Systoles and Lagrangians of random complex algebraic hypersurfaces

Abstract

Let $n\ge 1$ be an integer, and $\mathcal{L}\subset {\mathbb{R}}^n$ be a compact smooth affine real hypersurface, not necessarily connected. We prove that there exist $c>0$ and $d_0\ge 1$ such that for any $d\ge d_0$, any smooth complex projective hypersurface $Z$ in $\mathbb{C}{P}^n$ of degree $d$ contains at least $c\dim H_{\ast }(Z,\mathbb{R})$ disjoint Lagrangian submanifolds diffeomorphic to $\mathcal{L}$, where $Z$ is equipped with the restriction of the Fubini–Study symplectic form (Theorem 1.1). If moreover all connected components of $\mathcal{L}$ have non-vanishing Euler characteristic, which implies that $n$ is odd, the latter Lagrangian submanifolds form an independent family in $H_{n-1}(Z,\mathbb{R})$ (Corollary 1.2). These deterministic results are consequences of a more precise probabilistic theorem (Theorem 1.23) inspired by a 2014 result by J.-Y. Welschinger and the author on random real algebraic geometry, together with quantitative Moser-type constructions (Theorem 3.4). For $n=2$, the method provides a uniform positive lower bound for the probability that a projective complex curve in $\mathbb{C}{P}^2$ of given degree equipped with the restriction of the ambient metric has a systole of small size (Theorem 1.6), which is an analog of a similar bound for hyperbolic curves given by M. Mirzakhani (2013). In higher dimensions, we provide a similar result for the $(n-1)$-systole introduced by M. Berger (1972) (Corollary 1.14). Our results hold in the more general setting of vanishing loci of holomorphic sections of vector bundles of rank between 1 and $n$ tensored by a large power of an ample line bundle over a projective complex $n$-manifold (Theorem 1.20).