Real versus Complex Volumes on Real Algebraic Surfaces
Real versus Complex Volumes on Real Algebraic Surfaces
复制标题
实代数曲面上的实体积与复体积
DOI：
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发表时间：
2011
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通讯作者：
Arnaud Moncet
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作者：
Arnaud Moncet
设X是一个实代数曲面。通过比较充裕因子D的实轨迹和复轨迹的体积，我们定义了一致性，它是一个介于0和1之间的数字。当Picard数为1时，这个数等于1，并且对于一些具有“相当简单”内锥的曲面，例如Del Pezzo曲面。对于阿贝尔曲面，它是1/2或1，取决于x上是否存在正熵自同构。一般情况下，正熵自同构的存在给出了一致性的上界，即x的实轨迹与复轨迹的熵之比，并且当皮卡德数为2时，一致性等于这个比值。这个不等式的一个有趣的结果是实曲面X(R)的微分同态群中自同态的非密度，只要一致性是正的。最后，我们证明，由于这个上界，存在任意小一致性的K3曲面，考虑到三次奇异曲面（2,2,2）在三条投影线积中的变形。
Let X be a real algebraic surface. The comparison between the volume of real and complex loci of ample divisors D brings us to define the concordance, which is a number between 0 and 1. This number equals 1 when the Picard number is 1, and for some surfaces with a "quite simple" nef cone, e.g. Del Pezzo surfaces. For abelian surfaces, it is 1/2 or 1, depending on the existence or not of positive entropy automorphisms on X. In the general case, the existence of such an automorphism gives an upper bound for the concordance, namely the ratio of entropies in real and complex loci of X. Moreover the concordance is equal to this ratio when the Picard number is 2. An interesting consequence of the inequality is the non-density of the automorphisms in the group of diffeomorphisms of the real surface X(R), as soon as the concordance is positive. Finally we show, thanks to this upper bound, that there exist K3 surfaces with arbitrary small concordance, considering a deformation of a singular surface of tridegree (2,2,2) in the product of three projective lines.