论文标题
最大真实代数超曲面的指数罕见
Exponential rarefaction of maximal real algebraic hypersurfaces
论文作者
论文摘要
给定一个真正的Hermitian Holomorthic Line Bundle $ l $上的真正的hegebraic品种$ x $，这是$ l^{\ otimes d} $的真实全态部分的空间，继承了天然高斯的概率措施。我们证明，$ l^{\ otimes d} $的真实全体形态$ s $ s $ s $ s $的零基因座定义了最大高度表面的可能性趋于$ 0 $，因为$ d $ $ d $转到infinity。这延伸至任何维度的gayet和Welschinger的结果，对最大真实代数曲线有效。 起点是一个低度近似属性，它与$ l^{\ otimes d} $的真实全体形态部分的真实消失基因座的拓扑与真正消失的基因座的拓扑结合了$ l^{\ otimes dimimes d'} $的真实塑料部分的拓扑。这种声明的灵感来自Diatta和Lerario的最新作品。
Given an ample real Hermitian holomorphic line bundle $L$ over a real algebraic variety $X$, the space of real holomorphic sections of $L^{\otimes d}$ inherits a natural Gaussian probability measure. We prove that the probability that the zero locus of a real holomorphic section $s$ of $L^{\otimes d}$ defines a maximal hypersurface tends to $0$ exponentially fast as $d$ goes to infinity. This extends to any dimension a result of Gayet and Welschinger valid for maximal real algebraic curves inside a real algebraic surface. The starting point is a low degree approximation property which relates the topology of the real vanishing locus of a real holomorphic section of $L^{\otimes d}$ with the topology of the real vanishing locus a real holomorphic section of $L^{\otimes d'}$ for a sufficiently smaller $d'<d$. Such a statement is inspired by a recent work of Diatta and Lerario.
