PomeLemma 13.2.1.6632853A40C9 Let $R$ be a positive current of bidegree $(p,p)$ on an open set $W\subset \mathbb C^ n$, and let $E\subset W$ be a Borel set. Then the current $\mathbf1_ ER$, obtained by multiplying the coefficient measures of $R$ by $\mathbf1_ E$, is positive. If $R$ is strongly positive, that is, $R(\varphi )\geq 0$ for every test form $\varphi $ that is weakly positive at every point, then so is $\mathbf1_ ER$.
Lemma 13.2.16632853A40C9
Proof. By theorem 5.3.7, $R$ has measure coefficients. Let $\varphi $ be a test form of bidegree $(n-p,n-p)$ that is strongly positive, respectively weakly positive, at every point. The complex measure $R\wedge \varphi $, whose integral against a test function $\chi $ is $R(\chi \varphi )$, defines a positive distribution, because $\chi \varphi $ has the same positivity as $\varphi $ whenever $\chi \geq 0$; by the uniqueness in theorem 2.5.6, it is a positive measure. Hence $(\mathbf1_ ER)(\varphi )=(R\wedge \varphi )(E)\geq 0$. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.