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Item type:Publication, On singular real analytic Levi-flat foliations(International Press, Inc., 2020-12-01)A singular real analytic foliation $\mathcal{F}$ of real codimension one on an $n$-dimensional complex manifold $M$ is Levi-flat if each of its leaves is foliated by immersed complex manifolds of dimension $n-1$. These complex manifolds are leaves of a singular real analytic foliation $\mathcal{L}$ which is tangent to $\mathcal{F}$. In this article, we classify germs of Levi-flat foliations at $(\mathbb{C}^{n},0)$ under the hypothesis that $\mathcal{L}$ is a germ holomorphic foliation. Essentially, we prove that there are two possibilities for $\mathcal{L}$, from which the classification of $\mathcal{F}$ derives: either it has a meromorphic first integral or is defined by a closed rational $1-$form. Our local results also allow us to classify real algebraic Levi-flat foliations on the complex projective space $\mathbb{P}^{n} = \mathbb{P}^{n}_{\mathbb{C}}$. - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Chow's theorem for real analytic Levi-flat hypersurfaces(Elsevier Masson s.r.l., 2022-10-01)In this article we provide a version of Chow's theorem for real analytic Levi-flat hypersurfaces in the complex projective space Pn, n≥2. More specifically, we prove that a real analytic Levi-flat hypersurface M⊂Pn, with singular set of real dimension at most 2n−4 and whose Levi leaves are contained in algebraic hypersurfaces, is tangent to the levels of a rational function in Pn. As a consequence, M is a semialgebraic set. We also prove that a Levi foliation on Pn — a singular real analytic foliation whose leaves are immersed complex manifolds of codimension one — satisfying similar conditions — singular set of real dimension at most 2n−4 and all leaves algebraic — is defined by the level sets of a rational function.