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Register a new userRigid equivalences of $5$-dimensional $2$-nondegenerate rigid real hypersurfaces $M^5 subset mathbb{C}^ 3$ of constant Levi rank $1$
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Wei Guo Foo
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We study the local equivalence problem for real-analytic ($mathcal{C}^omega$) hypersurfaces $M^5 subset mathbb{C}^3$ which, in coordinates $(z_1, z_2, w) in mathbb{C}^3$ with $w = u+i, v$, are rigid: [ u ,=, Fbig(z_1,z_2,overline{z}_1,overline{z}_2big), ] with $F$ independent of $v$. Specifically, we study the group ${sf Hol}_{sf rigid}(M)$ of rigid local biholomorphic transformations of the form: [ big(z_1,z_2,wbig) longmapsto Big( f_1(z_1,z_2), f_2(z_1,z_2), a,w + g(z_1,z_2) Big), ] where $a in mathbb{R} backslash {0}$ and $frac{D(f_1,f_2)}{D(z_1,z_2)} eq 0$, which preserve rigidity of hypersurfaces. After performing a Cartan-type reduction to an appropriate ${e}$-structure, we find exactly two primary invariants $I_0$ and $V_0$, which we express explicitly in terms of the $5$-jet of the graphing function $F$ of $M$. The identical vanishing $0 equiv I_0 big( J^5F big) equiv V_0 big( J^5F big)$ then provides a necessary and sufficient condition for $M$ to be locally rigidly-biholomorphic to the known model hypersurface: [ M_{sf LC} colon u ,=, frac{z_1,overline{z}_1 +frac{1}{2},z_1^2overline{z}_2 +frac{1}{2},overline{z}_1^2z_2}{ 1-z_2overline{z}_2}. ] We establish that $dim, {sf Hol}_{sf rigid} (M) leq 7 = dim, {sf Hol}_{sf rigid} big( M_{sf LC} big)$ always. If one of these two primary invariants $I_0 otequiv 0$ or $V_0 otequiv 0$ does not vanish identically, we show that this rigid equivalence problem between rigid hypersurfaces reduces to an equivalence problem for a certain $5$-dimensional ${e}$-structure on $M$.
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Consider a $2$-nondegenerate constant Levi rank $1$ rigid $mathcal{C}^omega$ hypersurface $M^5 subset mathbb{C}^3$ in coordinates $(z, zeta, w = u + iv)$: [ u = Fbig(z,zeta,bar{z},bar{zeta}big). ] The Gaussier-Merker model $u=frac{zbar{z}+ frac{1}{2}z^2bar{zeta}+frac{1}{2} bar{z}^2 zeta}{1-zeta bar{zeta}}$ was shown by Fels-Kaup 2007 to be locally CR-equivalent to the light cone ${x_1^2+x_2^2-x_3^2=0}$. Another representation is the tube $u=frac{x^2}{1-y}$. Inspired by Alexander Isaev, we study rigid biholomorphisms: [ (z,zeta,w) longmapsto big( f(z,zeta), g(z,zeta), rho,w+h(z,zeta) big) =: (z,zeta,w). ] The G-M model has 7-dimensional rigid automorphisms group. A Cartan-type reduction to an e-structure was done by Foo-Merker-Ta in 1904.02562. Three relative invariants appeared: $V_0$, $I_0$ (primary) and $Q_0$ (derived). In Pocchiolas formalism, Section 8 provides a finalized expression for $Q_0$. The goal is to establish the Poincare-Moser complete normal form: [ u = frac{zbar{z}+frac{1}{2},z^2bar{zeta} +frac{1}{2},bar{z}^2zeta}{ 1-zetabar{zeta}} + sum_{a,b,c,d atop a+cgeqslant 3}, G_{a,b,c,d}, z^azeta^bbar{z}^cbar{zeta}^d, ] with $0 = G_{a,b,0,0} = G_{a,b,1,0} = G_{a,b,2,0}$ and $0 = G_{3,0,0,1} = {rm Im}, G_{3,0,1,1}$. We apply the method of Chen-Merker 1908.07867 to catch (relative) invariants at every point, not only at the central point, as the coefficients $G_{0,1,4,0}$, $G_{0, 2, 3, 0}$, ${rm Re} G_{3,0,1,1}$. With this, a brige Poincare $longleftrightarrow$ Cartan is constructed. In terms of $F$, the numerators of $V_0$, $I_0$, $Q_0$ incorporate 11, 52, 824 differential monomials.
Inspired by an article of R. Bryant on holomorphic immersions of unit disks into Lorentzian CR manifolds, we discuss the application of Cartans method to the question of the existence of bi-disk $mathbb{D}^{2}$ in a smooth $9$-dimensional real analytic real hypersurface $M^{9}subsetmathbb{C}^{5}$ with Levi signature $(2,2)$ passing through a fixed point. The result is that the lift to $M^{9}times U(2)$ of the image of the bi-disk in $M^{9}$ must lie in the zero set of two complex-valued functions in $M^{9}times U(2)$. We then provide an example where one of the functions does not identically vanish, thus obstructing holomorphic immersions.
In this note we generalize the methods of [1][2][3] to 5-dimensional Riemannian manifolds M. We study the relations between the geometry of M and the number of solutions to a generalized Killing spinor equation obtained from a 5-dimensional supergravity. The existence of 1 pair of solutions is related to almost contact metric structures. We also discuss special cases related to $M = S1 times M4$, which leads to M being foliated by submanifolds with special properties, such as Quaternion-Kahler. When there are 2 pairs of solutions, the closure of the isometry sub-algebra generated by the solutions requires M to be S3 or T3-fibration over a Riemann surface. 4 pairs of solutions pin down the geometry of M to very few possibilities. Finally, we propose a new supersymmetric theory for N = 1 vector multiplet on K-contact manifold admitting solutions to the Killing spinor equation.
We show that the boundary of any bounded strongly pseudoconvex complete circular domain in $mathbb C^2$ must contain points that are exceptionally tangent to a projective image of the unit sphere.
Let X be a smooth complex projective variety with basepoint x. We prove that every rigid integral irreducible representation $pi_1(X,x)to SL (3,{mathbb C})$ is of geometric origin, i.e., it comes from some family of smooth projective varieties. This partially generalizes an earlier result by K. Corlette and the second author in the rank 2 case and answers one of their questions.
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