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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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