ADDITIVE GROUP ACTIONS ON DANIELEWSKI VARIETIES AND THE CANCELLATION PROBLEM

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ADDITIVE GROUP ACTIONS ON DANIELEWSKI VARIETIES AND THE CANCELLATION PROBLEM ADRIEN DUBOULOZ Prepublication de l'Institut Fourier no 680 (2005) www-fourier.ujf-grenoble.fr/prepublications.html Abstract. The cancellation problem asks if two complex algebraic varieties X and Y of the same dimension such that X ? C and Y ? C are isomorphic are isomorphic. Iitaka and Fujita [15] established that the answer is positive for a large class of varieties of any dimension. In 1989, Danielewski [4] constructed a famous counter-example using smooth affine surfaces with additive group actions. His construction was further generalized by Fieseler [10] and Wilkens [22] to describe a larger class of affine surfaces. Here we construct higher dimensional analogues of these surfaces. We study algebraic actions of the additive group C+ on certain of these varieties, and we obtain counter-examples to the cancellation problem in every dimension n ≥ 2. Keywords: Danielewski varieties, Cancellation Problem, additive group actions, Makar- Limanov invariant. Resume. Le probleme dit de simplification demande si deux varietes algebriques complexes X et Y telles X ?C et Y ?C soient isomorphes sont isomorphes. Iitaka et Fujita ont montre a la fin des annees 70 que la reponse est affirmative pour une large classe de varietes. Les varietes affines-reglees ne font pas partie de cette classe, et, en 1989, Danielewski a construit un contre-exemple a partir de deux surfaces affines de ce type.

  • additive group

  • danielewski varieties

  • makar limanov invariant

  • bundle over

  • actions de groupes additifs

  • then x?1i


Publié le : mardi 19 juin 2012
Lecture(s) : 41
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Source : www-fourier.ujf-grenoble.fr
Nombre de pages : 15
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ADDITIVE GROUP ACTIONS ON DANIELEWSKI VARIETIES AND THE CANCELLATION PROBLEM ADRIEN DUBOULOZ
PrepublicationdelInstitutFouriern o 680 (2005) www-fourier.ujf-grenoble.fr/prepublications.html Abstract. The cancellation problem asks if two complex algebraic varieties X and Y of the same dimension such that X  C and Y  C are isomorphic are isomorphic. Iitaka and Fujita [15] established that the answer is positive for a large class of varieties of any dimension. In 1989, Danielewski [4] constructed a famous counter-example using smooth ane surfaces with additive group actions. His construction was further generalized by Fieseler [10] and Wilkens [22] to describe a larger class of ane surfaces. Here we construct higher dimensional analogues of these surfaces. We study algebraic actions of the additive group C + on certain of these varieties, and we obtain counter-examples to the cancellation problem in every dimension n  2. Keywords : Danielewski varieties, Cancellation Problem, additive group actions, Makar-Limanov invariant.
Resume. Leproblemeditdesimpli cationdemandesideuxvarietesalgebriquescomplexes X et Y telles X  C et Y  C soientisomorphessontisomorphes.IitakaetFujitaontmontre ala ndesannees70quelareponseestarmativepourunelargeclassedevarietes.Les varietesanes-regleesnefontpaspartiedecetteclasse,et,en1989,Danielewskiaconstruit uncontre-exempleapartirdedeuxsurfacesanesdecetype.Danscetarticle,ongeneralise laconstructiondeDanielewskipourobtenirdesvarietesanesquisontlesespacestotauxde br es principaux sous le groupe additif, de base un schema non separe, en l’occurrence, un espaceanedontleshyperplansdecoordonnesonetemultiplies.Graˆceaunetechniquede deformationequivariantedeveloppeeparKalimanetMakar-Limanov,ondetermineensuite touteslesactionsdegroupesadditifssurcertainesdecesvarietes.Celaconduit nalementa desgeneralisationsnaturellesducontre-exampledeDanielewski,valablesentoutedimension n  2. Mots clefs :varietesdeDanielewski,ProblemedeSimpli cation,groupesadditifs,invariant de Makar-Limanov. Mathematics Subject Classi cation (2000) : 14R10,14R20.
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2 ADRIEN DUBOULOZ Introduction The Cancellation Problem, which is sometimes referred to as Zariski’s Problem although Zariskisoriginalquestionwasdi erent(seee.g.[21]),hasbeenalreadydiscussedintheearly seventiesasthequestionofuniquenessofcoecientsrings.Theproblematthattimewas to decide for which rings A and B an isomorphism of the polynomials rings A [ x ] and B [ x ] implies that A and B are isomorphic (see e.g. [8]). Using the fact that the tangent bundle of the real n -sphere is stably trivial but not trivial, Hochster [13] showed that this fails in general. A geometric formulation of the Cancellation Problem asks if two algebraic varieties X and Y such that Y  A 1 is isomorphic to X  A 1 are isomorphic. Clearly, if either X or Y does not contain rational curves, for instance X or Y is an abelian variety, then every isomorphism : X  A  Y  A 1 induces an isomorphism between X and Y . So the Cancellation Problem leads to decide if a given algebraic variety X contains a family of rational curves, where by a rational curve we mean the image of a nonconstant morphism f : C X , where C is isomorphic to A 1 or P 1 . Iitaka and Fujita carried a geometric attack to this question using ideas from the classi cation theory of complete varieties. Every complex algebraic variety X embeds as an open subset of complete variety  X for which the boundary D =  X \ X is a divisor with normal crossing. By replacing the usual sheaves of forms q  X on  X by the sheaves q (log D ) of rational q -forms having at worse logarithmic poles along D , Iitaka [14] introduced, among others invariants, the notion of logarithmic Kodaira dimension   ( X ) of a noncomplete variety X , which is an analogue of the usual notion of Kodaira diamension for complete varieties. Iitaka anf Fujita [15] established the following result. Theorem. Let X and Y be two nonsingular algebraic varieties and assume that either  ( X )  0 or   ( Y )  0 . Then every isomorphism  : X  C  Y  C induces an iso-morphism between X and Y . The hypothesis   ( X )  0 above guarantees that X cannot contain too many rational curves. For instance, there is no cylinder-like open subset U ' C  A 1 in X , for otherwise we would have   ( X ) = 1 . It turns out that this additional assumption is essential, as shown by the following example due to Danielewski [4]. Example. The surfaces S 1  S 2  C 3 with equations xz  y 2 + 1 = 0 and x 2 z  y 2 + 1 = 0 are not isomorphic but S 1  C and S 2  C are . In the construction of Danielewski, these surfaces ˜ appear as the total spaces of principal homogeneous C + -bundles over A , the ane line with a double origin, obtained by identifying two copies of A 1 along A 1 \ { 0 } . The isomorphism S 1  C ' S 2  C isobtainedbyformingthe berproduct S 1  ˜ A S 2 , which is a principal C + -bundle over both S 1 and S 2 ,andusingthefactthateverysuchbundleoveranane variety is trivial. On the other hand, S 1 and S 2 are not even homeomorphic when equipped with the complex topology. More precisely, Danielewski established that the fundamental groups at in nit y of S 1 and S 2 are isomorphic to Z / 2 Z and Z / 4 Z respectively. Fieseler [10] studied and classi ed algebraic C + -actions on normal ane surfaces. As a consequence of his classi cation, he obtained many new examples of the same kind (see also [22]). Here we construct higher dimensional analogues of Danielewski’s counter-example. The paper is organized as follows. In the rst section, we introduce a natural generalization of 1 Actually,anonsingularanesurfacehaslogarithmicKodairadimension if and only if its contains a cylinder-like open set (see e.g. [20]).
ADDITIVE GROUP ACTIONS ON DANIELEWSKI VARIETIES AND THE CANCELLATION PROBLEM 3 the surfaces S 1 and S 2 aboveintheformofanevarietieswhicharethetotalspacesof ˜ certain principal homogeneous C + -bundle over A n , the ane n -space with a multiple system of coordinate hyperplanes. We call them Danielewski varieties . For instance, for every multi-index [ m ] = ( m 1  . . .  m n ) Z n> 0 the nonsingular hypersurface X [ m ]  C n +2 with equation x 1 m 1    x nm n z = y 2  1 is a Danielewski variety. As a generalization of a result of Danielewski (see also [10]), we establish that the total space of a principal homogeneous C + -bundle over ˜ A n is a Danielewski variety if and only if it is separated. This leads to a simple description of these varieties in terms of Cech cocycles (see Theorem 1.18). In a second part, we study algebraic C + -actions on a certain class of varieties which con-tains the Danielewski varieties X [ m ] as above. In particular we compute the Makar-Limanov invariant [17] of these varieties, i.e. the set of regular functions invariant under all C + -actions. We obtain the following generalization of a result due to Makar-Limanov [19] for the case of surfaces (see Theorem 2.8 ). Theorem. If ( m 1  . . .  m n ) Z n> 1 then the Makar-Limanov invariant of a variety X  C n +2 with equation r  1 x 1 m 1    x m n z = y r + X a i ( x 1  . . .  x n ) y i where r  2 i =0 is isomorphic to C [ x 1  . . .  x n ] . Asaconsequence,weobtainin nitefamiliesofcounter-examplestotheCancellationProblem in every dimension n  2. Theorem. Let [ m ] = ( m 1  . . .  m n ) Z n> 1 and [ m 0 ] = ( m 0 1  . . .  m 0 n ) Z n> 1 be two multi-indices for which the subsets { m 1  . . .  m n } and { m 0 1  . . .  m 0 n } of Z are distint, and let  1  . . .   r , where r  2 be a collection of pairwise distinct complex numbers. Then the Danielewski va-rieties X and X 0 in C n +2 with equations r r x 1 m 1    x nm n z  Y ( y   i ) = 0 and x 1 m 0 1    x nm 0 n z  Y ( y   i ) = 0 i =1 i =1 are not isomorphic, but the varieties X  C and X 0  C are isomorphic.
Acknowledgement. The author is very grateful to Shulim Kaliman for valuable discussions on the art of computing Makar-Limanov invariants using weigth degree functions.
1. Danielewski varieties Danielewski’s construction can be easily generalized to produce examples of ane varieties X and Y such that X  C and Y  C areisomorphic.Indeed,ifwecanequiptwoanevarieties X and Y with structures of principal homogeneous C + -bundle  X : X Z and  Y : Y Z over a certain scheme Z , then the b er product X  Z Y will be a principal homogeneous C + -bundle over X and Y , whence a trivial principal bundle X  C ' X  Z Y ' Y  C as X and Y are both ane. The base scheme Z which arises in Danielewski’s counter-example is the ane with a double origin. The most natural generalization is to consider an anespace C n with a multiple system of coordinate hyperplanes as a base scheme.
4 ADRIEN DUBOULOZ Notation 1.1 . In the sequel we denote the polynomial ring C [ x 1  . . .  x n ] by C [ x ], and the al-gebra C x 1  x  11 . . .  x n  x  n 1 of Laurent polynomials in the variables x 1  . . .  x n by C x x  1 . For every multi-index [ r ] = ( r 1  . . .  r n ) Z n , we let x [ r ] = x r 1 1    x rn n C x x  1 . We denote by H x = V ( x 1    x n ) the closed subvariety of C n consisting of the union of the n coordinate hyperplanes. Its open complement in C n , which is isomorphic to ( C  ) n , will be denoted by U x . De nition 1.2. We let Z n,r be the scheme obtained by gluing r copies  i : Z i   C n of the ane space C n = Spec ( C [ x 1  . . .  x n ]) by the identity along ( C  ) n . We call Z n,r the ane n -space with an r -fold system of coordinate hyperplanes. We consider it as a scheme over C n via the morphism  : Z n,r C n restricting to the  i ’s on the canonical open subset Z i of Z n,r , i = 1  . . .  n . 1.3. We recall that a principal homogeneous C + -bundle over a base scheme S is an S -scheme  : X S equipped with an algebraic action of the additive group C + , such that there exists an open covering U = ( S i ) i I of S for which   1 ( S i ) is equivariantly isomorphic to S i  C , where C + acts by translations on the second factor, for every i I . In particular, the total space of a principal homogeneous C + -bundle has the structure of an A 1 -bundle over S . The set H 1 ( S C + ) of isomorphism classes of principal homogeneous C + -bundles over S is isomorphic to the rst cohomology group  H 1 ( S O S ) ' H 1 ( S O S ). De nition 1.4. A Danielewski variety is an ane variety of dimension n  2 which is the total space  : X Z n,r of a principal homogeneous C + -bundle over Z n,r for a certain r  1. Example 1.5. The Danielewski surfaces S 1 = xz  y 2 + 1 = 0 and S 2 = x 2 z  y 2 + 1 = 0 above are Danielewski varieties. Indeed, the projections pr x : S i C , i = 1 2, factor through structural morphisms  i : S i Z 2 , 1 of principal C + -bundlesovertheanelinewithadouble origin. More generally, the Makar-Limanov surfaces S  C 3 with equations x n z  Q ( x y ) = 0, where n  1 and Q ( x y ) is a monic polynomial in y , such that Q (0  y ) has simple roots are Danielewski varieties. Remark 1.6 . The scheme Z n,r over which a Danielewski variety X becomes the total space of a principal homogeneous C + -bundle is unique up to isomorphism. Indeed, we have necessarily n = dim Z = dim X  1. On the other hand, it follows from 1.7 below that X is obtained by gluing r copies of C n  C along ( C  ) n  C . So we deduce by induction that H n +1 ( X Z ) is isomorphic to the direct sum of r copies of H n (( C  ) n  C Z ) ' H n (( C  ) n Z ) ' Z , whence to Z r . Therefore, if X admits another structure of principal homogeneous C + -bundle  0 : X Z n 0 ,r 0 then ( n 0  r 0 ) = ( n r ). However, we want to insist on the fact that this does not imply that the structural morphism  : X Z n,r on a Danielewski variety is unique, even up to automorphisms of the base . This question will be discussed in 1.13 below. 1.7. A principal homogeneous C + -bundle  : X Z n,r becomes trivial on the canonical open covering U of Z n,r be means of the open subsets Z i ' C n , i = 1  . . .  r (see de nition 1.2 above). So there exists a Cech 1-cocycle r g = { g ij } i,j =1 ,...,g C 1 U O Z n,r ' M C x x  1 i =1 representing the isomorphism class [ g ] H 1 Z n,r O Z n,r '  H 1 U O Z n,r of X such that X is equivariantly isomorphic to the scheme obtained by gluing r copies Z i  C = Spec ( C [ x ] [ t i ]) of
ADDITIVE GROUP ACTIONS ON DANIELEWSKI VARIETIES AND THE CANCELLATION PROBLEM 5 C n  C , equipped with C + -actions by translations on the second factor, outside H x  C  Z i  C by means of the equivariant isomorphisms ij : Z j \ H x  C   Z i \ H x  C ( x t j ) 7→ x t j + g ij x x  1   i 6 = j.

X Z 2 , 2
H x C 2 Figure 1.1. A Danielewski threefold X . 1.8. Since a Danielewski variety X is ane, the corresponding transition cocycle is not arbitrary. For instance, the trivial cocycle corresponds to the trivial C + -bundle Z n,r  C which is not even separated if r  2. More generally, if one of the rational functions g ij n is regular at a point  = (  1  . . .   n ) H x  C n , then for every germ of curve C  C intersecting H x transversely in  , (    )  1 ( C )  X is a nonseparated scheme. On the other hand, Danielewski established that the total space of a principal homogeneous C + -bundle  : X Z n, 2 de nedbyacocycle g 12 = x  [ r ] a ( x ), where [ r ] Z  n 1 , such that a ( x ) C x ] + x [ r ] C [ x ] = C x ] is ane, isomorphic to the variety X  C n +2 with equation x [ r ] z  y 2  a ( x ) y = 0. More generally, we have the following result. Theorem 1.9. For the total space of a principal C + -bundle  : X Z n,r de ne d by a transition cocycle g = g ij x x  1  i,j =1 ,...,r the following are equivalent. (1) For every i 6 = j , g ij = x  [ m ij ] a ij ( x ) for a certain multi-index [ m ij ] Z n> 0 and a polynomial a ij ( x ) such that a ij ( x ) C [ x ] + x (1 ,..., 1) C [ x ] = C x ] , (2) X is separated (3) X is ane. Proof. We deduce from I.5.5.6 in [12] that X is separated if and only if g ij C x x  1 generates C x x  1 as a C x ]-algebra for every i 6 = j . Letting g ij = x  [ m ] a ( x ), where [ m ] Z  n 0 and where a ( x ) C x ], this is the case if and only if x  [ m ] generates C x x  1 as a C x ]-algebra and a ( x ) C [ x ] + x [ m ] C x ] = C [ x ]. Indeed, the condition is sucien t as it guarantees that C x x  1 = C [ x ] x  [ m ]  C x ] [ g ij ]. Conversely, if C x x  1 = C x ] [ g ij ] then g ij = x  [ m ] a ( x ) for a certain multi-index [ m ] = ( m 1  . . .  m n ) Z  n 1 and a polynomial a C x ] not divisible by x i for every i = 1  . . .  r . Indeed, if there exists an indice i such that m i  0 then x i  1 6∈ C [ x ] [ g ij ] which contradicts our hypothesis. Furthermore, since
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