Dynamics of bimeromorphic maps of surfaces
Dynamics of bimeromorphic maps of surfaces
复制标题
表面双同构图的动力学
DOI：
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发表时间：
2001
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通讯作者：
C. Favre
中科院分区：
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作者：
J. Diller;C. Favre
本文根据紧致Kahler曲面X的双半纯自映射f *：H1，1（X）[inline-graphic xmlns：xlink =“http：//www.w3.org/1999/xlink”xlink：href =“02 i”/]在上同调上的作用，对它们进行了分类. http://www.w3.org/1999/xlink我们观察到，在双纯共轭下，f n * n的增长率是不变的，并且通过共轭总是可以安排f n * = f * n。我们证明了序列n * n * n可以是有界的，线性增长，二次增长，或指数增长。在前三种情况下，我们表明，共轭后，f是一个自同构几乎同位素的身份，f保持合理的纤维化，或f保持椭圆纤维化，分别。在最后一种情况下，我们证明了f * 存在唯一的（按比例）扩张特征向量θ+，θ+是nef，并且f与一个自同构双亚纯共轭当且仅当θ 2 + = 0。在这种情况下，我们继续构造一个代表θ+的动态自然正电流，并研究f的周期轨道的增长率。最后，我们用一个特定的例子来说明我们的结果。
We classify bimeromorphic self-maps f : X [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] of compact Kahler surfaces X in terms of their actions f *: H 1,1 ( X ) [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i "/] on cohomology. We observe that the growth rate of ║ f n *║ is invariant under bimeromorphic conjugacy, and that by conjugating one can always arrange that f n * = f * n . We show that the sequence ║ f n *║ can be bounded, grow linearly, grow quadratically, or grow exponentially. In the first three cases, we show that after conjugating, f is an automorphism virtually isotopic to the identity, f preserves a rational fibration, or f preserves an elliptic fibration, respectively. In the last case, we show that there is a unique (up to scaling) expanding eigenvector θ+ for f *, that θ+ is nef, and that f is bimeromorphically conjugate to an automorphism if and only if θ 2 + = 0. We go on in this case to construct a dynamically natural positive current representing θ+, and we study the growth rate of periodic orbits of f . We conclude by illustrating our results with a particular family of examples.