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Proposition 10.30.25F6B6BF8B562

Proposition 10.30.2.5F6B6BF8B562 If $g:Z'\to Z$ is a compact modification over $Y$, pushforward and pullback induce maps

\[ g_*:\mathscr A_{Z'}(\theta )\to \mathscr A_ Z(\theta ), \qquad g^*:\mathscr A_ Z(\theta )\to \mathscr A_{Z'}(\theta ), \qquad g_*g^*=\mathrm{id}. \]

Pushforward preserves classes with positive current representatives.

Proof. Write $\pi '=\pi \circ g$. Translation by the pulled-back background identifies the required maps with the Bott–Chern maps of subsection 20.4.9. Indeed, for a closed current $C'$ on $Z'$ and a smooth closed form $\beta $ on $Z$,

\[ g_*(\pi '^*\theta +C')=\pi ^*\theta +g_*C',\qquad g^*(\pi ^*\theta +\beta )=\pi '^*\theta +g^*\beta , \]

where the first equality uses the change of variables formula for the degree-one map $g$ on the smooth background. Since the Bott–Chern maps are well defined on the closed terms, these identities define the affine maps independently of the representatives. Their composition is the identity because $g_*g^*$ is the identity both on the background and on Bott–Chern classes. Finally, pushforward under a proper map preserves positive currents. $\square$

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