Consider a normal complex analytic space $X$ and a projective birational resolution $f:Y\rightarrow X$. Let $T$ be a closed positive current of bidimension $(p,p)$ on $X$. Is there always a closed positive current $S$ on $Y$ such that $f_*S=T$?
1 Answer
I believe the answer is no.
If I am reading the article below by Méo correctly, he shows that if $\pi : Y \to X$ is the blowup of the polydisc $X=D^m$ along $Z := \{ z_1 = \dots = z_k = 0 \}$, and if $k \leq p \leq n2$, then there exists a closed positive $(p,p)$current $T$ on $X$ such that $T' := \pi^* (T_{X\setminus Z})$ does not have locally finite mass near $\pi^{1}(Z)$. If such $S$ exists, then since $\pi$ is a biholomorphism on $Y' := \pi^{1}(X\setminus Z)$, one would have $S_{Y'} = T'$, which would thus have locally finite mass.
Reference
Michel Méo, "Image inverse d'un courant positif fermé par une application analytique surjective", C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 12, 1141–1144, MR1396655, Zbl 0858.32012.

$\begingroup$ Tack, Richard! That's indeed what I'm looking for. $\endgroup$ Mar 26, 2022 at 8:19