), \(\hat F\) a closed real \((1,1)\)-form, and \(\psi_t\) a real-valued smooth function on \(X\) and solution of the line bundle mean curvature flow \(\displaystyle\frac{d\psi_t}{dt}=\Theta_\alpha-\hat\Theta\) such that \(\Theta_\alpha\) stands for the Lagrange phase and \(\Theta_\alpha\equiv\hat\Theta\ (2\pi)\). Then, by assuming that \(\psi_0\) is hypercritical, \(\cot\hat\Theta_\alpha+\hat F\ge 0\), and \(\displaystyle\hat\Theta\ge \frac{\pi}{2}\), the author deduces that there is a family of curves \((C_j)_{j\in\mathcal{J}}\) (\(\mathcal{J}\) is a finite subset of \(\mathbb N\)) with negative self-intersection such that \(\displaystyle\lim_{t\to\infty}\psi_t=\psi_\infty\in C_{\mathrm{loc}}^\infty(X\setminus\cup_{j\in\mathcal{J}}C_j)\) and \(F_\infty=\hat F+\sqrt{-1}\partial\overline\partial\psi_\infty\) is a smooth Kähler current on \(X\setminus\cup_{j\in\mathcal J} C_j\) and satisfying \(\Im\left(\exp(-\sqrt{-1}\hat\Theta)(\alpha+\sqrt{-1}F_{\psi_\infty})^2\right)=0\) (Theorem 1).