PomeTheorem 23.2.64.403FA017D33B Let $X$ and $Y$ be bimeromorphic connected compact complex manifolds. Under the natural identification of their first de Rham cohomology groups, and the Poincaré-dual identification of their de Rham cohomology groups of degree $2n-1$, the spaces $K_ X,W_ X$ and the cones $\mathcal C_ X,\operatorname {LG}(X)$ correspond to those of $Y$.
Theorem 23.2.64403FA017D33B
Proof. First let $\pi :Y\to X$ be a modification. Holomorphic pullback commutes with $\mathrm d^{\mathrm c}$, so
\[ j_ Y(\pi ^*\xi )=\pi ^*j_ X(\xi ). \]
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
\[ \pi _*\operatorname {LG}(Y)=\operatorname {LG}(X). \]
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. $\square$
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