Spherical extension property no longer true for domains in algebraic variety with isolated singularity. (English) Zbl 1194.32022
Let \(D_1\) and \(D_2\) be simply connected bounded domains in \({\mathbb C}^n\) with real analytic spherical boundary. The boundary is called spherical if it is locally CR-equivalent to a portion of the unit sphere. By the generalization of the Riemann mapping theorem proved by S.-S. Chern and S. Ji [Ann. Math. 144, 421–439 (1996; Zbl 0872.32016)], every locally biholomorphic map between \(\partial D_1\) and \(\partial D_2\) extends to a biholomorphic map from \(D_1\) onto \(D_2\). If the boundary of the domains is algebraic, then the condition of simply connectedness can be removed [see S. M. Webster, Invent. Math. 43, 53–68 (1977; Zbl 0348.32005)].
In this paper the authors show that for simply connected domains with algebraic spherical boundary in an algebraic variety with isolated singularities the above result does not hold. As a counterexample they exhibit a pair of domains in the algebraic variety \(\{z_1z_2=z_3^2\}\subset {\mathbb C}^3\).
In this paper the authors show that for simply connected domains with algebraic spherical boundary in an algebraic variety with isolated singularities the above result does not hold. As a counterexample they exhibit a pair of domains in the algebraic variety \(\{z_1z_2=z_3^2\}\subset {\mathbb C}^3\).
Reviewer: Laura Geatti (Roma)
MSC:
32V25 | Extension of functions and other analytic objects from CR manifolds |
32V15 | CR manifolds as boundaries of domains |
14R20 | Group actions on affine varieties |
References:
[1] | Chern S-S, Ji S. On the Riemann mapping theorem. Ann of Math, 1996, 144: 421–439 · Zbl 0872.32016 · doi:10.2307/2118596 |
[2] | Huang X J. On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimensions. Ann Inst Fourier (Grenoble), 1994, 44: 433–463 · Zbl 0803.32011 |
[3] | Huang X, Ji S. Global holomorphic extension of a local map and a Riemann mapping theorem for algebraic domains. Math Res Lett, 1998, 5: 247–260 · Zbl 0912.32010 |
[4] | Mumford D. The topology of normal singularities of an algebraic surface and a criterion for simplicity. Publ Math Inst Hautes Etudes Sci, 1961, 9: 5–22 · Zbl 0108.16801 · doi:10.1007/BF02698717 |
[5] | Webster S M. On the mapping problem for algebraic real hypersurfaces. Invent Math, 1977, 43: 53–68 · doi:10.1007/BF01390203 |
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