[Paper Review] Deformation quantization of a Kaehler-Poisson structure vanishing on a Levi nondegenerate hypersurface

Alexander Karabegov|ArXiv.org|Aug 2, 2006

Advanced Topics in Algebra10 references3 citations

TL;DR

This paper provides an elementary proof of the existence of deformation quantization with separation of variables on a complex manifold equipped with a Kähler-Poisson structure that vanishes on a Levi nondegenerate hypersurface and is nondegenerate elsewhere. By constructing a star product via a formal deformation on a lifted space and projecting it back, the authors extend the standard star product from the complement of the hypersurface to the full manifold, preserving separation of variables and ensuring smoothness across the singular locus.

ABSTRACT

We give an elementary proof of the result by Leichtnam, Tang, and Weinstein that there exists a deformation quantization with separation of variables on a complex manifold endowed with a Kaehler-Poisson structure vanishing on a Levi nondegenerate hypersurface and nondegenerate on its complement.

Motivation & Objective

• To provide an elementary, self-contained proof of the existence of deformation quantization with separation of variables on Kähler-Poisson manifolds where the Poisson structure vanishes on a Levi nondegenerate hypersurface.
• To extend the standard star product of separation of variables from the nondegenerate complement of the hypersurface to the full manifold, ensuring smoothness and consistency across the singular locus.
• To demonstrate that such quantization can be constructed without relying on advanced tools like para-Kähler Lie algebroids, as used in prior work by Leichtnam, Tang, and Weinstein.
• To establish a global construction that respects local properties of the star product and allows for invariance under group actions, as illustrated in the example of CP^n.

Proposed method

• Lift the manifold locally to a product space $\tilde{U} = U \times \mathbb{C}^\times$ using a defining function $\psi$ of the hypersurface $S$, introducing a new coordinate $u$.
• Define a Hermitian matrix $\Gamma$ from the second derivatives of $\psi$, and assume its invertibility to construct a pseudo-Kähler structure on $\tilde{U}$.
• Introduce a formal star product $*$ on the space $\mathcal{F}(\tilde{U})$ of formal Laurent series in $u$, using the standard deformation quantization with separation of variables on the lifted space.
• Define a subalgebra $\mathcal{F}_{\geq 0}(\tilde{U})$ of non-negative powers of $\frac{h}{u\bar{u}}$, and construct a projection $\sigma$ to $C^\infty(U)[[\nu]]$ via $\sigma(\sum_r (\frac{h}{u\bar{u}})^r f_r) = \sum_r N_r(\nu) \psi^r f_r$, where $N_r(\nu)$ are normalization factors.
• Show that the pullback of the star product $*$ via $\sigma$ defines a new star product $\star$ on $U$, which agrees with the original standard product on $U \setminus S$ and extends smoothly across $S$.
• Globalize the construction by requiring local defining functions $\psi$ such that $\log|\psi|$ is a potential for the pseudo-Kähler form $\omega$ on $M \setminus S$, ensuring compatibility across overlapping charts.

Experimental results

Research questions

• RQ1 Can deformation quantization with separation of variables be constructed on a Kähler-Poisson manifold where the Poisson structure vanishes on a Levi nondegenerate hypersurface and is nondegenerate elsewhere?
• RQ2 Does the standard star product of separation of variables on the nondegenerate complement extend smoothly to the full manifold, preserving the separation of variables property?
• RQ3 Can this extension be achieved without using advanced geometric tools like para-Kähler Lie algebroids, as in prior work?
• RQ4 Is there a canonical way to lift the star product from the base manifold to a higher-dimensional space where the Poisson structure becomes nondegenerate, enabling a constructive extension?
• RQ5 Can such a quantization be globally defined and invariant under group actions, as in the case of $\mathbb{C}P^n$ with $SU(n,1)$ symmetry?

Key findings

• The standard deformation quantization with separation of variables on the pseudo-Kähler manifold $M \setminus S$ extends to a well-defined deformation quantization with separation of variables on the full Kähler-Poisson manifold $M$, even though the Poisson structure vanishes on the Levi nondegenerate hypersurface $S$.
• The extension is achieved via a projection $\sigma$ from a formal star product on a lifted space $\tilde{U} = U \times \mathbb{C}^\times$, ensuring that the resulting product $\star$ on $U$ is smooth and agrees with the original product on $U \setminus S$.
• The bidifferential operators $C_r(f,g)$ in the star product expansion satisfy $f \star g - fg = 0$ on $S$, confirming that the product vanishes to first order on the singular locus, as required.
• The construction is local and compatible across overlapping charts, allowing for a global extension under the condition that $\log|\psi|$ is a local potential for the pseudo-Kähler form $\omega$ on $M \setminus S$.
• The method provides an elementary alternative to the para-Kähler Lie algebroid approach used in [LTW], relying only on formal power series and coordinate lifting techniques.
• In the example of $\mathbb{C}P^n$ with the unit sphere $S$, the construction yields a global $SU(n,1)$-invariant star product that coincides with the standard separation-of-variables product on $\mathbb{C}^n \setminus S$, demonstrating the method’s effectiveness in symmetric settings.