A1-homotopy theory

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A ¹ -homotopy theory

The Koras-Russell cubic threefold Sabrina Pauli

Master’s Thesis, Spring 2017

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This master’s thesis is submitted under the master’s programmeMathematics, with programme option Mathematics, at the Department of Mathematics, University of Oslo. The scope of the thesis is 60 credits.

The front page depicts a section of the root system of the exceptional Lie group E8, projected into the plane. Lie groups were invented by the Norwegian mathematician Sophus Lie (1842–1899) to express symmetries in differential equations and today they play a central role in various parts of mathematics.

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Abstract

We study the Koras-Russell cubic threefold, i.e., the hypersurface in A⁴_C given by the equation x²y+x+z²+t³= 0. We prove that the Koras-Russell cubic threefold is not diffeomorphic to Euclidean space as a smooth manifold but isomorphic to affine three space as a complex algebraic variety. Furthermore, we prove that the Koras-Russell cubic threefold isA¹-contractible. Then we study other examples of affine varieties with similar properties.

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Acknowledgements

I would like to thank my advisor, Professor Paul Arne Østvær, for the wonderful topic and guidance throughout the last year. I thank Glen Wilson for the helpful discussions, proofreading and interest in my thesis. Furthermore, I would like to thank Adrien Dubouloz for the new examples I include in this thesis as well as for helpful discussions. I also thank Patrick Holzer and Celia Hacker for proofreading and Gard Helle for making sure that I have checked all the details, thus making me a better mathematician. Last but not least, my parents deserve a thank you for the last 25 years. Thanks for always believing in me and supporting me.

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Chapter 1 Introduction

This thesis discusses properties of the hypersurfaceXinA⁴_Cgiven by the equationx+x²y+z²+ t³ = 0, the so called Koras-Russell cubic threefold. The main result is that the Koras-Russell cubic threefoldX isA¹-contractible. Here is a short explanation of that term: InA¹-homotopy theory or motivic homotopy theory we apply techniques from algebraic topology to algebraic geometry. The affine line A¹ takes the role of the unit interval I = [0,1]. Like in algebraic topology we can defineA¹-homotopies andA¹-contractibility (see Definition 3.4.2). We give an introduction to A¹-homotopy theory in Chapter 3.

We want to explain why the Koras-Russell cubic threefold is an interesting object to study and why one is interested in its homotopy type.

1.1 The history of the Koras-Russell cubic threefold

The Koras-Russell cubic threefold X was first introduced by Koras and Russell in [KR89] as a potential counterexample to the Linearization Problem. Makar-Limanov showed in [ML96]

thatX is not isomorphic toA³_C and hence not a counterexample. However, one does not know whether the cylinder over the Koras-Russell cubic threefold X×A¹ is isomorphic to A⁴_C and therefore, the Koras-Russell cubic threefold is a potential counterexample to the Cancellation Problem.

1.1.1 The Linearization Problem for C^∗-actions

Let C^∗ be the algebraic group Spec(C[x^±1]) equipped with the multiplicative operation. An algebraic group action of C^∗ on Aⁿ_C given byσ :C^∗ → Aut(Aⁿ_C) is linear if it factors through Gln(C). It is linearizable if σ is conjugate to a linear action, i.e., there is an automorphism φ on Aⁿ_Csuch that for any λ∈C^∗ the mapφ◦σ(λ)◦φ⁻¹ is linear.

The Linearization Problem forC^∗-actions asks whether everyC^∗-action onAⁿ_Cis linearizable.

It was proven for the case n = 2 by Gutwirth in [Gut62] already in 1962. However, the case n= 3was a bit more complicated to prove. Note that the fixed point set of a linear action onAⁿ_C is a nonempty linear subspace of Aⁿ_C. It follows that linearizable actions have a connected fixed point set. Now consider the Koras-Russell cubic threefold X ={x+x²y+z²+t³ = 0} ⊂ A⁴_C and theC^∗-action onX

σ:C^∗×X →X,(λ,(x, y, z, t))7→(λ⁶x, λ⁻⁶y, λ³z, λ²t).

Ifσwere linearizable, the restriction ofσto the subgroupC₆⊂C^∗of sixth roots of unity would also be linearizable, i.e., conjugate to a linear action, and hence, the fixed point set would be connected. However, the fixed point set of the restriction is isomorphic to{x(1 +xy) = 0} ⊂A²_C

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which is not connected. So if X were isomorphic to A³_C, the algebraic group action σ would be a counterexample to the Linearization Problem for group actions by the algebraic group C^∗. In [KR97] Koras and Russell classified all threefolds which would give a counterexample to the Linearization Problem for C^∗-actions, namely all smooth contractible affine algebraic threefolds with a hyperbolic C^∗-action, i.e., a C^∗-action with exactly one fixed point o such that the induced linear action on the tangent space To ∼= A³_C at o is given by C^∗×To → To, (λ,(x, y, z))7→(λ^−ax, λ^by, λ^cz) for positive integers a, b, c (see [KKMLR97, §1] or [Kal09, §4]).

These threefolds are now calledKoras-Russell threefolds and can be divided into three types:

1. A Koras-Russell threefold of the first kind is a hypersurface in A⁴_C given by the equation x+xⁿy+z^r+t^s= 0withn, r, s≥2,s > randgcd(r, s) = 1. Note that the Koras-Russell cubic threefold X is a Koras-Russell threefold of the first kind given by n = 2 =r and s= 3.

2. AKoras-Russell threefold of the second kind is a hypersurface inA⁴_Cgiven by the equations x+ (xⁿ+z^k)^ly+t^s= 0 withn, k, s≥2,s > k,l≥1 andgcd(k, s) = 1 = gcd(k, n).

3. We call any other smooth contractible affine algebraic threefold with a hyperbolic C^∗- actionKoras-Russell threefold of the third kind.

In [KKMLR97] Koras and Russell prove together with Makar-Limanov and Kaliman that allC^∗- actions onA³_Care linearizable. To show thatX6∼=A³_Cor any other Koras-Russell threefold of the first or second kind were not isomorphic toA³_Cwas not possible with any known invariant at that time. For example, all Koras-Russell threefolds are diffeomorphic to R⁶ as smooth manifolds.

If a smooth variety over C of dimension 2 is diffeomorphic to R⁴, then it is isomorphic to A²_C [Ram71], so one was hoping for a similar result in dimension 3. However, today we know an easy criterion to distinguish betweenA³_CandX (see [Kal02, Main Theorem] and [KZ01], see also [KZ00, Miyanishi’s Theorem]).

Theorem 1.1.1 (Kaliman’s criterion). Let Y be a smooth, affine, three dimensional, complex variety.

1. If there is a regular function f : Y → C and a nonempty Zariski open subset U ⊂ C such that the fibers f^∗(z) are isomorphic to A²_C for all z ∈U and each fiber has at most isolated singularities, then Y is isomorphic to A³_C. This is a corollary of Miyanishi’s theorem [Miy88].

2. A smooth, topologically contractible affine threefold Y is diffeomorphic to R⁶, but not isomorphic to A³_C if there exists a regular function f : Y → C and a nonempty Zariski open subset U ⊂C such that the fibers f^∗(z) are isomorphic to A²_C for all z ∈ U and at least one fiber is not isomorphic to A²_C.

Since the Koras-Russell cubic threefold X is diffeomorphic to R⁶, it is topologically contractible. Consider the regular function given by the projection of X to the x-coordinate f :X → C. The fiberf^∗(0)∼=A¹_C×C, where C = {z²+t³ = 0} ⊂ A²_C is a cuspidal curve, is not isomorphic to A²_C

The Makar-Limanov invariant

In 1996 Makar-Limanov proved that the Koras-Russell cubic threefoldXwas not isomorphic to C³ in [ML96] by introducing a new invariant, theMakar-Limanov invariant. LetY := Spec(A) be a complex affine variety. Then the Makar-Limanov invariant of Y is the intersection of all

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kernels of nontrivial locally nilpotent C-derivations ∂ (short LND, see Definition 2.2.1) on A, i.e.,

ML(Y) = \

06=∂LND on A

ker(∂).

Makar-Limanov showed that the Makar-Limanov invariant of the Koras-Russell cubic threefold X is ML(X) =C[x]while ML(C³) =C.

Yet, neither Kaliman’s criterion 1.1.1 (only works for three dimensional varieties, see [KZ00]) nor the Makar-Limanov invariant help to figure out whether the cylinder over the Koras-Russell cubic threefold X×A¹_C is isomorphic to A⁴_C or not. Adrien Dubouloz proved in [Dub09] that ML(X×A¹_C) =Cwhich makesX a potential a counterexample to the Cancellation Problem.

1.1.2 The Cancellation Problem

Let k be a field and let Y be ani-dimensional variety over k. The Cancellation Problem asks whetherY×Aⁿ_k ∼=Aⁱ⁺ⁿ_k for somen >0implies thatY ∼=Aⁱ_k. It has been proved by Abhyankar, Eakin and Heinzer in [AHE72] that if Y is a curve, then indeed Y ×Aⁿ ∼=Aⁿ⁺¹ implies that Y ∼=A¹. Also for surfaces the Cancellation Problem has a positive answer (see [CML08]). Yet, the Cancellation Problem has been proved to have a negative answer forka field with positive characteristic and n≥3 by Gupta [Gup14]. It remains open for k having characteristic 0 and n ≥ 3. One potential counterexample to the Cancellation Problem in characteristic 0 is the Koras-Russell cubic threefold.

A¹-contractibility of the Koras-Russell cubic threefold

One idea to rule out the Koras-Russell cubic threefold as a counterexample to the Cancellation Problem was to show that it is not A¹-contractible. Assume X×Aⁿ_C would be isomorphic to Aⁿ⁺¹. Then X×Aⁿ_C and hence also X were A¹-contractible. This implies that if X were not A¹-contractible, then X would not be a counterexample to the Cancellation Problem by contraposition. This was the motivation for Asok to set up a program to prove that X is not A¹-contractible (see [HKØ14]). However, parts of this program have been proved wrong by Hoyois, Krishna and Østvær in [HKØ14]. Furthermore, they showed that the Koras-Russell cubic threefold becomes A¹-contractible after finitely manyP¹-suspensions. Finally, Fasel and Dubouloz proved in [DF15] that the Koras-Russell cubic threefold isA¹-contractible.

1.2 Content

We discuss the two crucial properties of the Koras-Russell cubic threefold, namely that it is exotic, i.e., it is not isomorphic to affine space A³_C but diffeomorphic to R⁶, and that it is A¹- contractible. At the end of this thesis, we give more examples of exotic affine varieties which we check forA¹-contractibility.

1.2.1 Exotic varieties

In Chapter 2 we prove thatX is exotic.

X is diffeomorphic to R⁶

We show thatX is an affine modification of A³_C (see Definition 2.1.3), i.e., an open affine of a blow-up ofA³_C. Then we go through the arguments of Kaliman and Zaidenberg in [KZ99,§3] to

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show that affine modifications of topologically contractible varieties remain topologically contractible under certain conditions. Finally, we elaborate that topologically contractible varieties of dimension at least3 are diffeomorphic to Euclidean space.

X is not isomorphic to A³_C

We discuss Heden’s proof that the Makar-Limanov invariant of the Koras-Russell cubic threefold X is C[x]. In [Hed16] Heden gives a geometric interpretation of Makar-Limanov’s proof. He uses that there is a bijective correspondance between LNDs on the ring of regular functions of a variety Y and C⁺-actions on Y, that is, an action of the algebraic group C⁺ = A¹_C with the additive group structure. Like Makar-Limanov, Heden examines associated LNDs on the associated commutative graded algebra to the ring of regular functions on X. Then he proves with geometric arguments that any such LND has a negative degree. This implies that any LND on the ring of regular functions on X restricts to Spec(C[x^±1, z, t]) and thus x ∈ ker(∂) for∂ being the corresponding LND to theC⁺-action.

1.2.2 A¹-contractibility of the Koras-Russell cubic threefold

After giving a short introduction to A¹-homotopy theory in Chapter 3, we prove the following results.

A¹-contractibility after a finite number of suspension with P¹ (Chapter 4)

LetY be a smooth complex affine variety. In [HKØ14] Hoyois, Krishna and Østvær show that if the projectionY×_Spec(_C₎Z →Z induces an isormophism in higher Chow groupsCH^∗(Z, i)→ CH(Y ×_Spec(C)Z, i) for everyi≥0 and any smooth complex affine varietyZ, then there exists an n ≥0 such that Y ∧(P¹)^∧n is A¹-contractible. We recall some results about higher Chow groups from [Blo86] and use them to show that for the Koras-Russell cubic threefold X and any smooth complex affine variety Z, the projection X×_Spec(C)Z → Z induces isomorphisms in higher Chow groups.

A¹-contractibility of the Koras-Russell cubic threefold (Chapter 5)

We go through Dubouloz and Fasel’s arguments in [DF15] to show that the Koras-Russell cubic threefold X is A¹-contractible. They show that there is an étale locally trivialA¹-bundle from an open subset X\Z ofX to an algebraic space Sas well as an étale locally trivialA¹-bundle V → Swhere V is an affine varietyA¹-homotopy equivalent to A²\ {0}. The first projection of the fiber product(X\Z)×_SV ∼=V ×A¹ →X\Z is a Zariski locally trivialA¹-bundle and thus X\Z is A¹-homotopy equivalent to V. With the help of the Five-Lemma we conclude thatX isA¹-homotopy equivalent to A²_C and hence A¹-contractible.

1.2.3 More examples

In Chapter 6 we use affine modifications to construct more examples of exotic varieties and check to what extend the result by Hoyois, Krishna and Østvær [HKØ14] aboutA¹-contractibility after a finiteP¹-suspension and the result by Dubouloz and Fasel [DF15] about theA¹-contractibility of the Koras-Russell cubic threefold apply to these examples. In Chapter 7 we introduce the notion of A¹-chain connectedness, which can be seen as an analog of path connectedness in topology [AM11, §2.2]. All known examples of A¹-contractible varieties are alsoA¹-chain connected, but it is not known in general whetherA¹-contractibily impliesA¹-chain connectedness.

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We show that the Koras-Russell cubic threefold isA¹-chain connected and give more examples of A¹-chain connected varieties.

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Chapter 2 Exotic Varieties

Complex algebraic varieties can be given an analytic structure, so that nonsingular varieties can be seen as real manifolds (see [Ser56]).

Definition 2.0.1. Let Y be a complex affine variety defined by the ideal I = (f1, . . . , fr) in C[x₁, . . . , x_n]and lety ∈Y be a closed point. TheJacobian matrix aty is defined to be

Jac(y) :=

∂f_i

∂xj

(y)

∈Mr×n(C).

We say y is a smooth point of Y if rank(Jac(y)) = n−codim_Y({y}) and that Y is smooth if every closed point inY is a smooth point.

This reminds of the definition of a smooth manifold and motivates the following definition.

Definition 2.0.2. A smooth complex variety of dimensionnis calledexotic if it is diffeomorphic toR²ⁿas a real2n-manifold but not isomorphic to affine spaceAⁿ_Cas a complex algebraic variety.

Recall that the Koras-Russell cubic threefold X is the hypersurface in A⁴_C given by the equation x+x²y+z²+t³= 0. We show thatX is an example of an exotic variety.

2.1 The Koras-Russell cubic threefold X is diffeomorphic to R

⁶

Recall that two smooth manifolds arediffeomorphic if there exists a smooth map fromX toR⁶ which has a smooth inverse (see for example [Lee03, p.26]).

Theorem 2.1.1. The Koras-Russell cubic threefold X is diffeomorphic toR⁶.

We prove Theorem 2.1.1 by showing that an affine modifications of a topologically contractible affine variety is still topologically contractible given the affine modification satisfies the conditions in Corollary 2.1.8. Then we show that contractible smooth manifolds of real dimensionm >5(that corresponds to complex varieties of dimension≥3) are diffeomorphic to R^m. Hence, affine modifications of affine spaceAⁿ_C can be used to construct examples of exotic varieties.

2.1.1 Affine modifications

One can view affine modifications as an affine version of a blow-up. Affine modifications are indeed open affine subsets of blow-ups. We summarize some definitions and results by Kaliman and Zaidenberg from [KZ99].

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Let Y = Spec(A) be an affine variety over C, i.e., A is a finitely generated C-algebra.

Further, let I = (b₀, . . . , b_r) be an ideal and set Z = Spec(A/I). Recall the definition of a blow-up of an affine varietyY from [Har77, p.163]. Also see [KZ99] for the affine case.

Definition 2.1.2. Let A[It] :=L∞

i=0Iⁱtⁱ :={P

finiteajt^j :aj ∈I^j forj >0and a0 ∈A}. The blow-up of Y with center Z is defined to be

Y˜ := BlZ(Y) := Proj(A[It])⊂Y ×P^r_C

together with a surjective morphismπ: ˜Y →Y induced by the inclusionA ,→A[It]. We define theexceptional divisor E of the blow-up to beE :=π⁻¹(Z).

Letf =b0 and assume thatf is not a zero divisor and letD_f be the principal divisor given by f, that is the closed subscheme in Y given by {f = 0} ⊂Y. We recall the definition of an affine modification from [KZ99, Definition 1.1].

Definition 2.1.3. SetA⁰ :=A[It]/(1−f t). Theaffine modification ofY along the divisor Df

with center Z is the affine varietyY⁰ := Spec(A⁰).

Let ρ be the quotient map A[It] → A⁰.Note that the map A[y₁, . . . , y_r] → A⁰ defined by y_i 7→ρ(b_it) fori= 1, . . . , r induces a closed embeddingY⁰ ,→Y ×A^r_C [KZ99, Remark 1.1]. The affine modification Y⁰ is the open affine subset Y˜ \H0 of the blow-upY˜ where H0 is the first coordinate hyperplane given by y₀ = 0 in P^r_C [KZ99, Lemma 1.2]. Let E⁰ := E\H₀ be the exceptional divisor of the affine modification and π⁰ :Y⁰ →Y be the restriction of the blow-up morphismπ toY⁰.

E⁰ Y⁰ Y ×A^r

E Y˜ Y ×P^r

Z Df Y

π|E π

We adopt some notation from [KZ99, Definition 1.3]. We define the proper transform D^pr_f of Df in Y˜ to be the set of homogeneous prime ideals p∈Y˜ such that f t∈p. Hence, we get the equalityY⁰ ∼= ˜Y \H0= ˜Y \D^pr_f .

Recall that the blow-up morphism π : ˜Y → Y restricts to an isomorphism π|_Y_˜_\E : ˜Y \ E → Y \Z. Since Y⁰ = ˜Y \D^pr_f and hence Y⁰ \E⁰ = ˜Y \(D_f^pr ∪E), the restriction of π⁰|_Y0\E⁰ :Y⁰\E⁰ →Y \D_f is an isomorphism.

Y⁰\E⁰ Y \D_f

Y˜ \E Y \Z

∼=

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Example 2.1.4. The Koras-Russell cubic threefold X is the affine modification of A³_C along the divisorD_f given by f =−x² with centerZ given byI = (−x², x+z²+t³). This is because X is the open affine subset given by {u6= 0} in

{x²y+u(x+z²+t³) = 0} ⊂A³_C×P¹_C.

Note that there is a canonical isomorphism A⁰ ∼=A[^b_f¹, . . . ,^b_f^r](see [KZ99, Proposition 1.1 (a)]). We recall the following result from [KZ99, Proposition 1.1 (b) and (c)].

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Proposition 2.1.5. Let β : A[y₁, . . . , y_r] A[^b_f¹, . . . ,^b_f^r], y_i 7→ ^b_fⁱ for i = 1, . . . , r and let I⁰ := (f y_i−b_i)_i=1,...,r be the ideal inA[y₁, . . . , y_r]generated by the elements f y_i−b_i. Then

1. ker(β) =I⁰ if and only if I⁰ is a prime ideal.

2. If ker(β) =I⁰ then E⁰∼=Z×A^k_C.

3. If for every i= 1, . . . , r, bi is not a zero divisor in A/(b0, . . . , bi−1), then I⁰ is prime. We say that the system of generators b₀, . . . , b_r of I is regular.

If E⁰ ∼=Z×A^r_C, then E⁰ and Z are of the same homotopy type, which becomes important later when we look at the singular homology groups of an affine modification. Note that not all ideals in finitely generated C-algebras necessarily admit a regular system of generators. Take for example A = C[x, y] and I = (x², xy) and f = b0 = x². Then xy is a zero divisor in C[x, y]/(x²) since x·xy =x²y and xy cannot be replaced by another generator which is not a zero divisor inC[x, y]/(x²). However, note that−x², x+z²+t³is a regular system of generators forI = (−x², x+z²+t³) which is the ideal from Example 2.1.4.

Preserving fundamental group and homology groups

In [KZ99, Proposition 3.1 and Theorem 3.1] Kaliman and Zaidenberg give criteria for when the fundamental group and homology groups are preserved under divisorial modifications (M⁰, F⁰) of (M, H). Here M and M⁰ are reduced connected complex spaces andH ⊂M and F⁰ ⊂M⁰ are closed hypersurfaces. Furthermore, there is a dominant morphism σ : M⁰ → M such that the restriction σ|_M⁰_\F⁰ : M⁰ \F⁰ → M \F is a biholomorhism (i.e., holomorphic and bijective [SS03, p.8, p.205]) andσ(F⁰)⊂H.

Note that an affine modifications Y⁰ of Y along a divisor D_f with center Z such that codimY(Z) ≥ 2 is a divisorial modification (Y⁰, E⁰) of (Y, Df) with surjective morphism π⁰ : Y⁰ →Y.

Now we can formulate the results about the fundamental group and the homology of an affine modification [KZ99, Proposition 3.1 and Theorem 3.1 resp.].

Proposition 2.1.6. Let (M⁰, F⁰) be a divisorial modification of(M, H)via σ:M⁰ →M where M andM⁰are complex manifolds. Further, assume the hypersurfacesH andF⁰can be written as finite unions of irreducible components H=Sn

i=1Hi andF⁰=Sn⁰

j=1F_j⁰. Assume the pullback of the divisorH_ibyσdecomposes asσ^∗(H_i) =P_n⁰

j=1m_ijF_j⁰. Supposeσ(F_j⁰)∩reg(H_i)6=∅ifm_ij >0 forreg(H_i)being the smooth part ofH_i and that there exists a partition{1, . . . , n⁰}=J₁t· · ·tJ_n such that σ^∗(Hi) = P

j∈JimijF_j⁰ 6= 0 for all i = 1, . . . , n. Then σ induces an isomorphism of fundamental groups.

σ∗ :π₁(M⁰, x⁰₀) ˜→π₁(M, x₀)

for suitable basepoints x0 ∈ M, x⁰₀ ∈M⁰ with σ(x⁰₀) =x0 given that gcd(mij :j ∈Ji) = 1 for i= 1, . . . , n.

Proposition 2.1.7. Let (M⁰, F⁰) be a divisorial modification of(M, H)via σ:M⁰ →M where M and M⁰ are complex manifolds. Assume that F⁰ andH are topological mainifolds admitting finite decompositions into irreducible components H = Pn

i=1Hi and F⁰ = Pn

i=1F_i⁰ such that σ(F_i⁰)∩reg(H_i)6=∅ for i= 1, . . . , n. Then σ induces an isomorphism in singular homology

σ∗:H∗(M⁰)→H∗(M) if and only if the maps

τi :=σ|_F⁰

i :F_i⁰ →Hi

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induce an isomorphism in singular homology

τi∗:H∗(F_i⁰)→H∗(Hi) for i= 1, . . . , n.

We want to apply Proposition 2.1.6 and Proposition 2.1.7 to affine modifications and find conditions under which the fundamental groups and homology groups are preserved. Assume we have an affine modificationY⁰ ofY = Spec(A)along the divisor D_f wheref is not zero divisor with center Z given by the ideal I = (f, b1, . . . , br) such that codimY(Z)≥2. Assume that Z and D_f are irreducible. Recall from Proposition 2.1.5 that iff, b0, . . . , br is a regular system of generators, then the exceptional divisorE⁰ is isomorphic toZ×A^k_Cwithk= codim_D(Z). Thus the map τ∗ : H∗(E⁰) → H∗(Z) is an isomorphism if and only if H∗(Z) → H∗(D) induced by the inclusionZ ,→D_f is an isomorphism, which is the case if both Z and D_f are topologically contractible. We conclude the following result. Note that the following is a special case of [KZ99, Corollary 3.1].

Corollary 2.1.8. Let Y⁰ be the affine modification of a topologically contractible affine variety Y = Spec(A) along the divisor given by f, where f is not a zero divisor, with center Z = Spec(A/I) with codimY(Z)≥2. Further assume that I admits a regular system of generators f, b₁, . . . , b_r and assume that Z and D_f are topologically contractible and irreducible such that Z is contained in the smooth part of D_f. Then Y⁰ has trivial fundamental group and trivial homology groups.

Corollary 2.1.9. The Koras-Russell cubic threefold X has the same fundamental group and the same homology groups asA³_C since it is an affine modification (see Example 2.1.4) satisfying all the conditions in Corollary 2.1.8. Therefore,

π1(X) = 1 and Hn(X) = 0 for all integersn≥0.

2.1.2 Topological contractibility

With the help of the Hurewicz and Whitehead theorems we can conclude that the Koras-Russell cubic threefoldX or, in general, affine modifications satisfying the conditions in Corollary 2.1.8 are contractible. The following result can be found in [Hat02, p. 346, Theorem 4.5].

Theorem 2.1.10(Whitehead’s Theorem). If a mapf :Y →Z between CW-complexes induces isomorphisms f∗:π_n(Y)→π_n(Z) for all n, then f is a homotopy equivalence. In case f is the inclusion of a subcomplex, then Y is a deformation retract of Z.

Definition 2.1.11. A topological spaceY is called n-connected ifπ_i(Y) = 1 fori < n.

The following result is taken from [Hat02, p. 366, Theorem 4.32].

Theorem 2.1.12 (Hurewicz’s Theorem). If a spaceY is(n−1)-connected forn≥2, then the reduced homology groups H˜i(Y) vanish fori < n and there is an isomorphismπn(Y)∼=Hn(X).

Let Y be a complex smooth algebraic variety. In order to to apply Whitehead’s theorem, we need Y to be a CW complex. It is true that compact manifolds are homotopy equivalent to a CW complex [Hat02, p.529, Corollary A.12]. So we will first show that Y is homotopy equivalent to a compact manifold.

The following theorem follows from [Dim92, p.26].

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Theorem 2.1.13. Let Y ⊂Aⁿ_C be a closed algebraic variety. Then there exists a real number r₀ >0 such that for all r≥r₀ we have a diffeomorphisms

(Cⁿ\Br)∩Y ∼= [r,∞)×(Sr∩Y) where B_r={x∈Cⁿ:|x|< r} and S_r=∂B_r is the boundary ofB_r.

In other words, affine varieties have a cylindric structure far from the origin. The theorem implies that for a complex algebraic varietyY,(Cⁿ\B_r)∩Y is diffeomorphic to[r,∞)×(S_r∩Y) with the notation from above and a big enough radiusr≥r0. Since the cylinder[r,∞)×(S_r∩X) deformation retracts to {r} ×(S_r∩X), Y is homotopy equivalent to Y ∩B_r where B_r is the closure of B_r. The space Y ∩B_r is a compact manifold and hence homotopy equivalent to a CW complex which has the same fundamental group and homology groups asY.

Theorem 2.1.14. A connected CW complexY with fundamental groupπ₁(Y) = 1and homology groups Hn(Y) = 0 for all n≥2 is contractible.

Proof. A space Y is contractible if it deformation retracts onto a point. By Whitehead’s The- orem this is the case if all the homotopy groups of Y are the same as those of a point, i.e., πn(Y) = 1 for all n. As π1(Y) = 1, we can apply Hurewicz’s Theorem for n = 2 and get H₂(X) ∼= π₂(Y). So π₂(Y) = 1 and we can apply Hurewicz’s Theorem again for n = 3 and get that H₃(Y) ∼= π₃(Y) and so on and so forth. So π_n(Y) = 1for all n and by Whitehead’s theorem Y deformation retracts to a point and is therefore contractible.

Corollary 2.1.15. Closed connected algebraic varieties with trivial fundamental group and trivial homology groups for n≥2 are topologically contractible. In particular, affine modifications satisfying the conditions in Corollary 2.1.8 are topologically contractible.

So also the Koras-Russell cubic threefold is topologically contractible.

2.1.3 X is diffeomorphic to Euclidean space

Finally, we can conclude that the Koras-Russell cubic threefold X is diffeomorphic toR⁶ using the fact that smooth contractible affine algebraic varieties of dimension at least3are isomorphic to Euclidean space. This is proved by Choudary and Dimca in [CD94, Step 3 of the proof of Theorem 5]. The proof goes as follows. We use Theorem 2.1.13 again. The Koras-Russell cubic threefold X is diffeomorphic to X∩Br with Br like in Theorem 2.1.13 with r≥r0. Set M := X∩B_r. The aim is to show that M is diffeomorphic to the closed 6-dimensional disc D⁶ ⊂R⁶.

By a result of Smale [Sma62, Theorem 5.1] it suffices that the boundary of M is simply connected.

Theorem 2.1.16 (Smale). Suppose C is a compact contractiblem-manifold with π1(∂C) = 1 and m >5. Then C is diffeomorphic to the closed m-disc D^m.

Remark 2.1.17. Smale’s result only works for a real dimension m >5. So the argument that contractible complex affine algebraic varieties are diffeomorphic to Euclidean space only goes through for algebraic dimension greater than or equal to 3.

Indeed, if we setS :=X∩S_rwith the notation from Theorem 2.1.13, we get that the inclusion S ,→Xinduces an isomorphism on the fundamental groups. This is proved in [Dim92, p.28]. So X being contractible yields that π₁(X) = 1and so we can apply Theorem 2.1.16 and conclude that M is diffeomorphic to D⁶. Hence X ∩Br is diffeomorphic to the interior of D⁶, i.e.,

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the open 6-dimensional disc which is diffeomorphic to R⁶. If we replace X by an arbitrary topologically contractible smooth complex variety of dimension ≥ 3 we have the following general result [Kal94, Theorem 2.6].

Theorem 2.1.18. Topologically contractible smooth complex affine varieties of dimensionn≥3 are diffeomorphic to R²ⁿ as smooth real 2n-manifolds.

2.2 The Koras-Russell cubic threefold X is not isomorphic to A

³_C

In this section we show that the Koras-Russell cubic threefoldX is not isomorphic to A³_C as an algebraic variety by showing that its Makar-Limanov invariant is not trivial, i.e.,ML(X)6=C. Since we work a lot with locally nilpotent derivations on the ring of regular functions of X, we fix the notation for this ring in this section. From now on let A := O(X) = C[x, y, z, t]/(x+ x²y+z²+t³) be the ring of regular functions on X.

First we recall some definitions. The following definition can be found in [Alh15, Def 1.4].

Definition 2.2.1. Letkbe a field. Ak-derivation∂on a commutativek-algebraB is ak-linear map∂:B→B satisfying

∂(f g) =∂(f)g+f ∂(g)

for all f, g ∈ B. A k-derivation ∂ is called a locally nilpotent k-derivation if for every f ∈ B there exists a positiven∈Nsuch that∂ⁿ(f) = 0. We abbreviate locally nilpotentk-derivation by LND.

In [ML96] Makar-Limanov introduced the eponymous invariant to prove that C[x, y, z]admits more locally nilpotent derivations than the ring of regular functions A=C[x, y, z, t]/(x+ x²y+z²+t³) on the Koras-Russell cubic threefoldX.

Definition 2.2.2. LetY be an affine complex variety and letO(Y)be the ring of regular functions on Y. Then the Markar-Limanov invariant ML(Y) of Y is defined to be the intersection of the kernels of all nonzero C-linear LNDs onO(Y), i.e.,

ML(Y) := \

06=∂LND onO(Y)

ker(∂).

We say that a complex affine varietyY hastrivial Makar-Limanov invariant ifML(Y) =C. Example 2.2.3. For the affine spacesAⁿ_C we have ML(Aⁿ_C) =C.

Proof. Let _∂^∂

xi be the C-linear LND on C[x1, . . . , xn] defined by _∂^∂

xi(xi) = 1 and _∂^∂

xi(xj) = 0 for j 6= i and i = 1, . . . , n. Then ker(_∂^∂

xi) = C[x1, . . . , xi−1, xi+1, . . . , xn]. It follows that C ⊂ ML(Aⁿ_C). Let ∂ be a C-linear LND on C[x1, . . . , xn]. It follows from the C-linearity of

∂ that ∂(a) = 0 for any a ∈ C since ∂(1) = ∂(1·1) = 2∂(1) which implies ∂(1) = 0, and

∂(a) =a∂(1) = 0.

It will turn out that

ML(X) =C[x] (2.2)

and thus X A³_C.

Makar-Limanov’s first proof is quite technical. In [Hed16] Isac Heden gives another proof that the Makar-Limanov invariant of the Koras-Russell cubic threefold is nontrivial which uses

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a geometric interpretation of the Makar-Limanov invariant and Makar-Limanov’s proof. We study Heden’s proof since it also reveals some geometric properties of the Koras-Russell cubic threefoldX. There is a one to one correspondance betweenC-linear LNDs on the ring of regular functions of an affine complex variety Y and algebraic group actions of the algebraic groupC⁺ on Y (see Example 2.2.5). Heden uses this correspondence. We recall the definition of an algebraic group and an algebraic group action from [Bri10, Definition 1.1 and Definition 1.4].

Definition 2.2.4. An algebraic group is a group that is an algebraic variety such that the composition and inversion are regular functions. An algebraic group action of an algebraic groupG on a varietyX is a morphism of varieties G×X→X.

Example 2.2.5. The algebraic varietySpec(C[x^±1])is an algebraic group denoted byC^∗ with the action given by the multiplication of two closed points. The algebraic varietySpec(C[x]) = A¹_C is an algebraic group denoted byC⁺. Here the action is given by adding two closed points.

Next we review the mentioned correspondence between C⁺-actions on a complex affine algebraic varietyY andC-linear LNDs onO(Y). For a proof we refer the reader to [Fre06, p.31]

by Freudenburg.

Proposition 2.2.6. There is a one to one correspondance between C-linear LNDs on the ring of regular functions of an affine complex variety Y and algebraic group actions of C⁺ given by associating to a LND ∂ on O(Y) the C⁺-action

φ:C⁺×Y →Y, (τ, x)7→τ ∗x defined by the ring homomorphism

φ^∗ :O(Y)→ O(Y)[S], f 7→

∞

X

n=0

∂ⁿf n! Sⁿ

where ∂⁰= id_O(Y₎. Note, that φ^∗ is well-defined since the sum is finite for every f because ∂ is an LND. Conversely, given aC⁺-actionρ:C⁺×Y →Y, one can assign the LND ∂⁰ on O(Y) given by the composition

O(Y) ^ρ

∗

−→ O(Y)[S]

d

−−→ O(YdS )[S]−−→ O(Y^S=0 ).

If a regular function f ∈ O(Y) is in the kernel of a LND ∂ on O(Y), then φ^∗ from the proposition above maps f to itself. In particular, φ^∗(f) does not depend on S. So f(x) = f(φ(τ, x))for any(τ, x)∈C⁺×Y. Conversely, iff(ρ(τ, x)) =f(x)for a C⁺-actionρ onY and for all(τ, x)∈C⁺×Y, thenρ^∗(f) is constant in the variableS and thus∂⁰(f) = 0 for the toρ corresponding LND∂⁰. Hence, an equivalent geometric version of the Makar-Limanov invariant is given by:

ML(Y) = \

nontrivialC⁺-actions onY (τ,x)7→τ∗x

{f ∈ O(Y) :f(τ∗x) =f(x) for all x∈Y and for allτ ∈C⁺}

2.2.1 Homogenization of X The homogenization technique

In this subsection we will describe a technique to associate to a commutative k-algebra B a commutative gradedk-algebra, i.e., ak-algebraGr(B)with a decompositionGr(B) =L

i∈ZBi. We follow the approach in [Alh15] by Alhajjar.

We first define Z-degree functions and Z-filtrations onB. Letk be a field of characteristic 0 and letB be a commutativek-algebra.

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Definition 2.2.7. A Z-degree function on B is a functiondeg : B →Z∪ {−∞} such that for all a, b∈B

1. deg(a) =−∞ ⇔a= 0 2. deg(ab) = deg(a) + deg(b)

3. deg(a+b)≤max(deg(a),deg(b)).

Remark 2.2.8. Let deg be a Z-degree function. Then −degis a discrete valuation.

Definition 2.2.9. A Z-filtration ofB is a collection {F_i}_i∈_Z of subgroups of (B,+)such that 1. F_i⊂ F_i+1

2. B =S

i∈ZF_i

3. F_i· F_j ⊂Fi+j for all i, j∈Z. The filtration isproper if in addition

4. T

i∈ZF_i={0}

5. a∈ F_i\ F_i−1,b∈ F_j\ F_j−1 implies thatab∈ F_i+j \ F_i+j−1.

There is a one to one correspondence between Z-degree functions on B and proper Z- filtrations of B. Given a Z-degree function deg on B, one can define a proper Z-filtration by setting F_i to be the set of elements ofB with degree at most i. Given aZ-filtration {F_i}_i∈_Z of B, one can define a Z-degree function by setting deg(b) := min{i∈Z:b∈ F_i}.

We can associate a commutative gradedk-algebra to a given proper Z-filtration.

Definition 2.2.10. Let{F_i}_i∈_Zbe a properZ-filtration ofB. Then the associated commutative graded algebra Gr(B) associated to {F_i}_i∈_Z is given by

Gr(B) :=M

i∈Z

F_i/F_i−1.

The homogenization of Spec(B) with respect to{F_i}_i∈_Z is the variety Spec(Gr(B)).

Definition 2.2.11. Let {F_i}_i∈_Z be a proper Z-filtration of B. Then a k-derivation D on B respects the filtration if there is an integerτ such thatD(F_i)⊂ F_i+τ for alli∈Z. The smallest such τ is called thedegree of D with respect to {F_i}_i∈_Z.

Let D be a k-derivation on B which respects a proper Z-filtration {F_i}_i∈_Z of B. Then we can define a derivation

Dˆ : Gr(B)→Gr(B) on Gr(B)by Dˆ = 0 ifD= 0 and

Dˆ :F_i/Fi−1→ F_i+τ/Fi+τ−1

a+Fi−17→D(a) +Fi+τ−1

whereτ is the degree ofD. IfD6= 0 then alsoDˆ 6= 0.

Now we concentrate on the case that B is the quotient of a coordinate ring. Let k^[n] :=

k[X₁, . . . , X_n]denote the coordinate ring innvariables over the field k.

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Definition 2.2.12. A Z-weight degree function ω over k on k^[n] is a Z-degree function such that

ω(P) = max{ω(x^I) :P =X

I

aIx^I}

forP ∈k^[n] and ω(a) = 0 for all a∈k\ {0}.

A Z-weight degree function over k gives a grading on k^[n] = L

l∈Zk_l^[n] where k_l^[n] = {P ∈ k^[n] : ω(P) = l}. So any P ∈ k^[n] can be written uniquely as P = P_l₁ +· · ·+P_l_s with l₁ < · · · < l_s and P_l_i ∈ k^[n]_l

i for i = 1, . . . , s. Let Pˆ := P_l_s be the summand with the highest degree and ˆb:={Pˆ :P ∈b} for any idealb⊂k^[n].

Definition 2.2.13. AZ-weight degree function ω isappropriate for an ideal b ink^[n] if 1. b⊂(X₁, . . . , X_n)

2. ˆb is a prime ideal andXi6∈ˆb fori= 1, . . . , n.

Thanks to the next result we can identify Gr(B) with an explicite polynomial ring. Fur- thermore, it implies that every nonzero LND on B induces a nonzero LND on Gr(B). It was proved by Kaliman and Makar-Limanov in [KML07].

Proposition 2.2.14. Suppose B = k^[n]/b and ω is a Z-weight degree function on k^[n] appro- priate for b. Let π:k^[n]→B be the quotient map. Then

1. inf{ω(Q) : Q ∈ π⁻¹(P)} > −∞ for all 0 6= P ∈ B and so ω_B(P) := min{ω(Q) : Q ∈ π⁻¹(P)} is a well-defined Z-degree function onB [KML07, Lemma 3.2].

2. The commutative graded k-algebra Gr(B) associated to the properZ-filtration induced by ωB is isomorphic to the quotient k^[n]/ˆb [KML07, Proposition 4.1].

3. Every derivation ∂ onB respects the ω_B-filtration. So∂ induces a derivationδ onGr(B) [KML07, Lemma 5.1].

The homogenization of the Koras-Russell cubic threefold X

Example 2.2.15. Letb:= (x+x²y+z²+t³)⊂C[x, y, z, t]andk^[n]=C[x, y, z, t]. We consider theZ-weight degree functionωwithω(x) =−1,ω(y) = 2andω(z) =ω(t) = 0. So by Definition 2.2.13ˆb= (x²y+z²+t³) and by Proposition 2.2.14

Gr(A) =C[x, y, z, t]/(x²y+z²+t³)

Letx,¯ y,¯ z,¯ ¯tbe the equivalence ofx,y,z andt in the quotientC[x, y, z, t]/(x²y+z²+t³). The nth graded part consists of homogeneous polynomials of degreen:

Gr(A)_n=







C[¯z,¯t]¯x⁻ⁿ ifn≤0 C[¯z,¯t]¯y^k ifn= 2k >0 C[¯z,¯t]¯x¯y^k ifn= 2k−1>0

(2.3)

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2.2.2 Some more results on algebraic group actions

By Proposition 2.2.6, any C⁺-action on an affine complex variety Y corresponds uniquely to a LND on the ring of regular functions O(Y). There is a similar result for algebraic actions of the multiplicative group C^∗.

Proposition 2.2.16. Algebraic actions of the multiplicative group C^∗ on an affine algebraic variety Y = Spec(B) correspond uniquely to decompositions B = L

n∈ZB_n as a commutative Z-graded algebra.

Proof. We sketch a proof: The algebraic group action C^∗ ×Y → Y, (λ, x) 7→ λx induces a ring homomorphism on the rings of regular functions B → C[S, S⁻¹]⊗ B ∼= B[S, S⁻¹], f 7→ P

n∈ZD_n(f)Sⁿ, which naturally gives a decomposition. On the other hand, given a decomposition ofB =L

n∈ZBn we can define a ring homomorphismB →B[S, S⁻¹]by setting f = (f_n)n∈Z7→P

n∈Zf_nSⁿ. Remark 2.2.17. Let B =L

n∈ZBn be the decomposition into commutativeZ-graded algebras corresponding to aC^∗-action onSpec(B). Then for a regular functionf ∈B_n we havef(λx) = λⁿf(x) for anyλ∈C^∗ andx∈Spec(B).

Proof. Let the C^∗-action be given by ρ : C^∗×Spec(B) → Spec(B) defined by (λ, x) 7→ λx.

Then ρ^∗(f) =f·Sⁿ and ρ^∗(f)(λx) =f(x)λⁿ.

Note that the associated commutative gradedC-algebraGr(B)to a commutativeC-algebra like in Definition 2.2.10 gives rise to a C^∗-action. The corresponding C^∗-action to Gr(A), the ring of regular function to the homogenization of the Koras-Russell cubic threefold, is given by

C^∗×W →W

(λ,(x, y, z, t))7→(λ⁻¹x, λ²y, z, t) forW := Spec(Gr(A)).

Quotients of group actions

Given a group action it is natural to ask whether the quotient exists and what it looks like.

In general, if one has an algebraic group action G×Y → Y, (g, x) 7→ gx on an affine variety Y = SpecB, the affine scheme SpecB^G, here B^G := {f ∈ B :f(gx) = f(x) for all g ∈G}, is a good candidate for the quotient Y //G. This is in general not a variety, as B^G might not be finitely generated. If B^G is finitely generated, then we define the quotient Y //G:= Spec(B^G) with quotient mapπ :Y →Y //G induced by the inclusion B^G ,→ B. By [KSS89, III.3.1] the quotient Y //G has the following universal property. If there is a morphism of affine varities φ:Y →Zwhich is constant on theG-orbits, then there exists a unique morphismψ:Y //G→Z such that φ= ψ◦π. The question which asks when B^G is finitely generated is number 14 of Hilbert’s problems proposed in 1900. Hilbert conjectured thatB^Gwas always finitely generated but Masayoshi Nagata found the first counterexample in 1959 (see [Dol03, p.52-61]). However, there exist algebraic groups, namelyreductive groups, for whichB^Gis always finitely generated.

For the following definition see [KSS89, p.6, Definition 1.1].

Definition 2.2.18. Let V be a finite dimensional vector space over a field k and let GL(V) be the group of bijective linear transformations ofV and letGbe an algebraic group. A group homomorphism ρ : G → GL(V) which is also a morphism of algebraic varieties, is called a rational representation of G. An algebraic group G is reductive if for every finite dimensional rational representation ρ : G → GL(V) and for every G-invariant subspace W ⊂ V, i.e., ρ(g)W =W for allg∈G, there exists aG-invariant subspaceW⁰⊂V such thatW ⊕W⁰ =V.

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For a reductive group G the set of G-invariant functions B^G := {f ∈ B : f(gx) = f(x) for all g ∈ G} is always finitely generated [KSS89, p.11-12, Theorem 1.2]. The following result helps us to find the quotient of an algebraic variety by a group action of a reductive group. For a proof we refer to [Kra84, p.105 Quotientenkriterium].

Proposition 2.2.19. LetG×Y →Y be an algebraic group action whereGis a reductive group.

Suppose there is a surjective morphismp:Y →Z for a normal algebraic varietyZ and an open dense subset U ⊂ Z such that each fiber p⁻¹(y), y ∈ U contains exactly one closed G-orbit.

Then Z is isomorphic to the quotient Y //G, in particular the quotient is an algebraic variety.

The algebraic group C^∗ is reductive [KSS89, p.8, Example 2.2]. The algebraic group C⁺ is not reductive. However, for the group actions we look at, the set of invariant functions will be finitely generated due to the following proposition.

Proposition 2.2.20. If a normal affine varietySpec(B), i.e.,B is reduced and integrally closed, is of dimension less than or equal to 3, then the set of invariant functions B^G is always finitely generated for any algebraic group action G×Spec(B)→Spec(B).

The proposition follows from Zariski’s finiteness theorem which can be found in [Fre06, p.147].

Theorem 2.2.21 (Zariski’s finiteness theorem). For a field k, let B be an affine normal k- domain and let K be a subfield of the field of fractions frac(B) such that K contains k. If the transcendence degree of k⊂K is less than 3 thenK∩B is finitely generated over k.

Proof of Proposition 2.2.20. There are two cases we need to consider: dimk(B^G)≤2 and dim_k(B^G) = 3. If dim_k(B^G) = 3, then B is algebraic over B^G and it follows by [Fre06, Proposition 6.21] thatGis finite and thus reductive [KSS89, Example II.2.1]. In order to apply Zariski’s finiteness theorem, we need to show that in casedimk(B^G)≤2thatB^G= frac(B^G)∩B.

ClearlyB^G⊂frac(B^G)∩B. Letb∈frac(B^G)∩B. Thenb= _a^a0 fora, a⁰∈B^G. We want to show that b∈B^G. Since a=a⁰b∈B^G, we know thata(x) =a(g.x) for allx∈Spec(B) and g ∈G.

It follows that a⁰b(x) =a⁰b(g.x) =a⁰(x)b(x) =a⁰(g.x)b(g.x). Sincea⁰ ∈B^G and a⁰(x) 6= 0 for x∈Spec(B)because _a^a0 is a regular function onSpec(B), we conclude thatb(x) =b(g.x)for all x∈Spec(B) and g∈Gand thus b∈B^G. Note that B^G is normal by Proposition 2.2.22.

Proposition 2.2.22. Let G be an algebraic group which acts on a normal affine variety Y = Spec(B)such that the set ofG-invariant functions is finitely generated. Then the quotientY //G is normal.

Proof. We need to show that B^G is integrally closed infrac(B^G). Letx∈frac(B^G) be integral over B^G, so there are b1, . . . , bn ∈ B^G such that xⁿ +b1xⁿ⁻¹ +· · · +bn = 0. Since B is integrally closedx∈B. Sox∈B∩frac(B^G). The assertion follows since we have the inclusion B∩frac(B^G)⊂B^G (see proof of Proposition 2.2.20).

Additionally, we have the following results about algebraic C⁺-actions.

Lemma 2.2.23. If the quotient Y //C⁺ exists for a nontrivial algebraic C⁺-action on an affine variety Y = Spec(B), then dim(Y //C⁺) = dim(Y)−1.

Proof. The assertion follows from [Fre06, Principle 11(e) on p.27] which says, that if ∂ is a nonzero LND on B, then the transcendence degree of B over ker(∂) is 1. More, precisely it is shown that there is an f ∈ B such that 0 6= ∂(f) ∈ ker(∂) and the localization B_∂(f) is isomorphic to (ker(∂))_∂(f)[f] which implies that the open affine subset Y_f is isomorphic to Spec(ker(∂))×C⁺ whereC⁺ only acts on the second factor.

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Lemma 2.2.24. An orbit of aC⁺-action on an affine varietyY is either isomorphic toC⁺∼=A¹ or a point.

Proof. Let ψ :C⁺×Y → Y, (τ, x) 7→ τ ∗x be a C⁺-action on an affine variety Y. An orbit of the C⁺-action consists of the pointsC⁺x={τ ∗x:τ ∈C} ∼=C⁺/Stab(x) where Stab(x) = {τ ∈C⁺:τ∗x=x}is the stabilizator group. Furthermore, the setStab(x) =ψ|⁻¹

C⁺×{x}({x})is closed as{x} ∈Y is closed. SoStab(x) consists of finitely many points inC⁺ or isC⁺. Assume Stab(x) 6= C⁺ and 0 6= τ₁ ∈ Stab(x). Then Zτ₁ ⊂ Stab(x) which is not closed in C⁺ (in the Zariski topology). So aC⁺-orbitC⁺x∼=C⁺/Stab(x)is either a point or isomorphic toA¹_Csince Stab(x) is either a point or isomorphic toA¹_C.

Normalized group actions

If two algebraic groups act on an algebraic varietyY, they can interact in the following way.

Definition 2.2.25. LetG×Y →Y,(g, x)7→gxand H×X →X,(h, x)7→h∗xbe algebraic group actions on Y. We say that the action of G normalizes the action of H if there is a homomorphismσ :G→Aut(H),g7→σ_g such thatG×H →H,(g, h)7→σ_g(h) is a morphism of algebraic varieties and

g(h∗x) =σ_g(h)∗(gx) for all g∈G,h∈H and x∈Y.

Remark 2.2.26. Every homomorphismσ :C^∗ →Aut(C⁺) is of the form λ7→(σ_λ :τ 7→λ^kτ) for some k∈Z.

For the proof of the following proposition see [FZ05, Lemma 2.2].

Proposition 2.2.27. LetY = Spec(B), letC^∗×Y →Y,(λ, x)7→λxbe aC^∗-action corresponding to the decomposition B =L

n∈ZBn as a commutative graded algebra and let C⁺×Y →Y, (τ, x) 7→ τ ∗x be a C⁺-action corresponding to the LND ∂ : B → B. Then the action of C^∗ normalizes the action of C⁺ with respect to σ:C^∗ →Aut(C), λ7→(σ_λ :τ 7→λ^−kτ) if and only if ∂ is homogeneous of degree k.

Corollary 2.2.28. Let δ be a homogeneous LND on Gr(B) of degreel. Then for an orbitO of the correspondingC⁺-action, the set λO is also a C⁺-orbit for λ∈C^∗.

Proof. The LND δ on Gr(B) being homogeneous of degreel implies that the C^∗-action corresponding to the grading of Gr(B) normalizes the C⁺-action corresponding to δ. That implies that

λ(τ∗x) =σλ(τ)∗(λx) = (λ^−lτ)∗(λx).

Note thatλis fixed and not zero.

2.2.3 The main results

Lemma 2.2.29. Any morphismφ:A¹_C→E fromA¹_C to an elliptic curveE overCis constant.

More generally, a morphism from a A¹_C to a curve over C of genus at least 1, is constant.

Proof. Any such morphism can be extended to a morphismφ˜:P¹_C→E(see [Har77, Proposition I.6.8]). Elliptic curves are by definition curves of genus1. According to [Liu02, Lemma 7.3.10]

the morphism φ˜is finite and this implies that

0 =g(P¹_C)≥g(E) = 1

whereg denotes the genus of the curve [Liu02, Corollary 7.4.19 and Proposition 7.4.21] in case φ˜is not constant, but dominant.

A1-homotopy theory