Timeline for zero extension of positive currents are always positive

6 events

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May 11, 2020 at 9:37	comment	added	Invariance		@Henri, ok, now I get it, thanks for your patience and assistance again.
May 11, 2020 at 9:35	comment	added	Henri		The quantity $\langle T \wedge u ,f \rangle $ has to be a number (not $+\infty$) if you want to make $\widetilde T$ into a current.
May 11, 2020 at 9:07	comment	added	Invariance		@Henri, thanks for your help. One more question maybe sounds stupid, why we need $\langle T\wedge u,f \rangle $ is convergent, as when consdering the local representation, it seems like non-negative already?
May 11, 2020 at 8:08	comment	added	Henri		I meant that the $\lambda_I$'s are positive measures (they don't necessarily have a density wrt the Lebesgue measure) with locally finite mass near $E$, hence $\langle \lambda_I, f\rangle$ is well-defined.
May 11, 2020 at 7:44	comment	added	Henri		This is where the finite mass condition comes in. It ensures that, with your notations, $\langle T \wedge u ,f \rangle$ is convergent. To simplify notation, say $T= \sum \lambda_I dz_I \wedge d \bar {z_I}$ and $u=dz_J\wedge d \bar{z_J}$. Then $\langle T\wedge u,f\rangle = \int_{X\setminus E}\lambda_{J^c} fdV_{\mathbb C^n}$ while we know that the functions $\lambda_I$ are non-negative on $X\setminus E$ and locally integrable near $E$.
May 11, 2020 at 4:19	history	asked	Invariance	CC BY-SA 4.0