arXiv subjects

Joël Merker

Publications and source records attributed to Joël Merker.

At least 19 recordsLinked to original sources

arXiv2026

A Second Main Theorem for Entire Curves Intersecting Three Conics

We establish a Second Main Theorem for entire holomorphic curves $ f: \mathbb{C} \to \mathbb{P}^2 $ intersecting a generic configuration of three conics $\mathcal{C}= \mathcal{C}_1+ \mathcal{C}_2+ \mathcal{C}_3 $ in the complex projective plane $\mathbb{P}^2$. Using invariant logarithmic $2$-jet differentials with negative twists, we prove the estimate \[ T_f(r) \leqslant 5 \sum_{i=1}^3 N_f^{[1]}(r, \mathcal{C}_i) + o\big(T_f(r)\big)\quad\parallel, \] where $ T_f(r) $ is the Nevanlinna characteristic function, and $ N_f^{[1]}(r, \mathcal{C}_i) $ is the $1$-truncated counting function. The key innovation of our approach is establishing new vanishing lemmas of the form \[ H^0\bigl(\mathbb{P}^2,\, E_{2,m}T_{\mathbb{P}^2}^*(\log \mathcal{C}) \otimes \mathcal{O}_{\mathbb{P}^2}(-t)\bigr) = 0 \] for specific pairs $(m, t)$, achieved by combining algebro-geometric arguments with computer-assisted computations through a mod-$p$ reduction technique. This provides a systematic method for proving vanishing results for negatively twisted jet differentials--a key component in complex hyperbolic geometry--thereby filling a missing piece in two-dimensional complex hyperbolicity theory.

math.CV↗

arXiv2026

Vanishing of Invariant 2-Jet Differentials and Improved Hyperbolicity Degree Bounds in Dimension Two

This paper establishes new degree bounds for Kobayashi hyperbolicity in dimension two. Our main results are: -- A very generic surface in $\mathbb{P}^3$ of degree at least $17$ is Kobayashi hyperbolic. -- The complement of a generic curve in $\mathbb{P}^2$ of degree at least $12$ is Kobayashi hyperbolic. These bounds improve the long-standing records in the field, lowering the threshold from $18$ to $17$ for surfaces (Păun) and from $14$ to $12$ for complements (Rousseau). Central to the proofs are new vanishing results for certain negatively twisted invariant $2$-jet differentials, obtained through a novel combination of algebraic reduction and computer algebra. Since Demailly's Santa Cruz lectures in 1995, the thresholds for the existence of such differentials---and consequently the limits of what $2$-jet techniques can accomplish toward the Kobayashi conjecture in dimension two---have been recognized as $d = 15$ in the compact case and $d = 11$ in the logarithmic case. While previous approaches were unable to reach these targets, the present work provides both the theoretical foundations and the algorithmic framework required to access them, and has already improved the known bounds to $d = 17$ and $d = 12, 13$, respectively. As an unexpected byproduct, our computational method reveals the existence of nonzero negatively twisted invariant $2$-jet differentials with weighted degree $3$ for hyperelliptic-type equations of degree at least $11$ in the logarithmic case and degree at least $15$ in the compact case. Moreover, in the logarithmic setting, we establish an effective quantitative refinement via a Second Main Theorem in Nevanlinna theory.

math.CV↗

arXiv2023

Accidental CR structures

We noticed a discrepancy between Élie Cartan and Sigurdur Helgason about the lowest possible dimension in which the simple exceptional Lie group ${\bf E}_8$ can be realized. This raised the question about the lowest dimensions in which various real forms of the exceptional groups ${\bf E}_\ell$ can be realized. Cartan claims that ${\bf E}_6$ can be realized in dimension 16. However Cartan refers to the complex group ${\bf E}_6$, or its split real form $E_I$. His claim is also valid in the case of the real form denoted by $E_{IV}$. We find however that the real forms $E_{II}$ and $E_{III}$ of ${\bf E}_6$ can not be realized in dimension 16 à la Cartan. In this paper we realize them in dimension 24 as groups of CR automorphisms of certain CR structures of higher codimension. As a byproduct of these two realizations, we provide a full list of CR structures $(M,H,J)$ and their CR embeddings in an appropriate ${\bf C}^N$, which satisfy the following conditions: (1) they have real codimension $k>1$, (2) the real vector distribution $H$ proper for the action of the complex structure $J$ is such that $[H,H]+H=TM$, (3) the local group $G_J$ of CR automorphisms of the structure $(M,H,J)$ is simple, acts transitively on $M$ and has isotropy $P$ being a parabolic subgroup in $G_J$, (4) the local symmetry group $G$ of the vector distribution $H$ on $M$ coincides with the group $G_J$ of CR automorphisms of $(M,H,J)$. Because all the CR structures from our list satisfy the last property we call them accidental. Our CR structures of higher codimension with the exceptional symmetries $E_{II}$ and $E_{III}$ are particular entries in this list.

math.CV↗

arXiv2023

Infinitesimal CR Symmetries of Accidental CR Structures

In this companion paper to our article {\em Accidental CR structures} (arxiv.org, January 2023), thought of as an appendix not submitted for publication, we provide complete explicit lists of infinitesimal CR automorphisms for the concerned CR models having respective Lie algebra structures: $${\bf E}_{II}, \qquad\ {\bf E}_{III}, \qquad\ \mathfrak{so}(\ell-1,\ell+1), \qquad\ \mathfrak{su}(p,q).$$ We start from our lists of {\em quadric} CR submanifolds $M^{2n+c} \subset \mathbb{C}^{n+c}$ of codimension $c >1$ which are shown to be {\em accidental}, in the sense that their CR symmetry groups are {\em equal to} (and not smaller than) the symmetry groups of the underlying real distribution structures -- after forgetting the complex structure. Thanks to intensive symbolic computer explorations, we then determine embedded vector field generators of these CR symmetries Lie algebras, and we express them in {\em extrinsic} holomorphic coordinates, because intrinsic formulas would be too extended to be shown.

math.CV↗

arXiv2021

Normal Forms of second order Ordinary Differential Equations $y_{xx}=J(x,y,y_{x})$ under Fibre-Preserving Maps

We study the equivalence problem of classifying second order ordinary differential equations $y_{xx}=J(x,y,y_{x})$ modulo fibre-preserving point transformations $x\longmapsto φ(x)$, $y\longmapsto ψ(x,y)$ by using Moser's method of normal forms. We first compute a basis of the Lie algebra ${\frak{g}}_{{\{y_{xx}=0\}}}$ of fibre-preserving symmetries of $y_{xx}=0$. In the formal theory of Moser's method, this Lie algebra is used to give an explicit description of the set of normal forms $\mathcal{N}$, and we show that the set is an ideal in the space of formal power series. We then show the existence of the normal forms by studying flows of suitable vector fields with appropriate corrections by the Cauchy-Kovalevskaya theorem. As an application, we show how normal forms can be used to prove that the identical vanishing of Hsu-Kamran primary invariants directly imply that the second order differential equation is fibre-preserving point equivalent to $y_{xx}=0$.

math.DG↗

arXiv2021

Classification of simply-transitive Levi non-degenerate hypersurfaces in $\mathbb{C}^3$

Holomorphically homogeneous CR real hypersurfaces $M^3 \subset \mathbb{C}^2$ were classified by Élie Cartan in 1932. In the next dimension, we complete the classification of simply-transitive Levi non-degenerate hypersurfaces $M^5 \subset \mathbb{C}^3$ using a novel Lie algebraic approach independent of any earlier classifications of abstract Lie algebras. Central to our approach is a new coordinate-free formula for the fundamental (complexified) quartic tensor. Our final result has a unique (Levi-indefinite) non-tubular model, for which we demonstrate geometric relations to planar equi-affine geometry.

math.DG↗

arXiv2020

Affine Homogeneous Surfaces with Hessian rank 2 and Algebras of Differential Invariants

Consider a graphed holomorphic surface $u=F(x,y)$ in $\mathbb{C}^3_{x,y,u}$ under the action of the affine transformation group $A(3)$. In 1999, Eastwood and Ezhov obtained a list of homogeneous models by determining possible tangential vector fields. Inspired by Olver's recurrence formulas, we study the algebra of $A(3)$ differential invariants of surfaces. We obtain necessary conditions for homogeneity of algebraic nature. Solving these conditions, we organise homogeneous models in inequivalent branches.

math.DG↗

arXiv2020

On Differential Invariants of Parabolic Surfaces

The algebra of differential invariants under $SA_3(\mathbb{R})$ of generic parabolic surfaces $S^2 \subset \mathbb{R}^3$ with nonvanishing Pocchiola $4^{\text{th}}$ invariant $W$ is shown to be generated, through invariant differentiations, by only one other invariant, $M$, of order $5$, having $57$ differential monomials. The proof is based on Fels-Olver's recurrence formulas, pulled back to the parabolic jet bundles.

math.DG↗

arXiv2020

New Explicit Lorentzian Einstein-Weyl Structures in 3-Dimensions

On a $3$D manifold, a Weyl geometry consists of pairs $(g, A) =$ (metric, $1$-form) modulo gauge $\widehat{g} = {\rm e}^{2φ} g$, $\widehat{A} = A + {\rm d}φ$. In 1943, Cartan showed that every solution to the Einstein-Weyl equations $R_{(μν)} - \frac{1}{3} R g_{μν} = 0$ comes from an appropriate $3$D leaf space quotient of a $7$D connection bundle associated with a 3$^{\rm rd}$ order ODE $y''' = H(x,y,y',y'')$ modulo point transformations, provided $2$ among $3$ primary point invariants vanish $$ \text{Wünschmann}(H) \equiv 0\equiv \text{Cartan}(H). $$We find that point equivalence of a single PDE $z_y = F(x,y,z,z_x)$ with para-CR integrability $DF := F_x + z_x F_z \equiv 0$ leads to a completely similar $7$D Cartan bundle and connection. Then magically, the (complicated) equation $\text{Wünschmann}(H) \equiv 0$ becomes $$0\equiv\text{Monge}(F):=9F_{pp}^2F_{ppppp}-45F_{pp}F_{ppp}F_{pppp}+40F_{ppp}^3,\qquad p:=z_x, $$ whose solutions are just conics in the $\{p, F\}$-plane. As an ansatz, we take $$F(x,y,z,p):= \frac{α(y)(z-xp)^2+β(y)(z-xp)p+γ(y)(z-xp) +δ(y)p^2+\varepsilon(y)p+ζ(y)}{λ(y)(z-xp)+μ(y) p+ν(y)}, $$ with $9$ arbitrary functions $α, \dots, ν$ of $y$. This $F$ satisfies $DF \equiv 0 \equiv \text{Monge}(F)$, and we show that the condition $\text{Cartan}(H) \equiv 0 $ passes to a certain $\boldsymbol{K}(F) \equiv 0$ which holds for any choice of $α(y), \dots, ν(y)$. Descending to the leaf space quotient, we gain $\infty$-dimensional functionally parametrized and explicit families of Einstein-Weyl structures $\big[ (g, A) \big]$ in $3$D. These structures are nontrivial in the sense that ${\rm d}A \not\equiv 0$ and $\text{Cotton}([g]) \not \equiv 0$.

math.DG↗

arXiv2019

On the real-analyticity of rigid spherical hypersurfaces in ${\mathbb C}^2$

We prove that every smooth rigid spherical hypersurface in ${\mathbb C}^2$ is in fact real-analytic. As an application of this result, it follows that the classification of real-analytic rigid spherical hypersurfaces in ${\mathbb C}^2$ found by V. Ezhov and G. Schmalz applies in the smooth case.

math.CV↗

arXiv2018

Kobayashi hyperbolicity in degree > n^{2n}

For a generic hypersurface $\mathbb{X}^{n-1} \subset \mathbb{P}^n(\mathbb{C})$ of degree \[ d \,\geqslant\, n^{2n} \] (1) $\mathbb{P}^n \big\backslash \mathbb{X}^{n-1}$ is Kobayashi-hyperbolically imbedded in $\mathbb{P}^n$; (2) $\mathbb{X}^{n-1}$ is Kobayashi($\Leftrightarrow$ Brody)-hyperbolic. (1) improves Brotbek-Deng 1804.01719: $d \geqslant (n+2)^{n+3}\, (n+1)^{n+3} = n^{2n}\,n^6\, \big(e^3+{\rm O}(\frac{1}{n}) \big)$. (2) supersedes Demailly 1801.04765: $d \geqslant \frac{1}{3}\, \big( e^1(n-1) \big)^{2n} = n^{2n}\, e^{2n}\, \big( \frac{1}{3\, e^2} + {\rm O} (\frac{1}{n}) \big)$. The method gives in fact $d \geqslant \frac{n^{2n}}{{\sf const}^n}$ for $n \geqslant N({\sf const})$ with any ${\sf const} > 1$.

math.AG↗

arXiv2008

Demailly-Semple jets of orders 4 and 5 in dimension 2

In view of Kobayashi's hyperbolicity conjecture, Demailly-Semple jets of orders 4 and 5 in dimension 2 are studied, some expectations about their algebraic tameness being (dis)proved, after systematic, substantial, formal, manual calculations.

math.CV↗

arXiv2007

A geometrical proof of the Hartogs extension theorem

100 years ago exactly, in 1906, Hartogs published a celebrated extension phenomenon (birth of Several Complex Variables), whose global counterpart was stated in full generality later by Osgood (1929): holomorphic functions in a connected neighborhood V(bD) of a connected boundary bD contained in C^n (n >= 2) do extend holomorphically and uniquely to the domain D. It was a long-standing open problem to derive a proof using only analytic discs, as did Hurwitz (1897), Hartogs (1906) and E.E. Levi (1911) in some special, model cases. Quite unexpectedly, Fornaess in 1998 exhibited a topologically strange (nonpseudoconvex) domain D^F in C^2 that cannot be filled in by holomorphic discs, when one makes the additional requirement that discs must all lie entirely inside D^F. However, one should point out that the standard, unrestricted disc method usually allows discs to go outsise the domain (just think of Levi pseudoconcavity). Confirming these rather ancient expectations, we show that the global Hartogs extension theorem can be established in such a natural way.

math.CV↗

arXiv2007

Holomorphic extension of CR functions, envelopes of holomorphy and removable singularities

This is an extensive (published) survey on CR geometry, whose major themes are: formal analytic reflection principle; generic properties of Systems of (CR) vector fields; pairs of foliations and conjugate reflection identities; Sussmann's orbit theorem; local and global aspects of holomorphic extension of CR functions; Tumanov's solution of Bishop's equation in Hoelder classes with optimal loss of smoothness; wedge-extendability on C^2,a generic submanifolds of C^n consisting of a single CR orbit; propagation of CR extendability and edge-of-the-wedge theorem; Painlevé problem; metrically thin singularities of CR functions; geometrically removable singularities for solutions of the induced d-barre. Selected theorems are fully proved, while surveyed results are put in the right place in the architecture.

math.CV↗

arXiv2004

Symmetries of partial differential equations

We establish a link between the study of completely integrable systems of partial differential equations and the study of generic submanifolds in C^n. Using the recent developments of Cauchy-Riemann geometry we provide the set of symmetries of such a system with a Lie group structure. Finally we determine the precise upper bound of the dimension of this Lie group for some specific systems of partial differential equations.

math.CV↗

arXiv2004

Étude de la régularité analytique de l'application de réflexion CR formelle (French)

Searching normal forms for real analytic submanifolds of C^n involves convergence problems. In 1983, J.K. Moser and S.M. Webster provided examples of real analytic surfaces in C^2 having an isolated hyperbolic (in the sense of E. Bishop) complex tangency, which are formally but not holomorphically normalizable (because of the presence of small divisors), even if the normal form is itself real analytic or algebraic. On the contrary, it appears that such a nonconvergence phenomenon does not appear for submanifolds of C^n whose CR dimension is locally constant, in view of recent results by S.M. Baouendi, P. Ebenfelt and L.P. Rothschild. These results hold true with hypotheses which are relatively simple, but satisfied at a Zariski-generic. Notably, these authors establish that every invertible formal CR mapping between two submanifolds of C^n which are real analytic, generic, finitely nondegenerate and minimal (in the sense of J.-M. Trépreau and A.E. Tumanov) is convergent. In this paper, we establish a more general convergence theorem, which is valid without any nondegeneracy condition, and which confirms the rigidity of the CR category (see Theorem 1.23). This result may be interpreted as a formal Schwarz reflection principle for CR mappings. We deduce that every formal CR equivalence between two submanifolds of C^n which are real analytic, generic and minimal is convergent if and only if both submanifolds are holomorphically nondegenerate (in the sense of N. Stanton). Finally, we establish that two submanifolds of C^n which are real analytic, generic and minimal are formally CR equivalent if and only if they are biholomorphically equivalent.

math.CV↗

arXiv2004

On the local geometry of generic submanifolds of C^n and the analytic reflection principle (Part I)

We build an elementary analytico-geometric theory of Segre chains and their jets.

math.CV↗

arXiv2004

Nonalgebraizable real analytic tubes in C^n

We give necessary conditions for certain real analytic tube generic submanifolds in C^n to be locally algebraizable. As an application, we exhibit families of real analytic non locally algebraizable tube generic submanifolds in C^n. During the proof, we show that the local CR automorphism group of a minimal, finitely nondegenerate real algebraic generic submanifold is a real algebraic local Lie group. We may state one of the main results as follows. Let M be a real analytic hypersurface tube in C^n passing through the origin, having a defining equation of the form v = ϕ(y), where (z,w)= (x+iy,u+iv) \in C^{n-1} \times C. Assume that M is Levi nondegenerate at the origin and that the real Lie algebra of local infinitesimal CR automorphisms of M is of minimal possible dimension n, i.e. generated by the real parts of the holomorphic vector fields \partial_{z_1}, ..., \partial_{z_{n-1}}, \partial_w. Then M is locally algebraizable only if every second derivative \partial^2_{y_ky_l}ϕis an algebraic function of the collection of first derivatives \partial_{y_1} ϕ,..., \partial_{y_m} ϕ.

math.CV↗