D. Barlet, Espace analytique r'eduit des cycles analytiques complexes compacts d'un espace analytique complex de dimension finie, in Fonctions de Plusieurs Variables Complexes II (S'em. F. Norguet 74/75), vol. 482 of Lecture Notes in Mathematics, Sprinver-Verlag, 1975, pp. 1--158.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(and perhaps obstructing) our analysis. These deformation methods may involve comparing cycles from different families. Chow varieties provide a canonical place for such comparisons. For an elaboration of the properties of Chow varieties, see [8] and the references contained therein. Also [2] contains a discussion of Chow varieties in the analytic category, which suffices for our purposes. Positive cycles on X of a fixed dimension and degree are parameterized by a Chow variety of X. We suppress the dependence on dimension and degree and write ChowX for any Chow variety of X. The open ....

D. Barlet, Espace analytique r'eduit des cycles analytiques complexes compacts d'un espace analytique complex de dimension finie, in Fonctions de Plusieurs Variables Complexes II (S'em. F. Norguet 74/75), vol. 482 of Lecture Notes in Mathematics, Sprinver-Verlag, 1975, pp. 1--158.

....monoid d0 C r;d (Y ) We provide C r (Y ) with the analytic topology; so defined, the topological abelian monoid C r (Y ) is independent of the projective embedding of Y . Indeed, the algebraic structure of C r (Y ) does not depend upon the projective embedding of Y as shown by D. Barlet [B]. If V ae Y is a Zariski open subset of a projective variety Y with complement Y1 = Y Gamma V , then we define C r (V ) to be the quotient topological monoid C r (Y ) C r (Y 1 ) so defined C r (V ) is independent of the projective closure V ae Y (cf. LiF] F G] We begin with what we ....

D. Barlet, Espace analytique r'eduit des cycles analytiques complexes compacts d'un espace analytique complexe de dimension finite, Fonctions de plusieurs variables, II, Lecture Notes in Math 482, Spring-Verlag, 1975, pp. 1-158.

.... once a projective embedding of Y is chosen, then one has Chow Varieties C j;d (Y ) of effective j cycles on Y of degree d (for integers j; d 0) and one considers the Chow monoid C j (Y ) 1 a d=0 C j;d (Y ) whose isomorphism type is independent of the choice of projective embedding of Y [B]. We provide C j (Y ) with the analytic topology and form its naive group completion Z j (Y ) whose homotopy type is that of the homotopy theoretic group completion of the topological monoid C j (Y ) cf. LiF] F G] The underlying discrete group Z j (Y ) disc of Z j (Y ) is the group of ....

D. Barlet, Espace analytique r'eduit des cycles analytiques complexes compacts d'un espace analytique complexe de dimension finite, Fonctions de plusieurs variables, II, Lecture Notes in Math 482, Spring-Verlag, 1975, pp. 1-158.

....P N , then C r;d Y denotes the Chow variety whose rational points are the effective r cycles on Y of degree d. We shall consider the Chow monoid C r Y j d0 C r;d Y of all effective r cycles on Y , a monoid whose isomorphism type is independent of the projective embedding Y ae P N (cf. [B]) We are grateful to I.H.E.S. I.A.S. and Rutgers University for their hospitality during the writing of this paper. x1 Continuous algebraic maps One is naturally led to consider continuous algebraic maps to Chow varieties when one is confronted with their construction in terms of elimination ....

D. Barlet, Espace analytique r'eduit des cycles analytiques complexes compacts d'un espace analytique complexe de dimension finie, Lectures Notes in Math 482 (1981).

....or pluripolarity. Such examples really occur, see [Ka 2] La] In the end of this paper in x4 we give some open questions. 1. Continuity principle. 1.1. Cycle space associated to a meromorphic map. We shall need some notions and results from the theory of cycle spaces developed by D. Barlet, see [Ba]. Partially recalling thouse facts we adapt them to our situation. For the english spelling of the Barlet terminology we refer to [F] Recall that an analytic cycle of dimension k in complex space Y is a formal sum Z = P j n j Z j , where fZ j g is a locally finite sequence of analytic subsets ....

.... Sym d ( Delta q ) such that Phi(s 0 ; Delta) represents Z for some s 0 2 S. Then for some neighborhood S 0 3 s 0 V is adapted to all Z s ; s 2 S 0 . In particular this defines a map Psi : S 0 Theta U Sym d 1 (B) Theorem (D. Barlet) Mapping Psi is holomorphic. For the proof see [Ba], Chapitres I and II. Here finite dimensionality and normality of S are essential. Withough loss of generality we suppose that our mapping f is defined on Delta n Theta A k (r; b) with b 1. Now each Z 2 C f;C can be covered by a finite number of adapted neighborhoods (V ff ; j ff ) Their ....

Barlet D.: Espace analytique reduit des cycles analytiques complexes compacts d'un espace analytique complexe de dimension finie . Semiar Norguet IX, Lect. Notes Math., 482, 1-157, (1975).

....then Xij C C is flat and the canonical map of the Hilbert scheme to the Chow variety [37, x5.4] shows OEj C is a morphism. By Hartogs Theorem on separate analyticity, OE is in fact a morphism. For a discussion of Chow varieties in the analytic category (which suffices for our purposes) see [2]. Any positive cycle Y on X is rationally equivalent to a positive integral linear combination of Schubert cycles. This rational equivalence occurs within the closure of B Delta Y in ChowX since B stable cycles of X (B fixed points in B Delta Y ) are integral linear combinations of Schubert ....

D. Barlet, Espace analytique r'eduit des cycles analytiques complexes compacts d'un espace analytique complex de dimension finie, in Fonctions de Plusieurs Variables Complexes II (S'em. F. Norguet 74/75), vol. 482 of Lecture Notes in Mathematics, Springer-Verlag, 1975, pp. 1--158.

....of Lawson homology one needs the homotopy type independent of the embedding i. Hoyt [4] proved the homeomorphism type of d0 C r;d (X; i) to be independent of embeddings. The purpose of this note is to clarify and straighten out Hoyt s approach for arbitrary fields. In fact by Barlet [1] one knows that C r;d (X; i) is independent of i up to isomorphism of algebraic sets in case of K = C . On the other hand Nagata [5] shows that the Chow variety of 0 dimensional cycles of degree 2 in P 3 Z 2 is not normal, but there is an embedding into some P n Z 2 such that the Chow variety ....

D. Barlet, Espace analytique r'eduit des cycles analytiques complexes compacts d' un espace analytique complexe de dimension finie, Fonctions de Plusieurs Variables Complexes II (S'eminaire F. Norguet), LNM 482. Springer--Verlag, Berlin, Heidelberg, New York (1975), 1 - 158.

....can calculate the Chow form of a suspended cycle (Prop. 2.2) from the original Chow form. Thus suspension is an algebraic map. To prove that the join pairing is algebraic (Thm. 2. 3) hence continuous, one just has to combine the continuity of the complex suspension with classical results (Barlet [1]) Independently Barlet [2] obtained a proof of Thm. 2.3 using different methods. In [4] the join pairing was used to define an infinite loop space structure on K(Z;even) which is compatible with the infinite loop space structure on BU induced by Whitney sum. The required continuity checks ....

....able to define a map which will prove the second part of the complex suspension theorem: 3. 10 Definition: Psi : 1 ffi ffi ( Sigma= Phi) C r 1;d (C P n 1 ) Theta Div 0 k Theta [0; 1] Gamma C r 1;dk (C P n 1 ) where denotes the intersection of cycles in the sense of Barlet [1]. We want to check that Psi is well defined: First we have Phi(Div 0 k ; 0; 1] ae Div 0 k . Each irreducible component of a cycle in Sigma=C r 1;d (C P n 1 ) meets the point z 0 , and z 0 is not contained in the support of any element in Div 0 k . Therefore a pair of cycles in ....

D. Barlet, Espace analytique r'eduit des cycles analytiques complexes compacts d' un espace analytique complexe de dimension finie, Fonctions de Plusieurs Variables Complexes II (S'eminaire F. Norguet), LNM 482. Springer--Verlag, Berlin, Heidelberg, New York (1975), 1 - 158.

....(and perhaps obstructing) our analysis. These deformation methods may involve comparing cycles from different families. Chow varieties provide a canonical place for such comparisons. For an elaboration of the properties of Chow varieties, see [8] and the references contained therein. Also [2] contains a discussion of Chow varieties in the analytic category, which suffices for our purposes. Positive cycles on X of a fixed dimension and degree are parameterized by a Chow variety of X. We suppress the dependence on dimension and degree and write ChowX for any Chow variety of X. The open ....

D. Barlet, Espace analytique r'eduit des cycles analytiques complexes compacts d'un espace analytique complex de dimension finie, in Fonctions de Plusieurs Variables Complexes II (S'em. F. Norguet 74/75), vol. 482 of Lecture Notes in Mathematics, Sprinver-Verlag, 1975, pp. 1--158.

....set A in Cn (Z) See [Barlet 1978a, Prop. 1] 1 Multiplicities are counted as follows: locally we can assume that Z # U Y where U and Y are open polydiscs in C n and C p , such that Cs 0 #U #Y = because Cs 0 # Y is finite (compare to the definition of ecaille adapte in [Barlet 1975, Chapter 1] Then Cs 0 defines a branched coverings of U via the projection U Y # U and we have the following classification theorem for degree k branched coverings in such a situation [Barlet 1975, Chapter 0] There exists a natural bijection between degree k branched coverings of U in U Y ....

.... #U #Y = because Cs 0 # Y is finite (compare to the definition of ecaille adapte in [Barlet 1975, Chapter 1] Then Cs 0 defines a branched coverings of U via the projection U Y # U and we have the following classification theorem for degree k branched coverings in such a situation [Barlet 1975, Chapter 0] There exists a natural bijection between degree k branched coverings of U in U Y and holomorphic maps f : U # Sym k Y . So if Cs 0 corresponds to f and Y is t 0 Y in Z, the intersection Cs 0 # Y is the k uple f(t 0 ) HOW TO USE THE CYCLE SPACE IN COMPLEX GEOMETRY 27 ....

[Article contains additional citation context not shown here]

D. Barlet, "Espace analytique reduit des cycles analytiques complexes compacts d'un espace analytique complexe de dimension finie", pp. 1--158 in Fonctions de plusieurs variables complexes II, edited by F. Norguet, Lecture Notes in Math. 482, Springer, Berlin, 1975.

No context found.

Barlet, D. , Espace analytique r'eduit des cycles analytiques complexes compacts d'un espace analytique complex de dimension finie. pages 1-158 in Fonctions de plusieurs variables complexes II. (S'eminaire F. Norguet 74/75). Lectures Notes in Math 482, Springer-Verlag, 1975.

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