23.2.61 Bimeromorphic invariance5156B6A245EB

Proof. By proposition 10.11.2, both pullback and pushforward preserve pseudoeffectivity; the pullback of potentials is the one used in lemma 5.43.19. Since $\pi $ has degree one,

Indeed, the identity of currents $\pi _*\pi ^*\theta =\theta $ follows by integration over the locus where $\pi $ is an isomorphism, whose complement has volume zero. This left inverse proves injectivity and the reverse implication for pseudoeffectivity. $\square$

Proof. For a proper analytic subset $A$ of a connected complex manifold $M$, the restriction map $H^1(M,\mathbb R)\to H^1(M\setminus A,\mathbb R)$ is injective, and it is an isomorphism if $A$ has complex codimension at least two. Indeed, choose a Whitney stratification of $A$. General position moves every one-cycle off the strata, which have real codimension at least two. If those codimensions are at least four, it also moves every two-chain with boundary in $M\setminus A$ off the strata, relative to its boundary. Thus the map $H_1(M\setminus A,\mathbb R)\to H_1(M,\mathbb R)$ is surjective in the first case and bijective in the second, and dualizing over $\mathbb R$ gives the assertions about restriction.

Since $X$ is smooth, there is an analytic subset $A\subset X$ of complex codimension at least two such that $\pi $ is an isomorphism over $U=X\setminus A$. Restriction from $X$ to $U$ is therefore an isomorphism, whereas restriction from $Y$ to $\pi ^{-1}U$ is injective. Every class on $\pi ^{-1}U\simeq U$ extends to $X$ and hence pulls back to a class on $Y$. The commutative diagram of restriction maps now shows that $\pi ^*$ is an isomorphism, and Poincaré duality and the projection formula give the assertion for $\pi _*$. $\square$

Proof. First let $\pi :Y\to X$ be a modification. Holomorphic pullback commutes with $\mathrm d^{\mathrm c}$, so

By lemma 23.2.62, a class $\xi $ lies in $\mathcal C_ X$ if and only if $\pi ^*\xi $ lies in $\mathcal C_ Y$, and likewise for $K_ X$ and $K_ Y$, since the pullback is injective on Bott–Chern classes. By lemma 23.2.63, every degree-one class on $Y$ is of the form $\pi ^*\xi $. Thus the isomorphism $\pi ^*$ in degree one identifies the two cones and the two kernels. Its dual $\pi _*$ identifies $W_ Y$ with $W_ X$, and theorem 23.2.60 gives

For a bimeromorphic map between $X$ and $Y$, resolve its graph to obtain modifications $p:Z\to X$ and $q:Z\to Y$ from a compact complex manifold $Z$. The preceding argument applies to both maps and identifies the cones and kernels via $(q^*)^{-1}p^*$ and its Poincaré dual. These identifications do not depend on the resolution: they are compatible with restriction to a common open set on which the original map is an isomorphism, and by the injectivity of degree-one restriction proved above, this determines each class. The dual identifications are therefore canonical as well. $\square$