标定几何中的多重次调和函数与q-凸性
Plurisubharmonic functions in calibrated geometry and q-convexity
摘要：
设$(M,ω)$是一个Kahler流形. M上的可积函数称为$ω^q$ -多重次调和函数，如果它在所有q维复子簇上都是次调和函数.证明了光滑的$ω^q$ -多重次调和函数是q-凸的.一个连续的$ω^q$ -多重次调和函数可以用光滑的$ω^q$ -多重次调和函数局部逼近.对于任何闭子簇$Z\subset M$，$\dim Z < q$，在$Z$的邻域中存在严格$ω^q$ -多重次调和函数（这个结果对于q-凸函数是已知的）.利用这个定理给出了Sibony引理关于在余维> p的复子簇外可积的正闭（p，p）-形式的可积性的一个新证明。
英文摘要：
Let $(M,ω)$ be a Kahler manifold. An integrable function on M is called $ω^q$-plurisubharmonic if it is subharmonic on all q-dimensional complex subvarieties. We prove that a smooth $ω^q$-plurisubharmonic function is q-convex. A continuous $ω^q$-plurisubharmonic function admits a local approximation by smooth, $ω^q$-plurisubharmonic functions. For any closed subvariety $Z\subset M$, $\dim Z < q$, there exists a strictly $ω^q$-plurisubharmonic function in a neighbourhood of $Z$ (this result is known for q-convex functions). This theorem is used to give a new proof of Sibony's lemma on integrability of positive closed (p,p)-forms which are integrable outside of a complex subvariety of codimension >p.
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