Systoles and Lagrangians of random complex algebraic hypersurfaces - Gayet, Damien (Journal of the European Mathematical Society : JEMS, 2023)
Let n≥ 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≥ 1 such that for any d≥ 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 (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).