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Holomorphic chains of given boundary in \(\mathbb{C} P^n\). (Chaînes holomorphes de bord donné dans \(\mathbb{C} P^n\).) (French) Zbl 0942.32007
If \(M\) is a real, oriented, closed \(C^2\) subvariety of a complex analytic space (or variety) of complex dimension \(n\), where \(M\) has real dimension \(2p-1\) (\(0<p\leq n\)), the boundary problem is to find necessary and sufficient conditions so that \(M\) is the boundary of a holomorphic \(p\)-chain, that is, there exists a holomorphic \(p\)-chain of \(X\setminus M\) with a simple extension to \(X\) such that \(dT=M\). This problem has been solved in various case, notably for \({\mathbb{C}}^n\) and for \(\mathbb{C} P^n\setminus\mathbb{C} P^{n-r}\). In this paper the authors obtain solutions of the boundary problem in \(\mathbb{C} P^n\) for arbitrary \(p\).
The authors assume that \(X\) is a \((n-p+1)\)-concave domain in \(\mathbb{C} P^n\), which is a nonempty union of projective subspaces of dimension \(r\leq n-p+1\). A closed \(C^2\) subvariety \(M\) of \(X\) is said to have negligible singularities if there exists a closed subspace \(\tau\subset M\) of \((2p-1)\)-dimensional Hausdorff measure zero such that \(M\setminus\tau\) is a closed oriented \(C^2\) subvariety of dimension \(2p-1\) and locally finite \((2p-1)\)-dimensional volume, with \(dM=0\) and such that \(1\leq n-p+1\leq q\).
Theorem II: If \(M\) is a \(C^2\) subvariety with negligible singularities and \[ G(\xi,\eta)=(2\pi i)^{-1}\int_{\gamma_{\nu '}}\zeta g^{-1}dg, \] then the following two conditions are equivalent: (1) \(M\) is the boundary of a holomorphic \(p\)-chain of \(X\) with locally finite mass; (2) \(M\) is maximally complex and there exists a point \(\nu^*\) belonging to the complex Grassmannian \(G_{\mathbb{C}}(n-p+1,n+1)\) in a neighborhood of which there exist a finite number of holomorphic functions of \((\xi,\eta)\) satisfying the system of partial differential equations for shock waves for \((\xi,\eta)\), and such that the second derivatives with respect to \(\xi\) of a linear combination of these functions and of \(G\) are equal. Here \((\xi,\eta)\) is a suitable coordinate system on \(G_{\mathbb{C}}(n-p+1,n+1)\), \(\gamma_{\nu '}\) is the \(C^2\) curve formed by cutting \(M\) by the projective subspace \(P_{\nu '}\) corresponding to \(\nu '\in G_{\mathbb{C}}(n-p+2,n+1)\), contained in an affine subspace \(W\equiv {\mathbb{C}}^n\) of \(\mathbb{C} P^n\), \(W\) has coordinates \((z_1,\dots,z_n)\), \(\zeta=(z_1,\dots,z_{n-p})\) and \(g\) is a linear form on \({\mathbb{C}}^n\) such that the hyperplane \(\{g=0\}\) is transversal to \(P_{\nu '}\). The existence part of the second statement is slightly more delicate than indicated above.
The proof of this theorem uses the techniques established in a previous paper of the authors [Aspects Math. E 26, 163-187 (1994; Zbl 0821.32008)] for the case \(p=1\). Much care needs to be taken in defining the chain \(T\).
This Theorem II has as a consequence the following result.
Theorem I: Let \(X\) be a \(q\)-concave domain of \(\mathbb{C} P^n\) such that \(n-p+1\leq q\leq n\) and let \(M\) be a closed, oriented \(C^2\) subvariety of \(X\), of dimension \(2p-1\). Then the following two conditions are equivalent: (i) \(M\) is the boundary of a holomorphic \(p\)-chain of \(X\) of locally finite mass; (ii) \(M\) is maximally complex and there exists a matrix \(\nu ^{'\ast}\) (i.e., a point of the complex Grassmannian \(G_{\mathbb{C}}(n-p+2,n+1)\)) such that for \(\nu '\) in a sufficiently small neighborhood of \(\nu ^{'\ast}\), for every projective subspace \(P_{\nu '}\) of \(\mathbb{C} P^n\) contained in \(X\) such that \(M\cap P_{\nu '}\) is a curve \(\gamma_{\nu '}\) of \(P_{\nu '}\) with finite length, there exists a holomorphic \(1\)-chain \(S_{\nu '}\) of \(P_{\nu '}\), of finite mass, with boundary \(\gamma_{\nu '}\), which depends continuously on \(\nu '\).
The authors also discuss some examples and further problems.
Reviewer: J.S.Joel (Kelly)

MSC:
32C30 Integration on analytic sets and spaces, currents
32V40 Real submanifolds in complex manifolds
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
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References:
[1] AUDIN (M.) , LAFONTAINE (J.) ed. - Holomorphic curves in Symplectic Geometry , Progress in Math., Birkhäuser, t. 117, 1994 . MR 95i:58005 | Zbl 0802.53001 · Zbl 0802.53001
[2] BISHOP (E.) . - Conditions for the analyticity of certain sets , Michigan Math. J., t. 11, 1964 , p. 289-304. Article | MR 29 #6057 | Zbl 0143.30302 · Zbl 0143.30302
[3] BOCHNER (S.) , MARTIN (W.T.) . - Several complex variables , Princeton Math. Ser., t. 10, 1948 . MR 10,366a | Zbl 0041.05205 · Zbl 0041.05205
[4] CHIRKA (E.M.) . - Complex analytic sets, Mathematics and its applications , 46. - Kluwer Academic Publishers, 1989 ; Russian edition 1985 . Zbl 0683.32002 · Zbl 0683.32002
[5] DARBOUX (L.) . - Théorie des surfaces, I, 2e éd. - Gauthier-Villars, Paris, 1914 .
[6] DINH TIEN CUONG (P.) . - Chaînes holomorphes à bord rectifiable , C. R. Acad. Sci. Paris, t. 322, Série I, 1996 , p. 1135-1140. MR 97e:32007 | Zbl 0865.32007 · Zbl 0865.32007
[7] DINH TIEN CUONG (P.) . - Enveloppe polynômiale d’un compact de longueur finie et chaînes holomorphes à bord rectifiable , Institut de Mathématiques de Jussieu, prépublication 93, 1996 , à paraître dans Acta Mathematica.
[8] DOLBEAULT (P.) . - On holomorphic chains with given boundary in \Bbb CPn , Springer Lectures Notes, t. 1089, 1983 , p. 118-129. MR 85i:32013 | Zbl 0538.32014 · Zbl 0538.32014
[9] DOLBEAULT (P.) , HENKIN (G.) . - Surfaces de Riemann de bord donné dans \Bbb CPn , Contributions to complex analysis and analytic geometry, Aspects of Math., Vieweg, t. 26, 1994 , p. 163-187. MR 96a:32020 | Zbl 0821.32008 · Zbl 0821.32008
[10] DOLBEAULT (P.) , HENKIN (G.) . - Chaînes holomorphes de bord donné dans \Bbb CPn , Institut de Mathématiques de Jussieu, prépublication 76, 1996 .
[11] DOLBEAULT (P.) , POLY (J.B.) . - Variations sur le problème des bords dans \Bbb CPn , prépublication, 1995 .
[12] GINDIKIN (S.) , HENKIN (G.) . - Integral geometry for \partial -cohomology in q-linear concave domains in \Bbb CPn , Funct. Anal. and Appl., t. 12, 1978 , p. 247-261. MR 80a:32014 | Zbl 0423.32013 · Zbl 0423.32013
[13] GROMOV (M.) . - Pseudo-holomorphic curves in symplectic manifolds , Inv. math., t. 82, 1985 , p. 307-347. MR 87j:53053 | Zbl 0592.53025 · Zbl 0592.53025
[14] HARVEY (R.) . - Holomorphic chains and their boundaries , Proc. Symp. Pure Math., t. 30, vol. 1, 1977 , p. 309-382. MR 56 #5929 | Zbl 0374.32002 · Zbl 0374.32002
[15] HARVEY (R.) , LAWSON (B.) . - On boundaries of complex analytic varieties , I, Ann. of Math., t. 102, 1975 , p. 233-290. MR 54 #13130 | Zbl 0317.32017 · Zbl 0317.32017
[16] HARVEY (R.) , LAWSON (B.) . - On boundaries of complex analytic varieties, II , Ann. of Math., t. 106, 1977 , p. 213-238. MR 58 #17186 | Zbl 0361.32010 · Zbl 0361.32010
[17] HARVEY (R.) , LAWSON (B.) . - Complex analytic geometry and measure theory , Proc. Symp. Pure Math., t. 44, 1986 , p. 261-274. MR 87h:32021 | Zbl 0587.32019 · Zbl 0587.32019
[18] HARVEY (R.) , SHIFFMAN (B.) . - A characterization of holomorphic chains , Ann. of Math. (2), t. 99, 1974 , p. 553-587. MR 50 #7572 | Zbl 0287.32008 · Zbl 0287.32008
[19] HENKIN (G.) , LEITERER (J.) . - Andreotti-Grauert theory by integral formulas , Progress in Math., Birkhaüser, Boston, t. 74, 1988 . MR 90h:32002b | Zbl 0654.32002 · Zbl 0654.32002
[20] HENKIN (G.) , TUMANOV (A.E.) . - Local characterization of holomorphic automorphisms of Siegel domains , Funct. Anal. and Appl., t. 17, 1983 , p. 285-294. MR 86a:32063 | Zbl 0572.32018 · Zbl 0572.32018
[21] HÖRMANDER (L.) . - L2 estimates and existence theorems for the \partial -operator , Acta Math., t. 113, 1965 , p. 89-152. Zbl 0158.11002 · Zbl 0158.11002
[22] IVASHKOVICH (S.M.) . - The Hartogs-type extension theorem for meromorphic maps into compact Kähler manifolds , Inv. Math., t. 109, 1992 , p. 47-54. MR 93g:32016 | Zbl 0782.32009 · Zbl 0782.32009
[23] JÖRICKE (B.) . - Some remarks concerning holomorphically convex hulls and envelopes of holomorphy , Math. Z., t. 218, 1995 , p. 143-157. Article | MR 96b:32014 | Zbl 0816.32011 · Zbl 0816.32011
[24] KING (J.) . - Open problems in geometric function theory, Proceedings of the fifth international symposium , division of Math., p. 4, The Taniguchi foundation, 1978 .
[25] KOHN (J.J.) , ROSSI (H.) . - On the extension of holomorphic functions from the boundary of a complex manifold , Ann. of Math., t. 81, 1965 , p. 451-472. MR 31 #1399 | Zbl 0166.33802 · Zbl 0166.33802
[26] KOPPELMAN (W.) . - The Cauchy integral for functions of several complex variables , Bull. A.M.S., t. 73, 1967 , p. 372-377. Article | MR 35 #416 | Zbl 0177.11103 · Zbl 0177.11103
[27] LAWRENCE (M.G.) . - Polynomial hulls of rectifiable curves , Amer. J. Math., t. 117, 1995 , p. 405-417. MR 96d:32012 | Zbl 0827.32012 · Zbl 0827.32012
[28] LELONG (P.) . - Fonctions entières (n variables) et fonctions plurisousharmoniques d’ordre fini dans \Bbb Cn , J. Analyse Math., t. 12, 1964 , p. 365-407. MR 29 #3668 | Zbl 0126.29602 · Zbl 0126.29602
[29] LERAY (J.) . - Le calcul différentiel et intégral sur une variété analytique complexe (Problème de Cauchy III) , Bull. S.M.F., t. 87, 1959 , p. 81-180. Numdam | MR 23 #A3281 | Zbl 0199.41203 · Zbl 0199.41203
[30] LEVY (H.) . - On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for functions of two complex variables , Ann. of Math., t. 64, 1956 , p. 514-522. MR 18,473b | Zbl 0074.06204 · Zbl 0074.06204
[31] PORTEN (E.) . - A Hartogs-type theorem for meromorphic mappings , preprint 1997 . · Zbl 1258.32015
[32] ROSSI (H.) . - Continuation of subvarieties of projective varieties , Amer. J. of Math., t. 91, 1969 , p. 565-575. MR 39 #5830 | Zbl 0184.31401 · Zbl 0184.31401
[33] ROTHSTEIN (W.) . - Bemerkung zur theorie komplexen Raüme , Math. Ann., t. 137, 1959 , p. 304-315. MR 22 #4081 | Zbl 0088.05704 · Zbl 0088.05704
[34] ROTHSTEIN (W.) , SPERLING (H.) . - Einsetzer und analytischer Flächenstücke in Zyklen auf komplexer Raüme , Festshrift zur Gedächtnisfeier für Karl Weierstrass 1815 - 1965 (Behnke und Kopfermann, ed.). Westentsche Verlag, Köln, 1965 , p. 531-554. Zbl 0145.31803 · Zbl 0145.31803
[35] SACKS (J.) , UHLENBECK (K.) . - The existence of minimal 2-spheres , Ann. of Math., t. 113, 1981 , p. 1-24. MR 82f:58035 | Zbl 0462.58014 · Zbl 0462.58014
[36] SARKIS (F.) . - CR meromorphic extension and the non embedding of the Andreotti-Rossi CR structure in the projective space , Institut de Mathématiques de Jussieu, prébublication 116, 1997 . · Zbl 1110.32308
[37] SHIFFMAN (B.) . - On the removal of singularities of analytic sets , Michigan Math. J., t. 15, 1968 , p. 111-120. Article | MR 37 #464 | Zbl 0165.40503 · Zbl 0165.40503
[38] STOLL (W.) . - Über die Fortsetzbarkeit analytischer Mengen endlichen Oberflächeninhaltes , Arch. Math., t. 9, 1958 , p. 167-175. MR 21 #729 | Zbl 0083.30801 · Zbl 0083.30801
[39] STOUT (E.L.) . - The boundary values of holomorphic functions of several complex variables , Duke Math. J., t. 44, 1977 , p. 105-108. Article | MR 55 #10722 | Zbl 0351.32015 · Zbl 0351.32015
[40] TRÉPREAU (J.-M.) . - Sur le prolongement holomorphe des fonctions CR définies sur une hypersurface réelle de classe C2 dans \Bbb C2 , Inv. Math., t. 83, 1986 , p. 583-592. MR 87f:32035 | Zbl 0586.32016 · Zbl 0586.32016
[41] WERMER (J.) . - The hull of a curve in \Bbb Cn , Ann. of Math., t. 68, 1958 , p. 550-561. MR 20 #6536 | Zbl 0084.33402 · Zbl 0084.33402
[42] WU (H.-H.) . - The equidistribution theory of holomorphic curves , Ann. of Math. Studies, Princeton Univ. Press, t. 64, 1970 . MR 42 #7951 | Zbl 0199.40901 · Zbl 0199.40901
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