Corollary 13.5.10`7FDDF6925B80`

Corollary 13.5.10.7FDDF6925B80 Let $T,S$ be closed positive $(1,1)$-currents on a compact Kähler manifold $X$ of dimension $n\geq 1$. Let $\nu $ be a positive Radon measure on $X$, and let $f,g$ be nonnegative Borel functions on $X$ such that

\[ \langle T^ n\rangle \geq f\nu ,\qquad \langle S^ n\rangle \geq g\nu . \]

Then for every integer $1\leq k\leq n-1$,

\[ \langle T^ k\wedge S^{n-k}\rangle \geq f^{k/n}g^{(n-k)/n}\nu . \]

The endpoint cases $k=0$ and $k=n$ are the hypotheses; for $n=1$ there are no other cases.

Proof. Apply proposition 13.5.8 with $k$ factors equal to $T$, with density $f$, and $n-k$ factors equal to $S$, with density $g$. $\square$

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BibTeX citation

@misc{pome-tag-7FDDF6925B80,
  author = {{Pome contributors}},
  title = {Corollary 13.5.10},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/7FDDF6925B80/}
}

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Corollary 13.5.107FDDF6925B80

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Corollary 13.5.10`7FDDF6925B80`