柯西-黎曼复形的正则性

柯西-黎曼复形的Neumann问题是多复变和偏微分方程领域的基本问题并和复代数几何密切相关。我们计划研究具有实解析边界的弱拟凸域上次椭圆乘子理想层的有效生成问题以及具有光滑边界的弱拟凸域的边界几何如何控制次椭圆乘子理想层的问题，我们希望进一步对d-dbar-问题建立相应的理论。..已取得的相关成果：引入一个刻画除子奇性的新指标和奇异度量并利用负曲率L平方方法建立了Nevanlinna第二基本定理型的结果，把Ru-Vojta的最新结果推进到最佳(Math.Ann.2019，Asian.J.Math.2018)；L平方方法在微分几何中的应用(Adv.Math.2017，Adv.Math.2014)；L平方Cramer法则(Math.Ann.2013，Adv.Math.2012)。

The Neumann problem for the Cauchy-Riemann complex is a fundamental problem both in the theory of several complex variables and in the theory of partial differential equations which also is closely connected to complex algebraic geometry. This project aims at the effectiveness problem for the subelliptic multiplier ideal sheaves on weakly pseudo-convex domains with real analytic boundaries, the effect of the boundary geometry on the multiplier ideal sheaves, we also plan to develop an analogous theory for the d-dbar-problem...We have obtained the following results on related topics: we introduced a new index revealing certain geometry of divisors, via a negative curvature version of L^2-technique, established the sharp form of a newly obtained result of Min Ru and Paul Vojta(Math.Ann.2019, Asian.J.Math.2018); Applications of L^2 theory to differential geometry(Adv.Math.2017, Adv.Math.2014); L^2-Cramer rule(Math.Ann.2013, Adv.Math.2012).