We give a new proof of Makar-Limanov's theorem, which states that Russell's
hypersurface is not isomorphic to affine three space.Comment: 6 page

Hedén, Isac

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RUSSELL’S HYPERSURFACE FROM A GEOMETRIC POINT OF VIEW

HEDEN, ISAC

Osaka City University Repository

Hedén, I.Osaka J. Math.53 (2016), 637–644RUSSELL’S HYPERSURFACE FROMA GEOMETRIC POINT OF VIEWISAC HEDÉN(Received November 4, 2014, revised May 11, 2015)AbstractThe famous Russell hypersurface is a smooth complex affine threefold which isdiffeomorphic to a euclidean space but not algebraically isomorphic to the three di-mensional affine space. This fact was first established by Makar-Limanov, using al-gebraic minded techniques. In this article, we give an elementary argument whichadds a greater insight to the geometry behind the original proof and which also maybe applicable in other situations.1. IntroductionRussell’s hypersurfaceX WD {(x , y, z, t) 2 C4 j x C x2 y C z3 C t2 D 0} ,! C4,is one of the most prominent examples of an exotic variety, i.e. a variety which isdiffeomorphic to an affine space ([1], [6, Lemma 5.1]), but not isomorphic to it. Thelatter is an immediate consequence of Theorem 1 below, which states that there are notsufficiently many actions of the additive group Ga on X , and the aim of this paper isto give an elementary argument for this theorem. It includes some important elementsof the original proof, but gives a greater geometrical insight to the situation.The study of exotic varieties goes back to a paper of Ramanujam [16], wherea nontrivial example of a topologically contractible smooth affine algebraic surface Sover C is constructed. Ramanujam observed that S  C is diffeomorphic to C3, andasked whether this product is also isomorphic to C3. This was later proven not to bethe case, and thus the algebraic structure on C3 coming from S  C2 is exotic [17].Later on, many other exotic structures on C3 have been constructed, see e.g. the intro-duction of [18] for a list. Note also that there are no exotic structures on affine spacein dimension  2 [16].2010 Mathematics Subject Classification. 14R05, 14R20.This work was done as part of PhD studies at the Department of Mathematics, Uppsala Univer-sity; the support from the Swedish graduate school in Mathematics and Computing (FMB) is gratefullyacknowledged. Many thanks also go to Karl-Heinz Fieseler for his supervision. Finally, I am thank-ful to the reviewer for many valuable comments and references, especially concerning the historicalbackground of the problem.638 I. HEDÉNThe motivation for studying Russell’s hypersurface originally came from the lin-earization conjecture for C3, which claims that each Gm-action on C3 is linearizable.In the proof of this result by Koras and Russell, they described a list of smooth affinethreefolds diffeomorphic to C3 which contains all the potential counterexamples to theconjecture, and thus it was reduced to determining whether all of these so called Koras–Russell threefolds are exotic [13]. Kaliman and Makar-Limanov established exoticityfor some of them [10], and Russell’s hypersurface is the “most simple” among the re-maining ones. The difficulty with Russell’s hypersurface was that all the usual alge-braic and geometric invariants failed to distinguish it from C3. Makar-Limanov finallyestablished exoticity of Russell’s hypersurface (Theorem 1), and later on Kaliman andMakar-Limanov were able to prove exoticity of the remaining Koras–Russell threefolds[8, 9] as well, elaborating on Makar-Limanov’s methods. This confirmed the lineariza-tion conjecture [7].From now on, we will focus on Makar-Limanov’s result, stated in the followingtheorem.Theorem 1 (Makar-Limanov, [14]). The projection pr1 W X ! C, (x , y, z, t) 7! xis invariant with respect to any Ga-action on X.Some years after Makar-Limanov proved Theorem 1, Kaliman proved, using non-elementary birational geometry, that morphisms C3 ! C with generic fiber C2 can-not have any other fibers [5]. Since all the fibers of pr1 W X ! C are C2 except thezero fiber pr 11 (0), it follows also from Kaliman’s result that X 6 C3. In 2005, Makar-Limanov gave another proof of the exoticity of Russell’s hypersurface [15]; yet anotherproof was given by Derksen [3], and Crachiola also proved the exoticity in the posi-tive characteristic case [2]. The original proof of Theorem 1 used algebraic techniques,while we rather focus on a geometric approach using fibrations and quotient maps.An outline of our proof. In order to prove Theorem 1, we make use of an iso-morphism X  U  M with an open subset U of a blowup  W M ! C3, such thatD WD M nU is the strict transform of {0}  C2 ,! C3 andX  U  M ! C3is the map (x , y, z, t) 7! (x , z, t). That is, X is isomorphic to an affine modificationU of C3. The key-result is then that O(M)  O(X ) is invariant for any Ga-action onX . Since O(M)  O(C3), this allows us to conclude that for any given Ga-action onX , there is an induced Ga-action on C3 which makes  jX W X ! C3 equivariant. Then(U ) is obviously invariant, and it follows that its interior CC2 is invariant as well.Theorem 1 is obtained from this by observing that any Ga-action on C  C2 leavesthe first coordinate invariant: a nontrivial Ga-orbit is isomorphic to C, but there are nonon-constant morphisms from C to C.GEOMETRIC PERSPECTIVE ON RUSSELL’S HYPERSURFACE 6392. Russell’s hypersurface in a blowup of C3We recall the realization of Russell’s hypersurface as an affine modification of C3,see also [11, Example 1.5]. Let N ,! C2 D Spec(C[z, t]) denote the affine cuspidalcubic curve given byN WD {(z, t) 2 C2 j z3 C t2 D 0},and let I WD (g, h)  C[x , z, t] denote the ideal which is generated by the two relativelyprime polynomials g(x , z, t) D x2 and h(x , z, t) D x C z3 C t2. The zero set of I is{0}  N , and the blowupM WD BlI (C3)  {((x , z, t), [u W v]) 2 C3  P 1 j h(x , z, t)u C g(x , z, t)v D 0}of C3 along I is a hypersurface in C3  P 1 with singular locus of codimension two:Sing(M) D {0}  N  {[0 W 1]}. In particular, M is a normal variety.REMARK 2.1. Russell’s hypersurface is isomorphic to the open subset U of Mgiven by u ¤ 0, via the embedding X ,! M , (x , y, z, t) 7! ((x , z, t), [1 W y]).We denote the complement of U in M by D. Note that D is then given by u D 0,and that the image of U under the blowup morphism is (U ) D C C2 [ ({0}  N ).3. Additive group actions on Russell’s hypersurfaceIn order to see that O(M)  A WD O(X ) is invariant for every Ga-action on X , weshow the equivalent fact that O(M)  A is stable under every locally nilpotent deriva-tion W A ! A. This obviously holds for the trivial Ga-action on X , so we may assumethat  ¤ 0. The first step is to characterize O(M) in terms of a filtration on A.REMARK 3.1. With the filtrationAn WD OnD(M) D { f 2 C(M) j div( f )   nD} [ {0}we have A0 D O(M) D (O(C3)), so (O(C3)) is stable with respect to a locallynilpotent derivation  W A ! A if and only if (A0)  A0.In order to understand the above filtration, we treat A as a subset of C(x , z, t) andnote that multiplicities along D are simply multiplicities along {0}C2; so x , y, z andt have multiplicities 1,  2, 0 and 0, respectively. Now every element f 2 A n {0} canbe written in the formf DkXiD0yi pi (x , z, t),640 I. HEDÉNwhere each pi is at most linear in x for i  1 (and pk ¤ 0). ThusA D1MkD 1Akis a direct sum of free C[z, t]-modules Ak of rank 1 defined as(1) Ak WD8<:C[z, t]x jkj, if k  0,C[z, t]yl , if k D 2l > 0,C[z, t]xyl , if k D 2l   1 > 0,and one can check thatAn DMknAk .We thus obtain an explicit description of the associated graded algebraB WD Gr(A) DMn2ZBn with Bn WD An=An 1.It is generated by the elements gr(x) 2 B 1, gr(y) 2 B2, gr(z), gr(t) 2 B0 andW WD Spec(B)  {(x , y, z, t) 2 C4 j x2 y C z3 C t2 D 0}.In particular B0 D C[gr(z), gr(t)] ' C[z, t].REMARK 3.2. This grading was also used by M. Zaidenberg, see [18, Lemma 7.4].Let l D l() 2 Z be minimal with the property that (An)  AnCl for all n 2Z; the existence of such an l follows from the fact that both A0 and B are finitelygenerated graded algebras, and l ¤  1 since we consider a nontrivial Ga-action. Itfollows that  W A ! A induces a nontrivial homogeneous locally nilpotent derivationon B of degree l; we will denote it by Æ. With this notation it is enough to show thatl  0 in order to obtain (A0)  A0. In fact, more is true:Proposition 3.3. With B as above, any nontrivial locally nilpotent homogeneousderivation Æ W B ! B has degree l < 0.Before going into the proof, let us start with a discussion of the geometry of W ,!C4 and prove Lemma 3.6 below. As a hypersurface in C4, W is a normal varietysince its singular set Sing(W ) D {0}  C  {0}  {0} has codimension two. It admitstwo different group actions: the Ga-action ( , w) 7!  w corresponding to the locallyGEOMETRIC PERSPECTIVE ON RUSSELL’S HYPERSURFACE 641nilpotent derivation Æ W B ! B; and the Gm-action corresponding to the grading of B.The latter is given byGm  W ! W , (, (x , y, z, t)) 7! ( 1x , 2 y, z, t),and since B0 D C[z, t], the Gm-quotient morphism is given byp W W ! C2  Spec(C[z, t]), (x , y, z, t) 7! (z, t).It is trivial above C2 n N : the map(C2 n N ) Gm ! p 1(C2 n N ), ((z, t), ) 7! ( 1,  (z3 C t2)2, z, t),is a Gm-equivariant isomorphism with inversep 1(C2 n N )  ! (C2 n N ) Gm , (x , y, z, t) 7! ((z, t), x 1).As for N , we have p 1(N ) D F [ FC, where F and FCare the subsets of p 1(N )given by y D 0 and x D 0 respectively.REMARK 3.4. The set F consists exactly of the points w 2 W for whichlim!1w exists, and FCconsists exactly of the points w 2 W for which lim!0 wexists.REMARK 3.5. The above trivialization extends to a trivialization C2Gm ! W nFC, but for W n F there is no such trivialization since the Gm-isotropy group of apoint in FCn F has order 2.Now let us turn to the Ga-action Ga  W ! W, ( , w) 7!   w, corresponding toÆW B ! B. Since Æ is homogeneous of degree l, it is normalized by the Gm-action, i.e.for w 2 W ,  2 Ga and  2 Gm , we have( l )  (w) D (  w).In particular this implies that O is a Ga-orbit for any Ga-orbit O .Lemma 3.6. Let Æ W B ! B be a nontrivial locally nilpotent derivation, homo-geneous of degree l. Then either(1) l < 0 and FCis invariant, or(2) l > 0 and F is invariant.Proof. Since the locally nilpotent derivation ÆW B ! B is homogeneous, its kernelBÆ WD { f 2 B j Æ( f ) D 0} D { f 2 B j f (  w) D f (w), 8 2 Ga}642 I. HEDÉNis a graded subalgebra, i.e.:BÆ DMn2ZBÆn .Given f 2 Bk n {0}, we have Æ f 2 BÆkCl n {0} for a suitable  2 N. It follows that(1) if l D 0, we have BÆn ¤ {0} for all n ¤ 0,(2) if l > 0 we have BÆn ¤ {0} for some n > 0,(3) if l < 0 we have BÆn ¤ {0} for some n < 0.First assume that l > 0, so that BÆn ¤ 0 for some n, and let f 2 BÆn n {0}. Then fvanishes on F since f (x) D n f (x) and since lim!1x exists in W for x 2 F .It follows that F is invariant since it is an irreducible component of the invariant setV ( f )  W of dimension two. If l < 0, it follows analogously that FC W is invariant.It remains to show that l cannot be zero.If l D 0, both F and FCare invariant. So p 1(N ) is invariant and W n p 1(N ) aswell. Then for any nontrivial Ga-orbit O  W n p 1(N ) the map (z3C t2) Æ pjO has nozeros, and thus must be constant, say with value a 2 C, since O  C. However, anymorphism pjO W O ! V (C2I z3 C t2   a) from the complex line to the smooth affineelliptic curve V (C2I z3 C t2   a) is constant, so O is contained in a p-fiber. Sincep(O) 2 C2 n N , this p-fiber is isomorphic to Gm , as p is a Gm-principal bundle overC2n N . This gives a contradiction since C cannot be embedded into Gm .Proof of Proposition 3.3. By Lemma 3.6 it is enough to show show that F ,given by y D 0, is not invariant. Suppose to the contrary that F is invariant; thenits complement in W , given by y ¤ 0, is invariant as well. Since there is no non-constant invertible function on Ga-orbits, all Ga-orbits in W n F  are contained in levelhypersurfaces of y. In particular the hypersurface V  W which is given by y D 1 isinvariant and we have V ' {(x , y, z) 2 C3 j x2 C z3 C t2 D 0}. The restriction of theGm-quotient projection WD pjV W V ! C2, (x , z, t) 7! (z, t)is a two sheeted branched covering of C2 with branch locus   1(N )  N and decktransformation W V ! V , (x , z, t) 7! ( x , z, t),which is simply the action of  1 2Gm . In particular  (O) is a Ga-orbit for any Ga-orbitO . Assume for the moment that every nontrivial Ga-orbit intersects   1(N ) exactlyonce. Since the hypersurface V is a normal surface, there is a quotient map W V ! V==Ga WD Spec(O(V )Ga ),the generic fiber of which is a Ga-orbit [4, Lemma 1.1]. Thus the restriction j  1(N ) W   1(N ) ! V==GaGEOMETRIC PERSPECTIVE ON RUSSELL’S HYPERSURFACE 643is injective on a nonempty open subset of   1(N ). Hence, V==Ga being a smoothcurve, it follows from Zariski’s main theorem that this restriction is an open embed-ding. However, this is a contradiction since the affine cuspidal cubic curve   1(N )has a singular point.Finally, any nontrivial Ga-orbit O intersects   1(N ): otherwise  jO would be anon-constant morphism to an affine elliptic curve z3Ct2 D a for some a 2 C, which isimpossible. Note that a point in O \  1(N ) is a common point of the two Ga-orbits (O) and O , so  (O) D O . Choose an equivariant isomorphism C  O such that0 2 C corresponds to a point in   1(N ). Then the involution  W O ! O correspondsto C! C,  7!   , and as a consequence every nontrivial Ga-orbit O ,! V intersectsthe branch locus   1(N ) D W  (the fixed point set of  ) in exactly one point.References[1] A.D.R. Choudary and A. Dimca: Complex hypersurfaces diffeomorphic to affine spaces, KodaiMath. J. 17 (1994), 171–178.[2] A.J. Crachiola: The hypersurface x C x2 y C z2 C t3 D 0 over a field of arbitrary characteristic,Proc. Amer. Math. Soc. 134 (2006), 1289–1298.[3] H. Derksen: Constructive invariant theory and the Linearization problem, Ph.D. thesis, Univer-sity of Basel, 1997.[4] K.-H. Fieseler: On complex affine surfaces with CC-action, Comment. Math. Helv. 69 (1994),5–27.[5] S. Kaliman: Polynomials with general C2-fibers are variables, Pacific J. Math. 203 (2002),161–190.[6] S. Kaliman: Actions of C and CCon affine algebraic varieties; in Algebraic Geometry—Seattle 2005, Part 2, Proc. Sympos. Pure Math. 80, Part 2, Amer. Math. Soc., Providence, RI,2009, 629–654.[7] S. Kaliman, M. Koras, L. Makar-Limanov and P. Russell: C-actions on C3 are linearizable,Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 63–71.[8] S. Kaliman and L. Makar-Limanov: On the Russell–Koras contractible threefolds, J. AlgebraicGeom. 6 (1997), 247–268.[9] S. Kaliman and L. Makar-Limanov: AK-invariant of affine domains; in Affine Algebraic Geom-etry, Osaka Univ. Press, Osaka, 2007, 231–255.[10] S. Kaliman and L. Makar-Limanov: Affine algebraic manifolds without dominant morphismsfrom Euclidean spaces, Rocky Mountain J. Math. 27 (1997), 601–610.[11] S. Kaliman and M. Zaidenberg: Affine modifications and affine hypersurfaces with a very tran-sitive automorphism group, Transform. Groups 4 (1999), 53–95.[12] M. Koras and P. Russell: On linearizing “good” C-actions on C3; in Group Actions and In-variant Theory (Montreal, PQ, 1988), CMS Conf. Proc. 10, Amer. Math. Soc., Providence, RI,1989, 93–102.[13] M. Koras and P. Russell: Contractible threefolds and C-actions on C3, J. Algebraic Geom. 6(1997), 671–695.[14] L. Makar-Limanov: On the hypersurface x C x2 y C z2 C t3 D 0 in C4 or a C3-like threefoldwhich is not C3, Israel J. Math. 96 (1996), 419–429.[15] L. Makar-Limanov: Again x C x2 y C z2 C t3 D 0; in Affine Algebraic Geometry, Contemp.Math. 369, Amer. Math. Soc., Providence, RI, 2005, 177–182.644 I. HEDÉN[16] C.P. Ramanujam: A topological characterisation of the affine plane as an algebraic variety,Ann. of Math. (2) 94 (1971), 69–88.[17] M. Zaı˘denberg: An analytic cancellation theorem and exotic algebraic structures on Cn , n  3,Astérisque 217 (1993), 251–282.[18] M. Zaı˘denberg: Exotic algebraic structures on affine spaces, St. Petersburg Math. J. 11 (2000),703–760.Mathematisches InstitutUniversität BaselRheinsprung 214051 BaselSwitzerlande-mail: Isac.Heden@unibas.ch

RUSSELL'S HYPERSURFACE FROM A GEOMETRIC POINT OF VIEW

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Russell's hypersurface from a geometric point of view

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