Systoles and Lagrangians of random complex algebraic hypersurfaces

Let $n\geq 1$ be an integer, $\mathcal L \subset \mathbb{R}^n$ be a compact smooth affine real hypersurface, not necessarily connected. We prove that there exists $c>0$ and $d_0\geq 1$, such that for any $d\geq d_0$, any smooth complex projective hypersurface $Z$ in $\mathbb{C} P^n$ of degree $d$ contains at least $c\dim H_*(Z, \mathbb{R})$ disjoint Lagrangian submanifolds diffeomorphic to $\mathcal L$, where $Z$ is equipped with the restriction of the Fubini-Study symplectic form. If moreover the connected components of $\mathcal L$ have non vanishing Euler characteristic, which implies that $n$ is odd, the latter Lagrangian submanifolds form an independent family of $H_{n-1}(Z, \mathbb{R})$. We use a probabilistic argument for the proof inspired by a result by J.-Y. Welschinger and the author on random real algebraic geometry, together with quantitative Moser-type constructions. 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, which is an analog to a similar bound for hyperbolic curves given by M. Mirzakhani. Our results hold in the more general setting of vanishing loci of holomorphic sections of vector bundles of rank between 1 and $n$ tensoredby a large power of an ample line bundle over a projective complex $n$-manifold.