标题： 具有渐近大贝蒂数的实代数超曲面的补丁工作 显示英文标题

标题： Patchworking real algebraic hypersurfaces with asymptotically large Betti numbers

摘要： 在本文中，我们描述了一种递归方法，用于从环境维数为$k=1,\ldots,n-1$的实射影代数超曲面族构造环境维数为$n$的实射影代数超曲面族。 然后可以将所得族的实部分的渐近贝蒂数用作为原料的族的实部分的渐近贝蒂数来描述。 该算法基于维罗的拼接法，并受到伊滕伯格和奥·维罗在任意维数中构造渐近极大族的构造的启发。 使用它，我们证明了对于任何$n$和$i=0,\ldots,n-1$，存在一族渐近最大实射影代数超曲面$\{Y^n_d\}_d$在$\mathbb{R} \mathbb{P} ^n$中（其中$d$表示$Y^n_d$的次数），使得第$i$个贝蒂数$b_i(\mathbb{R} Y^n_d)$渐近严格大于第$(i,n-1-i)$个霍奇数$h^{i,n-1-i}(\mathbb{C} Y^n_d)$。 我们还构建了实射影代数超曲面族，其实部分在次数$d$的渐近情况下，贝蒂数在环境维数$n$的渐近情况下非常大。 显示英文摘要

摘要： In this article, we describe a recursive method for constructing a family of real projective algebraic hypersurfaces in ambient dimension $n$ from families of such hypersurfaces in ambient dimensions $k=1,\ldots,n-1$. The asymptotic Betti numbers of real parts of the resulting family can then be described in terms of the asymptotic Betti numbers of the real parts of the families used as ingredients. The algorithm is based on Viro's Patchwork and inspired by I. Itenberg's and O. Viro's construction of asymptotically maximal families in arbitrary dimension. Using it, we prove that for any $n$ and $i=0,\ldots,n-1$, there is a family of asymptotically maximal real projective algebraic hypersurfaces $\{Y^n_d\}_d$ in $\mathbb{R} \mathbb{P} ^n$ (where $d$ denotes the degree of $Y^n_d$) such that the $i$-th Betti numbers $b_i(\mathbb{R} Y^n_d)$ are asymptotically strictly greater than the $(i,n-1-i)$-th Hodge numbers $h^{i,n-1-i}(\mathbb{C} Y^n_d)$. We also build families of real projective algebraic hypersurfaces whose real parts have asymptotic (in the degree $d$) Betti numbers that are asymptotically (in the ambient dimension $n$) very large.