A stability theorem for projective CR manifolds

ABSTRACT

We consider smooth deformations of the CR structure of a smooth 2-pseudoconcave compact CR submanifold $M$ of a reduced complex analytic variety $X$ outside the intersection $D\cap M$ with the support D of a Cartier divisor of a positive line bundle $F_X.$ We show that nearby structures still admit projective CR embeddings. Special results are obtained under the additional assumptions that $X$ is a projective space or a Fano variety.

KEYWORDS:

• Pseudoconcave CR manifolds
• CR embeddability of abstract CR structures
• stability of embeddings
• projective embeddings

AMS SUBJECT CLASSIFICATIONS:

• 32V20
• 32V30
• 32G07
• 32F10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 We recall that, if $V$ is a vector space, the monomials of degree $\mathcal{q}$ of its exterior algebra ${\mathrm{\Lambda }}^{\ast }V$ are the exterior products ${\mathcal{v}}_1\wedge \cdots \wedge {\mathcal{v}}_{\mathcal{q}}$ of vectors ${\mathcal{v}}_1,\dots ,{\mathcal{v}}_{\mathcal{q}}$ of $V.$

2 This is obtained by identifying $F$ with its vertical bundle and first defining ${\overline{\mathrm{\partial }}}_M^F$ on the smooth local sections s of $F$ by setting ${\overline{\mathrm{\partial }}}_M^{F}s\left(X+i{J}_{M}X\right)=ds\left(X\right)+J_{F}ds\left(J_{M}X\right),\mathrm{\forall }X\in HM.$In fact, when s is a section, the right-hand side of () is vertical.

3 A real valued smooth function ϕ on an $N$-dimensional complex manifold $X$ is strongly $\mathcal{q}$-pseudoconvex in the sense of [29] at points where its complex Hessian has at least $\left(N-\mathcal{q}+1\right)$ positive eigenvalues. Then $X$ is called strictly $\mathcal{q}$-pseudoconvex if there is an exhaustion function $\varphi \in {\mathcal{C}}^{\mathrm{\infty }}(X,\mathbb{R})$ which is strictly $\mathcal{q}$-pseudoconvex outside a compact subset of $X$ and strictly $\mathcal{q}$-pseudoconcave if there is an exhaustion function $\varphi \in {\mathcal{C}}^{\mathrm{\infty }}(X,\mathbb{R})$ such that $\left(-\varphi \right)$ is strictly $\mathcal{q}$-pseudoconvex outside a compact subset of $X.$

Additional information

Funding

The first author was supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation, grant BR 3363/2-2).