2019/09/19 by Damien Gayet, Gayet, Damien
Mathematics · #Algebraic Geometry and Number Theory #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.AG #math.PR #math.SG
paper · pdf · doi:10.48550/arxiv.1909.09023
arxiv created 2019/09/19 · arxiv updated 2019/09/20
Let n≥ 1 be an integer, \mathcal L ⊂ ℝn be a compact smooth affine real hypersurface, not necessarily connected. We prove that there exists c>0 and d0≥ 1, such that for any d≥ d0, any smooth complex projective hypersurface Z in ℂ Pn of degree d contains at least cdim H_*(Z, 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 Hn-1(Z, 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 ℂ P2 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.