An application of $p$-adic integration to the dynamics of a birational transformation preserving a fibration
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2017
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| _version_ | 1866916497529503744 |
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| author | Bianco, Federico Lo |
| author_facet | Bianco, Federico Lo |
| contents | Let $f\colon X \dashrightarrow X$ be a birational transformation of a projective manifold $X$ whose Kodaira dimension $κ(X)$ is non-negative. We show that, if there exist a meromorphic fibration $π\colon X\dashrightarrow B$ and a pseudo-automorphism $f_B\colon B\dashrightarrow B$ which preserves a big line bundle $L\in Pic(B)$ and such that $f_B\circ π=π\circ f$, then $f_B$ has finite order.
As a corollary we show that, for projective irreducible symplectic manifolds of type $K3^{[n]}$ or generalized Kummer, the first dynamical degree characterizes the birational transformations admitting a Zariski-dense orbit. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1707_09534 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | An application of $p$-adic integration to the dynamics of a birational transformation preserving a fibration Bianco, Federico Lo Algebraic Geometry Let $f\colon X \dashrightarrow X$ be a birational transformation of a projective manifold $X$ whose Kodaira dimension $κ(X)$ is non-negative. We show that, if there exist a meromorphic fibration $π\colon X\dashrightarrow B$ and a pseudo-automorphism $f_B\colon B\dashrightarrow B$ which preserves a big line bundle $L\in Pic(B)$ and such that $f_B\circ π=π\circ f$, then $f_B$ has finite order. As a corollary we show that, for projective irreducible symplectic manifolds of type $K3^{[n]}$ or generalized Kummer, the first dynamical degree characterizes the birational transformations admitting a Zariski-dense orbit. |
| title | An application of $p$-adic integration to the dynamics of a birational transformation preserving a fibration |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/1707.09534 |