Regularity of varieties in strictly pseudoconvex domains

Publicacions Matemátiques, Vol 32 (1988), 145-150 . Abstract REGULARITY OF VARIETIES IN STRICTLY PSEUDOCONVEX DOMAINS FRANC FORSTNERIC We prove a theorem on the boundary regularity of a purely p-dimensional complex subvariety of a relatively compact, strictly pseudoconvex domain in a Stein manifold . Some applications describing the structure of the polynomial hull of closed curves in C " are also given . Introduction Let X be a complex manifold, M C X a connected (2p - 1)-dimensional submanifold of X of class Crk (k >_ 1, p > 1), and A a closed complex subvariety of X\M of pure dimension p such that A C A U M . Then either A is a complex subvariety of X or else there exists a closed subset E C A of (2p - 1)- dimensional Hausdorff measure )l 2P _ 1(E) = 0 such that the pair (A\E, M\E) is a C k submanifold with boundary [2, p .190] . In the second case A has locally finite 2p dimensional volume in X, and M can be oriented si4ch that the pair (A, M) satisfies the theorem of Stokes [2, p .192], [6], [8] . Consequently M is a maximally complex submanifold of X, Le ., the maximal complex subspace Ti M of the real tangent space TM to M at z has real codimension one in TM . There is a converse of this due to Harvey and Lawson [6] : If X is a Stein manifold and M is a closed, compact, maximally complex submanifold of X of dimension 2p -1 (p > 2), then M bounds (in the sense of currents) a purely pdimensional complex subvariety A C X\M, with boundary regularity as above . We are interested in the boundary regularity of a purely p-dimensional complex subvariety of a relatively compact, strictly pseudoconvex domain 0 C X with C 2 boundary, We shall give a simple proof of the following The work was supported in part by a grant from the Science Foundation of the Republic of Slovenia . 14 6  F . FORSTNERIC Theorem 1 . Assume that (1) X is a Stein manifold ; (2) fl is a relatively compact, strictly pseudoconvex domain with C 2 boundary in X ; (3) M is a closed (2p-1)-dimensional submanifold of X of class C' (p >_ 1, k >_ 2) contained in the boundary bft of . 2 ;  _ (4) A is a purely p-dimensional complex subvariety of ft such that A C C A u M, and A intersects every connected component of M . Then there exists an open neighborho_od U of M such that the pair (A n U, M) is a Ck manifold with boundary, and A intersects bíl transversely in the set M . Consequently A has at most finitely many singularities in ft . The manifold M is maximally complex, and its tangent space T .M is not contained in the maximal complex tangent space Tz bft to the boundary of n for any z E M . We obtain an interesting consequence concerning holomorphic convexity of closed curves . We shall state the result only for X = C n . Recall that the polynomially convex hull of a compact set K C Cn is K = {z E C n : 1 f (z) 1 <_ sup 1 f 1 for all holomorphic polynomials f } . x If M is a rectifiable closed Jordan curve in en, then either M is polynomially convex, M = M, or else A= M\M is a purely one-dimensional analytic variety according to Wermer [10], [11, p .71j, Stolzenberg [9], and Alexander [1] . Corollary 2 . Let fl be a bounded C 2 ,strictly pse ;udoconvex domain in Cn with polynomially convex closure, and let M be asimple closed curve of class Ck, k >_ 2, contained in the boundary o[n . If M is not polynomially convex, then the one-dime nsional complex variety,A,-,M\M has at most finitely many singularities . Proof . Since '9 is polynomially convex, A is contained in 11 . Every point p E bft is a peak point for f], so the maxiimum principle iinplies that A is contained in n . Therefore the corollary follows from Theorem 1 . We shall say that a submanifold M C bíl of class C 1 is complex tangential at the point z E M if T Z M is contained in T°bfl . Here, Ti bft = Tzbft n Y/ 11 T z 6S1 . We shall say that M is complex transverse at z if it is not complex tangential . Corollary 3 . Let fZ C C n be as in Corollary 2 . If M . c bfl is a simple closed curve of class C 2 that is complex tangential at least at one point, then M is polynomially convex . Proof .. If M is not polynomially convex, Theorem 1 implies that the polynomial hull M = A u M C fZ is a complex variety with smooth boundary near REGULARITY OF VARIETIES  14 7 every point z E M, and M intersects bn transversely in M . This implies that M is complex transverse in bn and the corollary follows . Example . If M is a simple closed CZ curve in the sphere {z E C" : IzI = 1} parametrized by the map r (t) = (r l (t), . . . , rn (t» with nonvanishing derivative, and if n E r í (t)rj (t) = 0 j=i for some value of the parameter t, then M is polynomially convex It seems rather surprising that a condition at one point of the curve guaranties its polynomial convexity, as long as the curve stays inside the given strictly pseudoconvex boundary . Remarks . 1 . Theorem 1 is stated in [2, p .203], but the proof given there does not appear to be complete . 2 . If one knows already that M is the boundary of A = M\M in the sense of currents and if p > 2, then Theorem 1 is a specia1 case of Theorem 10 .3 in [6, p .275] . 3 . In the case when p = 1 and the variety A is a proper holomorphic image of the unit disc 0 = {z E C :Iz1 < 1}, Theorem 1 follows from the more general results of Cirka [3] concerning the regularity of one-dimensional complex varieties in the complement of a totally real submanifold of the ambient space . 4 . In the case p > 2, Theorem 1 was proved by the author in [4] . Ournew proof is simpler and includes the case p = 1 when M - is a curve . We first show that the pair (A, M) is a manifold with boundary in a neighborhood of each point z E M at which M is complex transversal, Le ., the condition (1) fails . The proof in this case is the same as in [4] . The main difficulty in [4] was to show that M can not be complex tangential at any point if it bounds a pdimensional variety . In this paper we prove this by a very simple perturbation argument . Acknowledgement . I wish to thank Josip Globevnik for several stimulating discussions on this subject . Proof of Theorem 1 By the embedding theorem of Fornaess and Khenkin [7, p .112] we may assume that X = C" and n is a strictly convex domain in C" . It suffices to prove that each point z ° E A n M has an open neighborhood U such that the pair (A n U, M n U) is a smooth manifold with boundary . We first prove this in the case when M is complex transverse at z° , Le ., condition (1) fails . This part of the argument is the same as in [4] . We include it for the conveniente of the reader . By an affine change of coordinates in C" we may assume that 148  F . FORSTNERI5 (i) z o = 0, (ii) T o bft = {ate z 1 = 0} and Tó bíl = { z 1 = 0}, and (iii) the domain S2 is contained in {% z 1 > 0} . Recall that T o M is a real (2p - 1)-dimensional subspace of {% z 1 = 0} that is not contained in {z 1 = 0} . Thus the orthogonal projection of ToM onto the z1 axis is a real line, and the intersection W =T o M n {z 1 = 0} has real dimension 2p - 2 . We can choose a complex (p -1)-dimensional subspace L contained in {z 1 = = 0} such that the orthogonal projection C" -+ L maps W surjectively onto L . After a unitary change of coordinates z Z . . . . , z" we may assume that L= {z1 = =zp+1 = . . .=z  =0} . Let 7r : C" -> Cp = { z p+1 = 0, . . . , z" = 0} be the orthogonal projection . Since bít is strictly convex, we can find an open polydisc neighborhood U= = U' x U" of 0 in C", with U' C Cp and U" C C"-p, such that 7r : U n íí -+ U' is a proper mapping . Our choice of L implies that 7r : ToM -> Cp is injective . Shrinking U if necessary it follows that 7r maps M n U diffeomorphically onto a real hypersurface I' C U' of class C' that splits U'\I' in two connected components F+ and F - . Let F+ be the region contained in {B?e z 1 > 0} . Since M n U is contained in the strictly convex boundary bfl n U and Cp x {0} contains the normal vector (1,0, . . . , 0) to bQ at 0, the projection 7r(Mn U) = I' is hypersurface in Cp which is strictly convex from the side r+, provided that the neighborhood U is sufficiently small . Since 7r : S2 n U -> U' is proper and the set (A U M) n U is closed in U, the restriction 7r : (AUM)nU->U' is also proper . The convexity of I'+ along r implies that 7r(A n U) is contained in r+ according to the maximum principle . Hence the mapping (2)  7r : An U -> F+ is an analytic cover [5, p .101] . Denote by s the number of sheets of this analytic cover, Le ., the number of points in the generic fiber . Notice that all sheets converge to the common edge M as we approach I' . We claim that this implies s = 1 . We only give a sketch of proof Since the detai1s can be found in [41 . Let z = (Z 1 , z"), where z' = ( z 1 , . . . ,z p ) and z" = ( z p+1 , . . . ,z") . There is a linear function w = w (z") that separates points of 7r -1 (z') nAn U for all points z' E I'+ outside a proper complex subvariety v C F+ . For each z' E 17+ w we denote by w, (z'), . . . , w, (z') the values of w at the points of 7r -1 (z') n A nU . Let P (w, z') be the polynomial in w defined by s P(w, z) = ll lw - wi (z )~ = ws + al (z' )w' -1 + . . . + ca ., (z'),  z' E I' + \a . i=1 REGULARITY OF VARIETIES  149 Its coefficients aj (z') are bounded holomorphic functions on r+ w, so they extend to bounded functions on F+ . The discriminant 6(z') of P( . , z') is also a bounded holomorphic function on r+ since it is a polynomial expression in the coefficients a ; of P . Recall that 6(z') = 0 if and only if P(,z') has multiple ro ots . If s > 1, the hypothesis A C A U M implies that the nontangential boundary values of 6 on I equal zero almost everywhere since the different sheets of (2) converge together to M . This implies 6 - 0 on I'+, a contradiction . Thus s = 1 as claimed . It follows that the projection (2) is a bijection, so (A U M) fl U is a graph of the form (A U M) fl U= {(Z', f (z')) : z' E ]P - ' - U F} . Since A is complex analytic and M is of class C' k , it follows that f is holomorphic on r+ and of class Ck on I' . Clearly f is also continuous on r+ U I' . The regularity theorem [6, p .2491 implies that f is of class C' ti on I P+ UI' . This proves that (A U M) fl U is aC k manifold with boundary intersecting bf] transversely . It remains to show that the manifold M is complex transverse at each point z E M n A so that the first part of the proof applies . The following argument is considerably simpler than the one in [4), and it also applies in the case p = 1 . Assume that the condition (1) is satisfied for some z = z ° E M fl A . Let A C Ti bn be the smallest complex subspace of C" containing T z oM . Since T z oM is not a complex subspace, there is a vector b E A\T z oM . We can choose a function h of class C 2 , supported on a neighborhood of z ° in C", such that hi m - 0, but the derivative of h at z° in the direction b is nonzero . Let p be a strictly convex defining function of class CZ for fl, so 11 -- {z E E C" : p(z) < 0} and dp 9~ 0 on b1l . If E > 0 is sufficiently small, the domain 12, = {z E C" : P(Z) - I - E h(z) < 0} is of class CZ and strictly convex . Fix such an E . Since h vanishes on M, M is contannnd in the boundary of SZ E . Thus we have A C M C 1Z E = íí,, and the maximum principie implies AC 11, Our choice of h implies that T z o bfl, does not contain A, so Tz RIE does not contain T z oM . This means that M is complex transverse in bO E a _ t the point z ° . By the first part of the proof, with n replaced by n E , the set A is a local Ck manifold with boundary M near z ° . We have proved that the pair (A, M) is a local manifold with boundary near every point z E A n M . This implies that A n M is an open and closed subset of M . Since w_ e assumed that A intersects every connected component of M, it follows that A =A U M . It remains to show that A intersects bn transversely . The restriction p' _  pis of the plurisubharmonic defining function p of 2 to A is a negative subharmonic 15 0  F . FORSTNERIC function of class C on the complex manifold with boundary A . The Hopf lemma implies p (z) < -c dist(z, M), . z E A for some c > 0 . Here, dist denotes the Euclidean distance . Since -p(z) is proportional to the distance of z to bS _ ], we conclude that dist(z, M) is proportional to dist(z, bft) for z E A . Hence A intersects bf] transversely at each point of M . Thus the condition (1) fails and M is everywhere complex transverse . References 1 .  H . ALEXANDER, Polynomial approximation and hulls in sets of finite linear measure in C", Amer . J . of Math . 93 (1971), 65-74 . 2 .  E .M . CIRKA, "Complex Analytic Varieties," (Russian) Nauka, Moskva, 1985 . 3 .  E .M . 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WERMER, Banach Algebras and Several Complex Variables, SpringerVerlag, New York 1976 . Institut of Mathematics, Physics, and Mechanics Jadrauska 19 YU-61000 Ljub1jana, YUGOSLAVIA . Rebut el 11, de Desembre de 1987