Suppose $(X,g,I)$ is a Hermitian (non kahler) complex manifold with small torsion, small derivative of torsion and small curvature. Let $\varphi$ be smooth PSH function satisfying $\sqrt{-1}\bar\partial_I\partial_I\varphi\geq\omega_I$ also assume that $X:=\{\varphi<1\}$. My question is that can we solve $\bar\partial u=v$, where $v$ is a $(0,1)$ form, with L2 estimate $$\|u\|_{L^2(e^{-\varphi)}}\leq C\|v\|_{L^2(e^{-\varphi})}.$$

With small torsion, derivative torsion and small curvature. I believe that, for any smooth compact $(0,1)$ form $\phi$, we are able to prove that $$\|\bar\partial\phi\|_{L^2}+\|\bar\partial^*\phi\|_{L^2}\geq \frac{1}{2}\|\phi\|_{L^2}.$$

Then my understanding for Hormander type estimate is do deal with the density issue. For example, Demailly's notes with complex metric and restrict application to $(n,1)$ form. Also Hormander's classical book use three different weights.

I wonder if Hormander's three weights thing can be applied here by using the Strictly PSH function $\varphi$.



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.