— Let α ∈ (0, 2) and d ∈ N. Consider the following stochastic differential equation (SDE) in Rd: dXt = b(t,Xt) dt+ a(t,Xt−) dL (α) t , X0 = x, where L(α) is a d-dimensional rotationally invariant α-stable process, b : R+ × Rd → Rd and a : R+ × Rd → Rd ⊗ Rd are Hölder continuous functions in space, with respective order β, γ ∈ (0, 1) such that (β ∧ γ) + α > 1, uniformly in t. Here b may be unbounded. When a is bounded and uniformly elliptic, we show that the unique solution Xt(x) of the above SDE admits a continuous density, which enjoys sharp two-sided estimates. We also establish sharp upper-bound for the logarithmic derivative. In particular, we cover the whole supercritical range α ∈ (0, 1). Our proof is based on ad hoc parametrix expansions and probabilistic techniques. Résumé (Noyau de la chaleur pour des EDS surcritiques à dérive non bornée) Soit α ∈ (0, 2) et d ∈ N. Considérons l’équation différentielle stochastique (EDS) suivante dans Rd : dXt = b(t,Xt) dt+ a(t,Xt−) dL (α) t , X0 = x, où L(α) est un processus α-stable isotrope de dimension d, b : R+×R → Rd et a : R+×R → Rd⊗Rd sont des fonctions Hölder continues en espace, d’indices respectifs β, γ ∈ (0, 1) tels que (β∧γ)+α > 1, uniformément en t. En particulier b peut être non bornée. Lorsque a est bornée et uniformément elliptique, nous montrons que la solution Xt(x) de l’EDS admet une densité continue, que l’on peut encadrer, à constante multiplicative près, par une même quantité. Nous obtenons également une borne supérieure précise pour la dérivée logarithmique de la densité. En particulier, nous traitons complètement le régime surcritique α ∈ (0, 1). Notre approche se base sur des développements parametrix ad hoc et des techniques probabilistes.
——让α∈(0,2)和d∈n .考虑下面的随机微分方程(SDE)理查德·道金斯:dXt = b (t, Xt) dt + a (t, Xt−)dL(α)t, X0 = x,在L(α)是一种采用旋转不变的α稳定过程,b: R +×Rd→Rd和:R +×Rd→Rd⊗Rd持有人连续函数在空间,与各自的订单β,γ∈(0,1)这样(β∧γ)+α> 1,统一在t。b可能是无限的。当a是有界的一致椭圆时，我们证明了上述SDE的唯一解Xt(x)具有连续密度，具有明显的双面估计。我们还建立了对数导数的明显上界。特别地，我们涵盖了整个超临界范围α∈(0,1)。我们的证明是基于特殊参数展开和概率技术。简历(果仁酒de la chaleur倒des EDS surcritiques推导非bornee)所以α∈(0,2)et d∈n .鉴于等式differentielle stochastique (EDS)下在理查德·道金斯:dXt = b (t, Xt) dt + a (t, Xt−)dL(α)t, X0 = x,或者l(α)是联合国突起α稳定均质德维d, b: R +×R→Rd et: R +×R→Rd⊗Rd是des函数持有人继续埃斯佩斯,d 'indices respectifsβ,γ∈(0,1)运输,(β∧γ)+α> 1,uniformement en t。en particulier b可能非bornee。(1)从广义上讲，我们用广义上讲，我们用广义上讲，我们用广义上讲，我们用广义上讲，我们用广义上讲，我们用广义上讲，我们用广义上讲，我们用广义上讲，我们用广义上讲，我们用广义上讲。现在，所有的人都被认为是超限的，所以他们都被认为是超限的。特别地，nous tratrons complitement le smodime surcritique α∈(0,1)。Notre -方法使用base sur - smodime parametric and hoc et of techniques probability。
{"title":"Heat kernel of supercritical nonlocal operators with unbounded drifts","authors":"S. Menozzi, Xicheng Zhang","doi":"10.5802/jep.189","DOIUrl":"https://doi.org/10.5802/jep.189","url":null,"abstract":"— Let α ∈ (0, 2) and d ∈ N. Consider the following stochastic differential equation (SDE) in Rd: dXt = b(t,Xt) dt+ a(t,Xt−) dL (α) t , X0 = x, where L(α) is a d-dimensional rotationally invariant α-stable process, b : R+ × Rd → Rd and a : R+ × Rd → Rd ⊗ Rd are Hölder continuous functions in space, with respective order β, γ ∈ (0, 1) such that (β ∧ γ) + α > 1, uniformly in t. Here b may be unbounded. When a is bounded and uniformly elliptic, we show that the unique solution Xt(x) of the above SDE admits a continuous density, which enjoys sharp two-sided estimates. We also establish sharp upper-bound for the logarithmic derivative. In particular, we cover the whole supercritical range α ∈ (0, 1). Our proof is based on ad hoc parametrix expansions and probabilistic techniques. Résumé (Noyau de la chaleur pour des EDS surcritiques à dérive non bornée) Soit α ∈ (0, 2) et d ∈ N. Considérons l’équation différentielle stochastique (EDS) suivante dans Rd : dXt = b(t,Xt) dt+ a(t,Xt−) dL (α) t , X0 = x, où L(α) est un processus α-stable isotrope de dimension d, b : R+×R → Rd et a : R+×R → Rd⊗Rd sont des fonctions Hölder continues en espace, d’indices respectifs β, γ ∈ (0, 1) tels que (β∧γ)+α > 1, uniformément en t. En particulier b peut être non bornée. Lorsque a est bornée et uniformément elliptique, nous montrons que la solution Xt(x) de l’EDS admet une densité continue, que l’on peut encadrer, à constante multiplicative près, par une même quantité. Nous obtenons également une borne supérieure précise pour la dérivée logarithmique de la densité. En particulier, nous traitons complètement le régime surcritique α ∈ (0, 1). Notre approche se base sur des développements parametrix ad hoc et des techniques probabilistes.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130672617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study intersection cohomology of character varieties for punctured Riemann surfaces with prescribed monodromies around the punctures. Relying on previous result from Mellit [Mel17b] for semisimple monodromies we compute the intersection cohomology of character varieties with monodromies of any Jordan type. This proves the Poincaré polynomial specialization of a conjecture from Letellier [Let13].
{"title":"Intersection cohomology of character varieties for punctured Riemann surfaces","authors":"Mathieu Ballandras","doi":"10.5802/jep.215","DOIUrl":"https://doi.org/10.5802/jep.215","url":null,"abstract":"We study intersection cohomology of character varieties for punctured Riemann surfaces with prescribed monodromies around the punctures. Relying on previous result from Mellit [Mel17b] for semisimple monodromies we compute the intersection cohomology of character varieties with monodromies of any Jordan type. This proves the Poincaré polynomial specialization of a conjecture from Letellier [Let13].","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"98 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126327884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $(X,d)$ be a geodesic Gromov-hyperbolic space, $o in X$ a basepoint and $mu$ a countably supported non-elementary probability measure on $operatorname{Isom}(X)$. Denote by $z_n$ the random walk on $X$ driven by the probability measure $mu$. Supposing that $mu$ has finite exponential moment, we give a second-order Taylor expansion of the large deviation rate function of the sequence $frac{1}{n}d(z_n,o)$ and show that the corresponding coefficient is expressed by the variance in the central limit theorem satisfied by the sequence $d(z_n,o)$. This provides a positive answer to a question raised in cite{BMSS}. The proof relies on the study of the Laplace transform of $d(z_n,o)$ at the origin using a martingale decomposition first introduced by Benoist--Quint together with an exponential submartingale transform and large deviation estimates for the quadratic variation process of certain martingales.
设$(X,d)$为测地线格罗莫夫-双曲空间，$o in X$为基点，$mu$为$operatorname{Isom}(X)$上的可数支持非初等概率测度。用$z_n$表示由概率测度$mu$驱动的$X$上的随机游走。假设$mu$具有有限的指数矩，我们给出了序列$frac{1}{n}d(z_n,o)$的大偏差率函数的二阶泰勒展开式，并证明了相应的系数由序列$d(z_n,o)$所满足的中心极限定理中的方差表示。这为cite{BMSS}中提出的问题提供了一个肯定的答案。该证明依赖于对原点处$d(z_n,o)$的拉普拉斯变换的研究，该变换使用了首先由Benoist—Quint引入的鞅分解，以及对某些鞅的二次变分过程的指数次鞅变换和大偏差估计。
{"title":"Random walks on hyperbolic spaces: second order expansion of the rate function at the drift","authors":"Richard Aoun, P. Mathieu, Cagri Sert","doi":"10.5802/jep.225","DOIUrl":"https://doi.org/10.5802/jep.225","url":null,"abstract":"Let $(X,d)$ be a geodesic Gromov-hyperbolic space, $o in X$ a basepoint and $mu$ a countably supported non-elementary probability measure on $operatorname{Isom}(X)$. Denote by $z_n$ the random walk on $X$ driven by the probability measure $mu$. Supposing that $mu$ has finite exponential moment, we give a second-order Taylor expansion of the large deviation rate function of the sequence $frac{1}{n}d(z_n,o)$ and show that the corresponding coefficient is expressed by the variance in the central limit theorem satisfied by the sequence $d(z_n,o)$. This provides a positive answer to a question raised in cite{BMSS}. The proof relies on the study of the Laplace transform of $d(z_n,o)$ at the origin using a martingale decomposition first introduced by Benoist--Quint together with an exponential submartingale transform and large deviation estimates for the quadratic variation process of certain martingales.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"106 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115812444","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we complete the program of relating the Laplace spectrum for rank one compact locally symmetric spaces with the first band Ruelle-Pollicott resonances of the geodesic flow on its sphere bundle. This program was started by Flaminio and Forni for hyperbolic surfaces, continued by Dyatlov, Faure and Guillarmou for real hyperbolic spaces and by Guillarmou, Hilgert and Weich for general rank one spaces. Except for the case of hyperbolic surfaces a countable set of exceptional spectral parameters always left untreated since the corresponding Poisson transforms are neither injective nor surjective. We use vector valued Poisson transforms to treat also the exceptional spectral parameters. For surfaces the exceptional spectral parameters lead to discrete series representations of $mathrm{SL}(2,mathbb R)$. In higher dimensions the situation is more complicated, but can be described completely.
{"title":"Spectral correspondences for rank one locally symmetric spaces: the case of exceptional parameters","authors":"Christian Arends, J. Hilgert","doi":"10.5802/jep.220","DOIUrl":"https://doi.org/10.5802/jep.220","url":null,"abstract":"In this paper we complete the program of relating the Laplace spectrum for rank one compact locally symmetric spaces with the first band Ruelle-Pollicott resonances of the geodesic flow on its sphere bundle. This program was started by Flaminio and Forni for hyperbolic surfaces, continued by Dyatlov, Faure and Guillarmou for real hyperbolic spaces and by Guillarmou, Hilgert and Weich for general rank one spaces. Except for the case of hyperbolic surfaces a countable set of exceptional spectral parameters always left untreated since the corresponding Poisson transforms are neither injective nor surjective. We use vector valued Poisson transforms to treat also the exceptional spectral parameters. For surfaces the exceptional spectral parameters lead to discrete series representations of $mathrm{SL}(2,mathbb R)$. In higher dimensions the situation is more complicated, but can be described completely.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128384150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate the structure of large uniform random maps with $n$ edges, $mathrm{f}_n$ faces, and with genus $mathrm{g}_n$ in the so-called sparse case, where the ratio between the number vertices and edges tends to $1$. We focus on two regimes: the planar case $(mathrm{f}_n, 2mathrm{g}_n) = (mathrm{s}_n, 0)$ and the unicellular case with moderate genus $(mathrm{f}_n, 2 mathrm{g}_n) = (1, mathrm{s}_n-1)$, both when $1 ll mathrm{s}_n ll n$. Albeit different at first sight, these two models can be treated in a unified way using a probabilistic version of the classical core-kernel decomposition. In particular, we show that the number of edges of the core of such maps, obtained by iteratively removing degree $1$ vertices, is concentrated around $sqrt{n mathrm{s}_{n}}$. Further, their kernel, obtained by contracting the vertices of the core with degree $2$, is such that the sum of the degree of its vertices exceeds that of a trivalent map by a term of order $sqrt{mathrm{s}_{n}^{3}/n}$; in particular they are trivalent with high probability when $mathrm{s}_{n} ll n^{1/3}$. This enables us to identify a mesoscopic scale $sqrt{n/mathrm{s}_n}$ at which the scaling limits of these random maps can be seen as the local limit of their kernels, which is the dual of the UIPT in the planar case and the infinite three-regular tree in the unicellular case, where each edge is replaced by an independent (biased) Brownian tree with two marked points.
{"title":"The mesoscopic geometry of sparse random maps","authors":"N. Curien, I. Kortchemski, Cyril Marzouk","doi":"10.5802/jep.207","DOIUrl":"https://doi.org/10.5802/jep.207","url":null,"abstract":"We investigate the structure of large uniform random maps with $n$ edges, $mathrm{f}_n$ faces, and with genus $mathrm{g}_n$ in the so-called sparse case, where the ratio between the number vertices and edges tends to $1$. We focus on two regimes: the planar case $(mathrm{f}_n, 2mathrm{g}_n) = (mathrm{s}_n, 0)$ and the unicellular case with moderate genus $(mathrm{f}_n, 2 mathrm{g}_n) = (1, mathrm{s}_n-1)$, both when $1 ll mathrm{s}_n ll n$. Albeit different at first sight, these two models can be treated in a unified way using a probabilistic version of the classical core-kernel decomposition. In particular, we show that the number of edges of the core of such maps, obtained by iteratively removing degree $1$ vertices, is concentrated around $sqrt{n mathrm{s}_{n}}$. Further, their kernel, obtained by contracting the vertices of the core with degree $2$, is such that the sum of the degree of its vertices exceeds that of a trivalent map by a term of order $sqrt{mathrm{s}_{n}^{3}/n}$; in particular they are trivalent with high probability when $mathrm{s}_{n} ll n^{1/3}$. This enables us to identify a mesoscopic scale $sqrt{n/mathrm{s}_n}$ at which the scaling limits of these random maps can be seen as the local limit of their kernels, which is the dual of the UIPT in the planar case and the infinite three-regular tree in the unicellular case, where each edge is replaced by an independent (biased) Brownian tree with two marked points.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129269090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper has two objectives. First, we study lattices with skew-Hermitian forms over division algebras with positive involutions. For division algebras of Albert types I and II, we show that such a lattice contains an"orthogonal"basis for a sublattice of effectively bounded index. Second, we apply this result to obtain new results in the field of unlikely intersections. More specifically, we prove the Zilber-Pink conjecture for the intersection of curves with special subvarieties of simple PEL type I and II under a large Galois orbits conjecture. We also prove this Galois orbits conjecture for certain cases of type II.
{"title":"Lattices with skew-Hermitian forms over division algebras and unlikely intersections","authors":"Christopher Daw, M. Orr","doi":"10.5802/jep.240","DOIUrl":"https://doi.org/10.5802/jep.240","url":null,"abstract":"This paper has two objectives. First, we study lattices with skew-Hermitian forms over division algebras with positive involutions. For division algebras of Albert types I and II, we show that such a lattice contains an\"orthogonal\"basis for a sublattice of effectively bounded index. Second, we apply this result to obtain new results in the field of unlikely intersections. More specifically, we prove the Zilber-Pink conjecture for the intersection of curves with special subvarieties of simple PEL type I and II under a large Galois orbits conjecture. We also prove this Galois orbits conjecture for certain cases of type II.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117192256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove stochastic homogenization for integral functionals defined on Sobolev spaces, where the stationary, ergodic integrand satisfies a degenerate growth condition of the form c|ξA(ω, x)| ≤ f(ω, x, ξ) ≤ |ξA(ω, x)| +Λ(ω, x) for some p ∈ (1,+∞) and with a stationary and ergodic diagonal matrix A such that its norm and the norm of its inverse satisfy minimal integrability assumptions. We also consider the convergence when Dirichlet boundary conditions or an obstacle condition are imposed. Assuming the strict convexity and differentiability of f with respect to its last variable, we further prove that the homogenized integrand is also strictly convex and differentiable. These properties allow us to show homogenization of the associated Euler-Lagrange equations.
我们证明了Sobolev空间上定义的积分泛函的随机齐次化，其中平稳遍历被积函数满足退化生长条件c|ξ a (ω， x)|≤f(ω， x， ξ)≤|ξ a (ω， x)| +Λ(ω， x)，对于某p∈(1，+∞)，并且具有平稳遍历对角矩阵a，使得其范数及其逆范数满足最小可积假设。我们还考虑了狄利克雷边界条件和障碍条件下的收敛性。假设f对其最后一个变量具有严格的凸性和可微性，进一步证明了齐次被积函数也是严格凸可微的。这些性质使我们能够证明相关欧拉-拉格朗日方程的均匀性。
{"title":"Stochastic homogenization of degenerate integral functionals and their Euler-Lagrange equations","authors":"M. Ruf, T. Ruf","doi":"10.5802/jep.218","DOIUrl":"https://doi.org/10.5802/jep.218","url":null,"abstract":"We prove stochastic homogenization for integral functionals defined on Sobolev spaces, where the stationary, ergodic integrand satisfies a degenerate growth condition of the form c|ξA(ω, x)| ≤ f(ω, x, ξ) ≤ |ξA(ω, x)| +Λ(ω, x) for some p ∈ (1,+∞) and with a stationary and ergodic diagonal matrix A such that its norm and the norm of its inverse satisfy minimal integrability assumptions. We also consider the convergence when Dirichlet boundary conditions or an obstacle condition are imposed. Assuming the strict convexity and differentiability of f with respect to its last variable, we further prove that the homogenized integrand is also strictly convex and differentiable. These properties allow us to show homogenization of the associated Euler-Lagrange equations.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"78 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116125893","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate the construction of real analytic Levi-flat hypersurfaces in K3 surfaces. By taking images of real hyperplanes, one can construct such hypersurfaces in two-dimensional complex tori. We show that"almost every"K3 surfaces contains infinitely many Levi-flat hypersurfaces of this type. The proof relies mainly on a recent construction of Koike and Uehara, ideas of Verbitsky on ergodic complex structures, as well as an argument due to Ghys in the context of the study of the topology of generic leaves. -- On s'int'eresse `a la construction d'hypersurfaces Levi-plates analytiques r'elles dans les surfaces K3. On peut en construire dans les tores complexes de dimension 2 en prenant des images d'hyperplans r'eels. On montre que"presque toute"surface K3 contient une infinit'e d'hypersurfaces Levi-plates de ce type. La preuve repose principalement sur une construction r'ecente due `a Koike-Uehara, ainsi que sur les id'ees de Verbitsky sur les structures complexes ergodiques et une adaptation d'un argument d^u `a Ghys dans le cadre de l''etude de la topologie des feuilles g'en'eriques.
研究了K3曲面上实解析列维平面超曲面的构造。通过获取实超平面的图像，可以在二维复环面中构造这样的超曲面。我们证明了“几乎每个”K3曲面包含无限多个这种类型的列维平面超曲面。这个证明主要依赖于Koike和Uehara最近的一个构造，Verbitsky关于遍历复杂结构的想法，以及Ghys在研究泛叶拓扑结构的背景下的一个论点。——关于s'int 'eresse ' a la construction d'超曲面列维板分析或'elles dans les曲面K3。在此基础上，我们构建了一种基于二维平面的复杂结构，并提出了“超平面”和“平面”的图像。在montre que“preque toute”曲面上，K3大陆上无限的超曲面列维板。拉可静止principalement关于建设r ' ecente由于 ' Koike-Uehara,依照ainsi, les id “ee de Verbitsky苏尔les结构复合物ergodiques等一个适应一个参数d ^你 '一个Ghys在干部de l ' '练习曲de La topologie des树叶味g eriques“en ”。
{"title":"Presque toute surface K3 contient une infinité d’hypersurfaces Levi-plates linéaires","authors":"F'elix Lequen","doi":"10.5802/jep.233","DOIUrl":"https://doi.org/10.5802/jep.233","url":null,"abstract":"We investigate the construction of real analytic Levi-flat hypersurfaces in K3 surfaces. By taking images of real hyperplanes, one can construct such hypersurfaces in two-dimensional complex tori. We show that\"almost every\"K3 surfaces contains infinitely many Levi-flat hypersurfaces of this type. The proof relies mainly on a recent construction of Koike and Uehara, ideas of Verbitsky on ergodic complex structures, as well as an argument due to Ghys in the context of the study of the topology of generic leaves. -- On s'int'eresse `a la construction d'hypersurfaces Levi-plates analytiques r'elles dans les surfaces K3. On peut en construire dans les tores complexes de dimension 2 en prenant des images d'hyperplans r'eels. On montre que\"presque toute\"surface K3 contient une infinit'e d'hypersurfaces Levi-plates de ce type. La preuve repose principalement sur une construction r'ecente due `a Koike-Uehara, ainsi que sur les id'ees de Verbitsky sur les structures complexes ergodiques et une adaptation d'un argument d^u `a Ghys dans le cadre de l''etude de la topologie des feuilles g'en'eriques.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"66 4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130921763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a new geometrical invariant of CR manifolds of hypersurface type, which we dub the"Levi core"of the manifold. When the manifold is the boundary of a smooth bounded pseudoconvex domain, we show how the Levi core is related to two other important global invariants in several complex variables: the Diederich--Forn{ae}ss index and the D'Angelo class (namely the set of D'Angelo forms of the boundary). We also show that the Levi core is trivial whenever the domain is of finite-type in the sense of D'Angelo, or the set of weakly pseudoconvex points is contained in a totally real submanifold, while it is nontrivial if the boundary contains a local maximum set. As corollaries to the theory developed here, we prove that for any smooth bounded pseudoconvex domain with trivial Levi core the Diederich--Forn{ae}ss index is one and the $overline{partial}$-Neumann problem is exactly regular (via a result of Kohn and its generalization by Harrington). Our work builds on and expands recent results of Liu and Adachi--Yum.
{"title":"The core of the Levi distribution","authors":"G. Dall’Ara, Samuele Mongodi","doi":"10.5802/jep.239","DOIUrl":"https://doi.org/10.5802/jep.239","url":null,"abstract":"We introduce a new geometrical invariant of CR manifolds of hypersurface type, which we dub the\"Levi core\"of the manifold. When the manifold is the boundary of a smooth bounded pseudoconvex domain, we show how the Levi core is related to two other important global invariants in several complex variables: the Diederich--Forn{ae}ss index and the D'Angelo class (namely the set of D'Angelo forms of the boundary). We also show that the Levi core is trivial whenever the domain is of finite-type in the sense of D'Angelo, or the set of weakly pseudoconvex points is contained in a totally real submanifold, while it is nontrivial if the boundary contains a local maximum set. As corollaries to the theory developed here, we prove that for any smooth bounded pseudoconvex domain with trivial Levi core the Diederich--Forn{ae}ss index is one and the $overline{partial}$-Neumann problem is exactly regular (via a result of Kohn and its generalization by Harrington). Our work builds on and expands recent results of Liu and Adachi--Yum.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"2013 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127393611","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Twisted cotangent bundle of Hyperkähler manifolds (with an appendix by Simone Diverio)","authors":"F. Anella, A. Höring","doi":"10.5802/jep.175","DOIUrl":"https://doi.org/10.5802/jep.175","url":null,"abstract":"","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117113507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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