Antoinette Baujard Don't Reduce

Amartya Sen

Antoinette Baujard

Don't reduce Amartya Sen to a single identity!

Keywords: Amartya Sen, Welfare, Discussion, Reason, Identities, Memoir B31, D63, D71, I31, I32 JEL. B31, D63, D71, I31, I32 Amartya Sen, Welfare, Discussion, Reason, Identities, Memoir

This paper reviews Amartya's Sen autobiography, Home in the World. A Memoir (Penguin

account of his entire life. Rather, he has two main objects in view in surveying his personal trajectory and the influences on him up to his thirties.

First of all, the memoir overcomes the limitations in the reception of his work arising from the partiality of commentary on it, whether by social choice theorists, standard economists, sociologists, leftists, Occidentals, or white males; too often such misreadings have led his readers to miss the major points he intended to make. Secondly, there is a constant reminder of his core propositions, notably that all women and men have plural identities, not one single, unchanging identity; and that only the consistent use of reasoned argument in an open democracy can guarantee respect for human agency (the value of the voice of each human being in their plurality, and their capacity to act in and on the world).

In this review I will leave aside the details of Sen's biography, together with the many illuminating anecdotes and uplifting pieces of poetry -these should be read in the context of the book itself. I shall instead focus essentially on the first objective: that the memoir helps us to understand his main contributions by circumventing third-party perspectives.

Once someone becomes famous she or he runs the risk that anything they say is reinterpreted to fit the preoccupations of others, or as if a theory had been summarized in "a one-line slogan" (which Sraffa urged Sen not to do, as we learn in chapter 16, p.261). This dilution through fame can compromise the robustness of arguments, their leading principles and underlying theoretical propositions; in the process of reception the original argument can even be subverted to such an extent that it appears to support that which it was intended to rebut.

What can be done to restore the interpretation first intended by the scholar? Let us consider three alternatives.

Repeating the arguments in an additional paper proves to be an inefficient option -regardless of the reams written on the subject, how many economists are actually aware that there is no such thing as an Adam Smith Problem?

Highlighting the bias through which commentators read the contribution is another solution to avoid misinterpretations. Muriel Gilardone and I did so regarding Sen's capability approach [START_REF] Baujard | Sen is not a capability theorist[END_REF]. Starting from his disregarded claim that he was not a capability theorist, we argued Sen had introduced the capability approach as a strategy to extend the consideration of the persons' advantages and disadvantages beyond the problematic confinement of welfarism -according to which social welfare just depends on individual utilities, as we measure them in economics. But from the perspective of readers biased by a welfarist framework, this line of argument is unfortunately straightforwardly reinterpreted as a capability theory, hence as a refinement of welfarism. Regrettably, a capability theory permits the substitution of the demanding requirements of thinking and listening in particular contexts -which was Sen's target -by a systematic appeal to standard statistics -however richer than the previous version of welfarism that might be. We cannot deny that this reception represents progress in the measurement of welfare, but it is fair to say that the message Sen wanted to send has only been partly received.

When the thinker is still alive, there is a third alternative for ensuring that the reception of his works coincides with his original intention, and Sen makes use of this in his memoir. He does recognize a considerable number of influences on his thinking -the name index runs from pages 426 to 442 (!) -and each influence is positively associated with a name deserving of praise, and as being exceptionally important to him. He describes, in varying details, their intellectual legacy (e.g. his grandfather Kshiti Mohan, or his mentors Maurice Dobb and Piero Sraffa). He unequivocally highlights the different potential interpretations of each of these thinkers (e.g. Tagore, Marx) who have been most influential for him. Among these he introduces his own interpretation for example, how Sen receives Tagore or Marx, by contrast with the way that Europeans read Tagore, how leftists or rightists read Marx compared with what Sen was himself most impressed by. If the reader had a different interpretation (of Tagore, or of Marx), they would certainly misunderstand Sen. Hence this memoir readers indirect evidence to guide or realign their reading.

Let me enumerate five significant potential misinterpretations highlighted in the memoir.

First, Sen admits he is an Indian. But he is also a Bengali: his early life was both in Dhaka with his parents, which is today located in Bangladesh, and in Santiniketan with his grand-parents, which is today located in India. And he lived in Burma for a couple of years as a child. He comes from a Hindu family; his grandfather joined the Kabir Panth, which combined Hindu and Muslim religious ideas; and he describes himself as being in his youth rather a-religious. He is not a nationalist -I believe the choice of the title of the memoir is owed to Tagore's novel The home and the World, which he describes as a "'strong and gentle' warning against the corruptibility of nationalism" (p.92). Amartya Sen studied at the educational institution created by Rabindranath Tagore called Sisva Bharathi, which means "uniting the world" and which was formed by the best of the world's cultural heritage. A great deal of the book refers to his many other geographical identities; and it is fair to say that already back in 1963, Cambridge UK and Cambridge USA are just two among many. He could hardly answer the journalist's question presented as the opening of the first chapter: "Where do you consider to be your home?" This would probably mean something different in English for Americans ("where do you normally live?"), or for Bengalis ("Where is/are your families from?"). In any case, the answer would not be straightforward. Sen not only comes from many places in the world, he wishes that his home were not restricted to any specific parish.

Second, Rabindranath Tagore has been a great influence upon Amartya Sen -Tagore was a close friend of his mother's family, but it was only after his death that Amartya began to recognize his major importance. Sen devotes at least four chapters to Tagore's thinking. He recalls that, when Tagore received the Nobel prize for Literature, the Europeans and the Americans presented him as a romantic dreamer, an attractive eastern Indian poet; they admired his alleged mysticism and his rejection of reason. Sen is emphatically surprised by how unlikely and ill-informed this reception was, without denying or ignoring the beauty of Tagore's literature, poetry, theater, music and painting. Sen insists that Tagore had an open mind, he primarily devoted his life to reason and freedom, implying his full respect for scientific methods, and the importance of disentangling ethics from science.

Third, Sen presents himself as a leftist, given his major commitment to the reduction of poverty, inequalities and his certitude that the wider provision of health and education is central for development. He devotes a specific chapter to the reinterpretation of Marx, and describes in detail what he does and does not take from him. He retains Marx's focus on needs, which may explain, he suggests, the welcome creation of the NHS and of the welfare state in many European countries. He appreciates the discussion of objective illusions: one may believe workers and capitalists participate in free exchange, but you may perceive things differently when considering the bargaining power of the worker -whether a poor worker is actually "free" to decline a job. A stark illustration of the illusion of freedom is given by a traumatic experience Sen had as a child. Sen witnessed a man almost dying from a stab wound in his garden. Kader Mia was a Muslim man. Despite the dangerous tensions between the communities in India he was formally free to stay home, or to go and work in the Hindu area. As a father, his desire to feed his family more than anything else, including his own security, almost got him killed. In this example, Sen also shows that those killed in the Hindu-Muslim riots of the 1940s shared a class identity (the workers and dispossessed) although they differed in their religious or communal identity. Whilst freedom can be an illusion, it is important per se, and Sen does say that Marx was sometimes badly interpreted in this regards. He quotes a passage of The German Ideology which Sen says most authors "often tended to miss": Marx is positive about "bringing 'the conditions for the free development and activity of individuals under their own control'" including the ability to "criticize after dinner" -a passage which, we might admit, leaves us with the impression that liberty of choice and freedom of expression might be important but that there is little to be found in Marx with which the sentiment might be defended. Sen observes that Marx did not in his claim regarding the "dictatorship of the proletariat" discuss what the proletariat's demand would be and could mean -for lack of a social choice approach. Not only was this absence "rather disappointing" (p.212), Sen overtly charges Marx responsibility for the authoritarian consequences of his lack of serious consideration of freedom and democratic issues. Sen recalls in several chapters that he was always dubious of Soviet practices (notably the show trials), insisting on the primary importance of democracy and freedom of speech.

Fourth, one of his identities is related to Trinity College Cambridge, renowned for its great intellectuals and its leftist tendencies. But when there he did not believe in the standard opposition between schools of thought that accompanied his years in Cambridge. Among the opposing camps there were neoclassical so-called mainstream economists and the neo-Keynesians rejecting the adequacy of the market economy -and also the Marxists: "The divisions between schools of economic thought seemed to play a mesmerizing role in Cambridge rhetoric, particularly in classifying economists into two distinct categories: friends or foes." (p.288) However Sen, along with a few others, appears very distant from this neverending academic battle. On the one hand, none of these camps would take seriously the fundamental issues he wanted to work on: poverty, inequality, the wider provision of health and education. These issues were "seen as a non-subject" either way, and everybody recommended that he not work on social choice theory and welfare economics. Joan Robinson would assert something that Sen calls the most profound error in development thinking -that "in terms of priorities, what you have to concentrate on is simply maximizing economic growth"; then "you can turn to health care, education…" (p.287-288). Sen concluded that leftist heterodox economists did not think so very differently than mainstream economists did on the issues he thought were essential. In passing, he makes the scathing observation that the most intelligent Professor of Economic Development he met in Cambridge, Peter Bauer, happened to be a rightist conservative neo-classical economist.

On the other hand, let me provide a second explanation for the acrimony we feel between the lines when Sen mentions Joan Robinson. The book devotes an impressive number of lines and pages to various compliments regarding Joan Robinson, noting also how familiar she was with India, how close she and her husband were with Sen. We also learn that he was not very impressed by her work on capital theory, but this theoretical assessment is not the central point he wants to make. Sen instead emphasizes that she never listened to any opposing arguments in any discussion, with any discussant. Her view on the importance Sen devoted to welfare issues a problem for him, not because he was not convinced by her arguments regarding this importance, but because she had never entered this debate in the first place. Sen belongs to no one camp; he belongs the argumentative tradition of Indian philosophy: "I could not help thinking that the argumentative tradition that had been so persistently championed in Indian philosophical debates, and which included careful listening, could have made something of a contribution to Joan's convictions about what makes a thesis powerful. Her neglect of mainstream theories seemed to me to lack a reasoned defence, as did her rapid dismissal of the Marxian perspectives carefully developed by Dobb, Sraffa and Hobsbawm."

For Sen, it is always necessary to provide reasoned argument in support of a thesis. A striking illustration is given in chapter 10. Sen spends pages and pages cautiously and respectfully addressing the thesis that the British Raj brought many advantages to India, before he concludes that he might not be convinced by this thesis after all. Should you (or Joan Robinson) find obvious or superfluous these long factual descriptions, analyses and caveats in elaborating the appropriate counterfactuals, you (she) would definitely miss what Sen finds essential. Assertions are only obvious in a certain context and from a certain perspective; hence it is worth thinking and weighing the reasons that make sense from a diversity of perspectives. Careful listening and consideration of every claim is indispensable, including the claims of the underprivileged, regardless of your institutional status, your worthy leftist sentiments or what is currently considered to be politically correct. The pleasure of any "agenda-less free discussion on any topic that might come up" (p.183) -called "adda" in Bengali -is nicely illustrated in the many pages in which Sen describes how he as a child was involved in discussion with each adult of his family, how while in Cambridge he spent his afternoons in cafés, in the Apostles Society, and in the socialist, liberal and conservative political clubs. Political and intellectual debate seems to be the most constant element of Sen's life, from his very early years with his family -his grand-father in particular -and whoever he encountered in his academic or his personal lives.

Fifth, Sen uncontroversially believes in reason. His knowledge of science, his practice of axiomatic and formal demonstration, is one element. Theorems and data analysis that he developed still play a prominent role in economics and development studies. But he is not what we could call a rationalist thinker, assuming there is a rational truth to be imposed from above. Sen was skeptical of the idea that the relevant questions were technical: "I found it difficult to believe that the downfall of capitalism, if that were to occur, would be caused by some sophisticated mistake in capital theory rather than because of the nasty way capitalism treats human beings" (p.287). Much relevant information may arise in discussion involving a range of stakeholders, not to mention the essential participation of subalterns. Not only that, sentiment is essential in these reasoned debates. Sen quotes A. C. Pigou (p.287): "It is not wonder, but rather the social enthusiasm which revolts from the sordidness of mean streets and the joylessness of withered lives, that is the beginning of economic science." An illustration is given by the Bengal famine in 1943 which killed nearly 3 million people. The economic analysis of the famine was flawed. Seen from London, the quantity of food was sufficient at the macroeconomic level. Eventually the political reaction of the Raj to the famine was only due to indignation on the part of the British public. Information about the famine was eventually published in British newspapers, making the Bengali people closer to the English people: when Bengalis become, in their conscience and sentiments, England's neighbors.

In the cases we have just described (and many others), Amartya Sen is not where we expect him to be, and this does not ease our understanding. He never conforms to either side of the standard "friend or foe" opposition. To take full advantage of his body of works we must avoid confining him to labeled boxes. This both implies we should accept that he has multiple identities (which this memoir very much helps to clarify), and that we need to be aware of the particular perspective and bias from which we read his works.

The memoir is focused on his 30 first years of life, ranging from 1933 to 1963. That it stops somewhat abruptly after 407 pages is striking. 1963 is when Sen starts to become well-known for his contribution to poverty and social welfare issues! I confess this sudden end saddened me. I would have like to read more of his reflections, about the way he looks back on his life after he began to attract wider attention for his thinking . When I closed the book, I was eager to stay in Amartya K. Sen's delightful company. Because I learned so much of the intellectual debates in reading his memoir; because all that I have read in this memoir has opened my mind and clarified my thoughts about the world's woes and possible cures; because the memoir has been as much a pleasure to read as swimming in warm open water in the light of a rising sun. Whether Amartya Sen will have the energy to write the sequel for our own pleasure is still an open question, as I learned from his recent interviews. However, as far as one's history sets a person's identities [START_REF] Giraud | Five decades of HOPE[END_REF]: p. 645, Forget 2005), let us recognize that the two main objectives elaborated above have so far been met.

This text shall be published as a book review in the journal History of Political Economy. I wish to thank Muriel Gilardone, Steve Medema and Keith Tribe for their kind comments on previous versions of this review, however I am only responsible for the personal views expressed in it.

Univ Lyon, UJM Saint-Etienne, GATE UMR 5824, F-42023 Saint-Etienne, France. antoinette.baujard@univst-etienne.fr; Orcid: 0000-0002-4156-7527.


S Richard

R Tiedra De Aldecoa

New formulae for the wave operators for a rank one interaction

We prove new formulae for the wave operators for a Friedrichs scattering system with a rank one perturbation, and we derive a topological version of Levinson's theorem for this model.

Introduction and main results

Let us consider the Hilbert space H := L 2 (R) with norm • and scalar product •, • , and let H 0 ≡ X be the operator of multiplication by the variable, i.e. (H 0 f )(x) = xf (x) for any f ∈ D(H 0 ) ≡ H 1 . For u ∈ H, we also consider the rank one perturbation of H 0 defined by

H u f := H 0 f + u, f u, f ∈ H 1 .
It is well known that the wave operators Ω ± := s-lim t→±∞ e iHut e -iH0t exist and are asymptotically complete, and that the scattering operator S := Ω * + Ω -is a unitary operator in H. In fact, S ≡ S(X) is simply an operator of multiplication by a function S : R → T, with T the set of complex numbers of modulus 1.

A rather explicit formula for the wave operators for this model was proposed in [START_REF] Grubb | Derivation of the wave and scattering operators for an interaction of rank one[END_REF]. Its expression involves singular integrals that have to be manipulated with some care. In this Note, we propose a simpler formula for the wave operators, and put into light a straightforward consequence of it. However, we stress that contrary to [START_REF] Grubb | Derivation of the wave and scattering operators for an interaction of rank one[END_REF], our formula and its corollary hold only if some additional (but weak) hypotheses on u are imposed.

To state our results, we need to introduce the even / odd representation of H. Given any function m on R, we write m e and m o for the even part and the odd part of m. We also set H := L 2 (R + ; C 2 ) and introduce the unitary map U : H → H given by

U f := √ 2 f e f o and U * f1 f2 (x) := 1 √ 2 [f 1 (|x|) + sgn(x)f 2 (|x|)] , f ∈ H, f1 f2 ∈ H , x ∈ R.
Now, observe that if m is a function on R and A the generator of dilations in H, then we have

U m(X)U * = me(X+) mo(X+) mo(X+) me(X+)
and

U AU * = A+ 0 0 A+ ,
where X + is the operator of multiplication by the variable in L 2 (R + ), and A + the generator of dilations in

L 2 (R + ), namely (e itA+ f )(x) := e t/2 f (e t x) for f ∈ L 2 (R + ), x ∈ R + .
In the sequel we assume that the vector u satisfies the following assumption.

Assumption 1.1. The function u ∈ H is Hölder continuous with exponent α > 0. Furthermore, if x 0 ∈ R satisfies u(x 0 ) = 0 and 1 -R dy |u(y)| 2 (x 0 -y) -1 = 0, then there exists an exponent α ′ > 1/2 such that

|u(x) -u(y)| ≤ Const. |x -y| α ′
for all x, y near x 0 .

This assumption clearly implies that u is bounded and satisfies lim |x|→∞ u(x) = 0. Furthermore, if u ∈ H is Hölder continuous with exponent α > 1/2, then the previous assumption holds. On the other hand, under Assumption 1.1, the operator H u satisfies the hypotheses of [1, Sec. 2], and so it has at most a finite number of eigenvalues of multiplicity one.

Our main result is the following representation of the wave operator Ω -in H .

Theorem 1.2. Let u satisfy Assumption 1.1. Then, one has

U Ω -U * = ( 1 0 0 1 ) + 1 2 1 -tanh(πA+)+i cosh(πA+) -1 -tanh(πA+)-i cosh(πA+) -1 1
Se(X+)-1 So(X+) So(X+) Se(X+)-1 + K, (1.1) where K is a compact operator in H .

Let us immediately mention that a similar formula holds for Ω + . Indeed, by using Ω + = Ω -S(X) one gets

U Ω + U * = ( 1 0 0 1 ) + 1 2 1 tanh(πA+)-i cosh(πA+) -1 tanh(πA+)+i cosh(πA+) -1 1 Se(X+)-1 So(X+) So(X+) Se(X+)-1 + K ′ ,
where K ′ is a compact operator in H . We also note that the proof of Theorem 1.2 will make clear why the minimal hypothesis u ∈ H is not sufficient in order to prove the claim. We now state a corollary of the theorem. Its proof is a straightforward consequence of formula (1.1), even if it will require the introduction of an algebraic framework. Such a result was already known for more general perturbations but under stronger regularity conditions [START_REF] Buslaev | Spectral identities and the trace formula in the Friedrichs model[END_REF][START_REF] Dreyfus | The determinant of the scattering matrix and its relation to the number of eigenvalues[END_REF] (see also [START_REF] Faddeev | On a model of Friedrichs in the theory of perturbations of the continuous spectrum[END_REF][START_REF] Yafaev | Mathematical scattering theory. General theory[END_REF] for general informations on the Friedrichs model). Our result does require neither the differentiability of the scattering matrix nor the differentiability of u. Nonetheless, if S is continuously differentiable, then the winding number can also be expressed in terms of an integral involving the (on-shell) time delay operator, which is the logarithmic derivative of the scattering matrix [START_REF] Tiedra De Aldecoa | Time delay for dispersive systems in quantum scattering theory[END_REF].

The content of this Note is the following. In Section 2 we prove Formula (1.1) and derive some auxiliary results. In Section 3 we give a description of the algebraic framework involved in the proof of the Corollary 1.3, which is proved at the end of the section.

Derivation of the new formula

We start by recalling some notations and results borrowed from [START_REF] Alsholm | Inverse scattering theory for perturbations of rank one[END_REF] and [START_REF] Grubb | Derivation of the wave and scattering operators for an interaction of rank one[END_REF]. We shall always suppose that u satisfies Assumption 1.1.

For x ∈ R and ε > 0 we set

I ε ± (x) := R dy |u(y)| 2 x -y ± iε .
The limit

I ± (x) := lim εց0 I ε ± (x) exists for all x ∈ R [1, Lemma 1.(a)
]. Furthermore, the set of x such that I ± (x) = 1 is finite [1, p. 396]. In consequence the expression [1 -I ± (x)] -1 is well defined for almost every x ∈ R, and the domain

D ± of [1 -I ± (X)] -1 in H is dense. Let F denote the Fourier transform in H, namely (F f )(x) := 1 √ 2π R dy e -ixy f (y), f ∈ H ∩ L 1 (R).
Given a Borel function m on R, we set m(D) := F * m(X)F . Finally, χ (-∞,0) stands for the characteristic function for the half-line (-∞, 0).

We are now in a position to recall the formula [8, eq. ( 56b)] for Ω -. One has

Ω -= 1 -2πiu(X)χ (-∞,0) (D)u(X)[1 -I + (X)] -1
on D + (note that we use a convention for the sign ± of the wave operators Ω ± which differs from the one of [START_REF] Grubb | Derivation of the wave and scattering operators for an interaction of rank one[END_REF]). So, if

K := -2πi[u(X), χ (-∞,0) (D)]u(X)[1 -I + (X)] -1 , then one gets on D + Ω --1 = -2πiu(X)χ (-∞,0) (D)u(X)[1 -I + (X)] -1 = χ (-∞,0) (D) -2πi|u(X)| 2 [1 -I + (X)] -1 + K = χ (-∞,0) (D){S(X) -1} + K, (2.1)
by using [8, Eq. (66b)] in the last equality.

In the next lemma, we determine the action of χ (-∞,0) (D) in H . For that purpose, we define

φ ∈ C([-∞, ∞]; T) by φ(x) := tanh(πx) + i cosh(πx) -1 for all x ∈ R. Lemma 2.1. One has U χ (-∞,0) (D)U * = Φ(A + ), where Φ(A + ) := 1 2 1 -φ(A+) -φ(A+) 1
.

Proof. The usual Hilbert transform H on R satisfies sgn(D) = iH. Thus

χ (-∞,0) (D) = 1 2 1 -sgn(D) = 1 2 (1 -iH). (2.2)
Using the expression for iH in terms of the generator of dilations in H given in [10, Lemma 3], one gets

U iHU * = 0 tanh(πA+)-i cosh(πA+) -1 tanh(πA+)+i cosh(πA+) -1 0 . (2.
3)

The claim follows then from (2.2) and (2.3).

We now recall some results on the scattering matrix. The last lemma deals with the remainder term K of Formula (1.1).

Lemma 2.3. Let u satisfy Assumption 1.1. Then the operator [u(X), χ (-∞,0) (D)]u(X)[1 -I + (X)] -1
, defined on D + , extends to a compact operator in H.

Proof. (i) Define for all x ∈ R the function ψ(x) := u(x)[1 -I + (x)] -1 . We know that u is bounded and that lim |x|→∞ u(x) = 0. We also know from [1, Lemma 1.(a)] that I + is Hölder continuous and that lim |x|→∞ I + (x) = 0. So, outside any neighbourhood of the finite set of points where I + equals 1, the function ψ is bounded. Furthermore, Assumption 1.1 and [1, Lemma 1.(c)], imply that ψ is locally square integrable (see also [8, p. 2423]). Therefore, ψ can be written as

ψ = ψ ∞ + ψ 2 , with ψ ∞ ∈ L ∞ (R) and ψ 2 ∈ L 2 (R)
with support in a small neighbourhood of the points where I + equals 1.

We now show the compacity of the operators [u(X),

χ (-∞,0) (D)]ψ ∞ (X) and [u(X), χ (-∞,0) (D)]ψ 2 (X). (ii) Choose a function ϕ 1 ∈ C ∞ (R) and a function ϕ 2 ∈ L ∞ (R) with compact support such that ϕ 1 +ϕ 2 = χ (-∞,0) . Then [u(X), ϕ 1 (D)] is compact due to [3, Thm. C], and [u(X), ϕ 2 (D)] is Hilbert-Schmidt due to [11, Thm. 4.1]. So [u(X), χ (-∞,0) (D)]ψ ∞ (X) = [u(X), ϕ 1 (D)]ψ ∞ (X) + [u(X), ϕ 2 (D)]ψ ∞ (X)
is a compact operator.

(iii) For each f ∈ H and almost every x ∈ R, define the operator

(K 0 f )(x) := i 2π R dy u(x) -u(y) y -x ψ 2 (y)f (y).
From the Assumption 1.1 we know that

|u(y + x) -u(y)| ≤ Const. |x| α ′ , α ′ > 1/2
for each y ∈ supp(ψ 2 ) and each x ∈ R with |x| small enough. In particular, there exists δ > 0 such that

4π 2 R 2 dx dy i 2π u(x) -u(y) y -x ψ 2 (y) 2 = R dy R dx |u(y + x) -u(y)| 2 x 2 |ψ 2 (y)| 2 = R dy R\[-δ,δ] dx |u(y + x) -u(y)| 2 x 2 |ψ 2 (y)| 2 + R dy δ -δ dx |u(y + x) -u(y)| 2 x 2 |ψ 2 (y)| 2 ≤ R dy R\[-δ,δ] dx 4 u 2 ∞ x 2 |ψ 2 (y)| 2 + Const. R dy δ -δ dx |x| 2(α ′ -1) |ψ 2 (y)| 2 < ∞.
Thus, K 0 is a Hilbert-Schmidt operator. Furthermore, we have for f ∈ D + and almost every

x ∈ R [u(X), χ (-∞,0) (D)]ψ 2 (X)f (x) = -i 2 [u(X), H]ψ 2 (X)f (x) = -i 2π R dy u(x) -u(y) x -y ψ 2 (y)f (y) = (K 0 f )(x).
Therefore, the operator [u(X), χ (-∞,0) (D)]ψ 2 (X) extends to an Hilbert-Schmidt operator.

Algebraic framework

This section is dedicated to the presentation of the algebraic framework leading to Corollary 1.3. Since a similar construction already appears in [START_REF] Kellendonk | On the structure of the wave operators in one dimensional potential scattering[END_REF] for the proof of Levinson's theorem in one dimensional potential scattering, we only sketch the construction and refer to this reference for more details.

We start by giving the definition of the Mellin transform associated with the generator of dilations A + in L 2 (R + ) (see [START_REF] Jensen | Time-delay in potential scattering theory, some "geometric" results[END_REF]Sec. 2] for a general presentation when the operator acts in L 2 (R n )). Let V : L 2 (R + ) → L 2 (R) be defined by (V f )(x) := e x/2 f (e x ) for x ∈ R, and remark that V is a unitary map with adjoint V * given by (V * g)(x) = x -1/2 g(ln x) for x ∈ R + . Then, the Mellin transform M : L 2 (R + ) → L 2 (R) is defined by M := F V . Its main property is that it diagonalizes the generator of dilations, namely, M A + M * = X. Formally, one also has M ln(X + )M * = -D.

Let us now recall from Remark 2.4 that under the Assumption 1.1 the wave operators Ω ± are isometries with range projection 1 -P p , where P p is the projection onto the subspace spanned by the finite number N of eigenvectors of H u . In particular, Ω -is a Fredholm operator with index(Ω -) = -Tr(P p ) = -N . Furthermore, we recall that any Fredholm operator F in H is invertible modulo a compact operator, that is, its image q(F ) in the Calkin algebra B(H)/K(H) is invertible. Now, assume that Ω -belongs to a norm-closed subalgebra E of B(H) containing K(H). Moreover, assume that E/K(H) is isomorphic to C S; M 2 (C) , the algebra of continuous functions over the circle with values in the 2 × 2 matrices. Then, viewing q(Ω -) as such a function, we can take pointwise its determinant to obtain a non-vanishing function over the circle. The winding number of that latter function can be related to the index of Ω -; this is essentially the content of Corollary 1.3.

The choice of E is suggested by the formula obtained in Theorem 1.2. Indeed, we consider the closure E in B(H ) of the algebra generated by elements of the form ϕ(A + )ψ(X + ), where ϕ is a continuous function on R with values in M 2 (C) which converges at ±∞, and ψ is a continuous function on R + with values in M 2 (C) which converges at 0 and at +∞.

Stated differently, ϕ ∈ C R; M 2 (C) with R = [-∞, ∞], and ψ ∈ C R + ; M 2 (C) with R + = [0, ∞].
Let J be the norm closed algebra generated by ϕ(A + )ψ(X + ) with functions ϕ and ψ for which the above limits vanish. Then, J is an ideal in E , and the same algebras are

obtained if ψ(X + ) is replaced by η(ln(X + )) with η ∈ C R; M 2 (C) or η ∈ C 0 R; M 2 (C) , respectively.
These algebras have already been studied in [START_REF] Georgescu | C * -algebras of quantum Hamiltonians[END_REF] in a different context. The authors constructed them in terms of the operators X and D on L 2 (R, E), with E an auxiliary Hilbert space, possibly of infinite dimension. In that situation, ϕ and η are norm continuous functions on R with values in K(E). The isomorphism between our algebras and the algebras introduced in [START_REF] Georgescu | C * -algebras of quantum Hamiltonians[END_REF]Sec. 3.5], with E = C 2 , is given by the Mellin transform M , or more precisely by M ⊗ 1, where 1 is identity operator in M 2 (C). For that reason, we shall freely use the results obtained in that reference, and refer to it for the proofs. For instance, it is proved that J = K(H ), and an explicit description of the quotient E /J is given, which we specify now in our context.

To describe the quotient E /J , we consider the square := R + × R, whose boundary ∂ is the union of four parts: [START_REF] Georgescu | C * -algebras of quantum Hamiltonians[END_REF]Thm. 3.20] that E /J is isomorphic to C ∂ ; M 2 (C) . This algebra can be seen as the subalgebra of

∂ ≡ B 1 ∪ B 2 ∪ B 3 ∪ B 4 , with B 1 := {0} × R, B 2 := R + × {∞}, B 3 := {∞} × R, and B 4 := R + × {-∞}. It is proved in
C R; M 2 (C) ⊕ C R + ; M 2 (C) ⊕ C R; M 2 (C) ⊕ C R + ; M 2 (C) (3.1)
given by elements γ ≡ (γ 1 , γ 2 , γ 3 , γ 4 ) which coincide at the corresponding end points, that is,

γ 1 (∞) = γ 2 (0), γ 2 (∞) = γ 3 (∞), γ 3 (-∞) = γ 4 (∞), and γ 4 (0) = γ 1 (-∞). Furthermore, for any ϕ ∈ C R; M 2 (C) and ψ ∈ C R + ; M 2 (C) , the image of ϕ(A + )ψ(X + ) through the quotient map q : E → C ∂ ; M 2 (C) is given by γ 1 = ϕψ(0), γ 2 = ϕ(∞)ψ, γ 3 = ϕψ(∞) and γ 4 = ϕ(-∞)ψ.
From what precedes we deduce that the subalgebra E of B(H), defined by E := U * E U , contains the ideal of compact operators on H and that the quotient E/K(H) is isomorphic to C ∂ ; M 2 (C) ∼ = C S; M 2 (C) . We are thus in the framework defined above, and the for any invertible element γ of C ∂ ; M 2 (C) , the winding number of its pointwise determinant is a well-defined quantity. So we are ready to give the proof of Corollary 1.3.

Proof of Corollary 1.3. We know from Theorem 1.2 and Lemma 2.2 that U Ω -U * ∈ E , or equivalently that Ω -∈ E. Due to Formula (1.1), the element γ belonging to (3.1) and associated with q(Ω -) is given by suitable restrictions of the function Γ : R + × R → M 2 (C), where Γ(x, y) := 1 + Φ(y) se(x)-1 so(x) so(x) se(x)-1 = 1 2 se(x)-φ(y)s o (x)+1 so(x)-φ(y)[se(x)-1] so(x)-φ(y)[se(x)-1] se(x)-φ(y)so(x)+1 . Namely, γ 1 = Γ(0, • ), γ 2 = Γ( • , +∞), γ 3 = Γ(+∞, • ), and γ 4 = Γ( • , -∞). The pointwise determinants of these functions are easily calculated by using the identity φ(±∞) = ±1: one gets det γ 1 (y) = s e (0), det γ 2 (x) = s(-x), det γ 3 (y) = 1 and det γ 4 (x) = s(x).

The precise relation between the winding number of the map det γ : ∂ → T and the index of Ω -has been described in [START_REF] Kellendonk | On the structure of the wave operators in one dimensional potential scattering[END_REF]Prop. 7]. However, the algebra corresponding to E in that reference was defined in terms of the operators A + and B + = 1 2 ln (D 2 ) + which satisfy the relation [iA + , B + ] = -1. In our case, the algebra E has been constructed with the operators A + and ln(X + ) which satisfy the relation [iA + , ln(X + )] = 1. Therefore, in order to apply [START_REF] Kellendonk | On the structure of the wave operators in one dimensional potential scattering[END_REF]Prop. 7] in our setting, one previously needs to apply the automorphism of C R; M 2 (C) defined by η(x) := η(-x) for all x ∈ R, or equivalently the automorphism of C R + ; M 2 (C) defined by ψ(x) := ψ(x -1 ) for all x ∈ R + . Therefore the pointwise determinants of the function γ j associated with q(Ω -) are det γ 1 (y) = 1, det γ 2 (x) = s(-x -1 ), det γ 3 (y) = s e (0) and det γ 4 (x) = s(x -1 ).

Now, [START_REF] Kellendonk | On the structure of the wave operators in one dimensional potential scattering[END_REF]Prop. 7] reads ω(det γ) = index(Ω -) = -N , where N is the number of eigenvalues of H u . The convention used in that reference for the calculation of the winding number implies that the contribution of x → det γ 2 (x) is from x = 0 to x = +∞ and the contribution of x → det γ 4 (x) is from x = +∞ to x = 0. This corresponds to the calculation of the winding number of x → det S(x) , from x = -∞ to x = +∞. Since the contributions of det γ 1 and det γ 3 are null because these terms are constant, the claim is proved.

Corollary 1 . 3 .

 13 Let u satisfy Assumption 1.1. Then S(±∞) = 1 and the following equality holds: ω(S) =number of eigenvalues of H u , where ω(S) is the winding number of the continuous map S : R → T.

Lemma 2 . 2 .

 22 Let u satisfy Assumption 1.1. Then the map S belongs to C([-∞, ∞]; T) and satisfies S(±∞) = 1. Proof. The continuity of S follows from [1, Thm. 1.(i)]. The asymptotic equalities S(±∞) = 1 follow from the formula S(x)-1 = -2πi|u(x)| 2 [1-I + (x)] -1 together with [1, Lemma 1.(a)] and the fact that lim |x|→∞ |u(x)| 2 = 0.

  Proof of Theorem 1.2. The operator K extends to a compact operator due to Lemma 2.3. So Equation (2.1) holds on H, and the claim follows from Lemma 2.1. The proof of Corollary 1.3 relies on the fact that the range of the wave operators is the orthocomplement of the subspace spanned by the eigenvectors of H u . Since the wave operators are complete, such a property holds if and only if H u has no singularly continuous spectrum. Now, by using the characterization of the singular spectrum recalled in[5, p. 299] and by taking into account Lemmas 1 and 2 of[START_REF] Alsholm | Inverse scattering theory for perturbations of rank one[END_REF] (which are valid since u satisfies Assumption 1.1), one easily gets that the singular spectrum of H u only consists of a finite set. So Assumption 1.1 implies the absence of singularly continuous spectrum for H u .

	Remark 2.4.

Acknowledgements

S. Richard is supported by the Swiss National Science Foundation. R. Tiedra de Aldecoa is partially supported by Núcleo Científico ICM P07-027-F "Mathematical Theory of Quantum and Classical Magnetic Systems" and by the Chilean Science Fundation Fondecyt under the Grant 1090008.


Adrien Dubouloz

Parag David R Finston
email: dfinston@nmsu.edu

Deepak Mehta
email: pmehta@muwci.net

Parag Deepak

Mehta Factorial

Parag Deepak Mehta

Deepak Parag

Mehta

Factorial threefolds with Ga-actions

come

Introduction

Let G a denote the additive group of complex numbers, and X a complex affine variety. Throughout, an action of G a on X means an algebraic action. Every such action can be realized as the exponential of some locally nilpotent derivation δ of the coordinate ring C[X] of X and every locally nilpotent derivation gives rise to an action. The ring C 0 of G a -invariants in C[X] is equal to the ring of constants of the generating derivation.

An action σ : G a × X → X is said to be equivariantly trivial if there is a variety Y for which X is G a -equivariantly isomorphic to Y × G a , where G a acts on Y × G a by translations on the second factor. In this case Y is affine and X the total space of a trivial G a -bundle over Y. A section for the bundle can be identified with the zero locus of a regular function s ∈ C[X] for which δs = 1. Such a function is called a slice; if one exists, C[X] = C 0 [s] and Y ∼ = Spec C 0 . The action is said to be locally trivial if a geometric quotient π : X → Y exists in the category of algebraic spaces for which π is a principal G a -bundle. The action is said to be proper if the morphism σ × id : G a × X → X × X 1991 Mathematics Subject Classification. Primary 14R20, Secondary 20G20.

1 is proper.
The main point of the paper is to investigate the class of hypersurfaces X m,n : x m vy n u = 1 in A 4 = Spec(C[x, y, u, v]), where m, n are positive integers. Since the polynomial x m vy n u = 1 is an invariant for the G a action t(x, y, u, v) = (x, y, u + tx m , v + ty n ) on A 4 , the X m,n are stable under this action. Moreover, with A 2 * = A 2 \{0}, the complement of the origin in the complex affine plane, the X m,n , or rather the cohomology classes in H 1 (A 2 * , O A 2 * ) of the Čech coycles x -m y -n ∈ C[x ±1 , y ±1 ], constitute a natural basis for the set of isomorphy classes of principal G a -bundles over A 2 * . As seen below, these varieties can be viewed as globalizing the local structure of arbitrary smooth threefolds with locally trivial G a -action.

Similarly constructed threefolds were investigated in [START_REF] Finston | The automorphism group of certain factorial threefolds and a cancellation problem[END_REF] as factorial counterexamples to a generalized cancellation problem:

If X × A 1 ∼ = Y × A 1 is X ∼ = Y ?
Those examples arise from principal G a -bundles over the smooth locus of factorial surfaces of logarithmic Kodaira dimension 1, hence admit no nonconstant morphism from A 2 . As such they differ substantially from affine three space. As total spaces for G a -bundles over A 2 * the varieties X m,n considered here are closer in spirit to affine spaces and also have isomorphic cylinders. For distinct pairs (m, n) the corresponding varieties X m,n represent nonisomorphic G a -bundles, nevertheless some of them turn out to be isomorphic as varieties. We do not know as yet whether any pair of them provides a counterexample to cancellation.

For factorial affine varieties, i.e. those whose coordinate ring is a unique factorization domain (UFD), local triviality is equivalent with the intersection of C 0 with the image of δ generating the unit ideal in C[X] [2]. In this case the geometric quotient has the structure of a quasiaffine variety. To see this explicitly, consider a locally trivial G a -action on a factorial affine variety X and let δ(a 1 ), ..., δ(a n ) ∈ C 0 generate the unit ideal in C[X]. The geometric quotient is embedded as an open subvariety in Spec R for an integrally closed ring R constructed as follows. Set R

i = C[X, 1 δ(a i ) ] Ga . Note that C[X, 1 δ(a i ) ] = R i [ a i δ(a i ) ] so that R i is a finitely generated C-algebra, say R i = C[b i1 , ..., b im , 1 δ(a i ) ], with b ij ∈ C 0. Define R to be the integral closure of C[b ij , δ(a i )|1 ≤ i ≤ n, 1 ≤ j ≤ m].
The map π : X → Spec(R) induced from the ring inclusion is smooth, hence open, and endows its image U with the structure of a geometric quotient. Moreover, for the open cover

U = {U i = Spec(R i )} 1≤i≤n of U , the Čech cocycle ( a 1 δ(a 1 ) - a 2 δ(a 2 ) , . . . , a n-1 δ(a n-1 ) - a n δ(a n ) ) ∈ C 1 (U, O U )
defines the structure of π : X → U as a principal G a -bundle.

Recall that for an integral domain R with quotient field K, the transform of R with respect to the ideal I, written

T I (R) is ∪ n>0 {α ∈ K|αI n ⊂ R}.
The ring of G a -invariants is isomorphic to the ring of sections of the structure sheaf of Spec(R) on U , hence to T I (R) where I defines the closed subscheme Z = Spec(R) \ U equipped with its reduced structure. It is clear from the construction that

I = (δ(a 1 ), ..., δ(a n )). Since R is normal, R = C 0 if all components of Z have codimension ≥ 2 (equivalently if I is contained in no height one prime ideal of R).
In case the dimension of X is less than or equal to three, a theorem of Zariski [14, p.45] implies that C 0 is a finitely generated C-algebra. Combined with the fact that the complement of a pure codimension one subvariety of a two dimensional normal affine variety is again an affine variety [14, p.45], the following ring theoretic criterion for triviality of a G a -bundle with a three dimensional affine factorial total space X is obtained:

Let X be an affine factorial threefold with a locally trivial G a -action generated by the locally nilpotent derivation δ of C[X]. With the ring R, ideal I,and Z, U as above, let J = ∩ I⊂p ht p=1 p . Then C 0 = T J R and the action is equivariantly trivial if and only if J = I. Indeed, since the complement W of the zero locus of J in Spec(R) is affine, every principal G a -bundle over W is trivial. If J properly contains I then U is equal to the complement of a finite but nonempty subset of Spec(C 0 ) (since I is a radical ideal, the Nullstellensatz implies that it has no embedded primes). In particular, U is not affine, and therefore neither is U × A 1 .

More generally, a set-theoretically free action of an algebraic group on a normal variety admits a geometric quotient in the category of algebraic spaces, and the quotient map is a locally trivial fiber bundle in the étale topology. In this context, the properness of the action is then equivalent to the separatedness of the algebraic space quotient. Restricting now to proper G a -actions on normal quasiaffine threefolds, we obtain that the quotient exists as a quasiprojective variety and that the quotient map is locally trivial in the Zariski topology. Indeed, the quotient embeds as an open subset of a normal two dimensional analytic space. By Chow's lemma we know that a desingularization is quasiprojective, hence the quotient is the image of a quasiprojective variety under a proper morphism. As such it is again quasiprojective. That the quotient morphism is a Zariski locally trivial G a -bundle follows for instance from [START_REF] Mumford | Geometric Invariant Theory 3 rd edition[END_REF]Prop. 0.9] All proper G a -actions on quasiaffine surfaces are known to be equivariantly trivial. For normal surfaces this was observed in [START_REF] Deveney | Local triviality of proper Ga actions[END_REF], but again, an application of [13, Prop. 0.9] gives the result in general. All fixed point free actions on contractible threefolds are equivariantly trivial [START_REF] Kaliman | C+-Actions on contractible threefolds Michigan Math[END_REF]. Examples of a nonlocally trivial proper action on A 5 and on a smooth factorial fourfold are also discussed in [START_REF] Deveney | Local triviality of proper Ga actions[END_REF]. The latter example is isomorphic to a cylinder over an affine variety which itself has the structure of the total space of a principal G a -bundle over the punctured affine plane A 2 * , the topic of the subsequent sections of the present work. Except for special kinds of actions [START_REF] Deveney | Twin Triangular actions[END_REF], the issue of local triviality of proper G a -actions on the affine four-space A 4 is unsettled.

Replacing the smoothness hypothesis on the affine threefold X by factoriality and smoothness of the ring of G a -invariants, the following lemma of Miyanishi [START_REF] Miyanishi | Algebraic characterizations of the affine 3-space[END_REF] leads to a local description of X.

Y = Spec(O). Let U = Y \{m}. Assume that f U : f -1 (U ) → U is an A 1 - bundle. Then either X ∼ = Y × A 1 or f -1 (m) = ∅ (the latter is only possible if n = 2).
Corollary 1. Let X be a smooth affine factorial threefold with a locally trivial G a -action that is not equivariantly trivial. Let δ be the derivation generating the action and

I = √ C 0 ∩ ker δC[X]. Assume that C 0 is regular. For a maximal ideal m of C 0 , set S = C 0 \ m. Then with u, v denoting algebraically independent indeterminates, (1) S -1 C[X] ∼ =(C 0 ) m [u] if I m and (2) S -1 C[X] ∼ =(C 0 ) m [u, v]/(au-bv -1) for some regular sequence (a, b) in m if I ⊆ m.
Proof. Since S ⊂ C 0 , the action extends to S -1 C[X]. Since the action is not equivariantly trivial ,

IC 0 = C 0 . If I m, then S -1 C[X] contains a slice. If I ⊆ m, then apply the lemma to O = (C 0 ) m and A = S -1 C[X] to see that mS -1 C[X] = C[X]. On the other hand, no proper principal ideal of (C 0 ) m blows up in S -1 C[X].
The X m,n above represent a class of global versions of case 2.

Example 1. Using a cover by the principal open subsets x = 0, y = 0, one sees that a "basis" for the principal G a -bundles over A 2 * is given by the affine varieties X m,n ⊂ A 4 defined by x m vy n u -1 = 0 where m, n are positive integers. These correspond to the Čech 1-cocycles x -m y -n relative to the given cover of A 2 * . The X m,n are smooth factorial threefolds with locally trivial G a -action generated by the locally nilpotent derivation

δ : u → x m → 0, v → y n → 0. Since A 2 * is a geometric quotient, C 0 = C[x, y] ∼ = Γ(A 2 * , O A 2 * ).
Regularity of C 0 is not a necessary condition for the conclusion of the corollary.

Example 2. Let A = C[x, y, z]/(x 2 + y 3 + z 5 ), the coordinate ring for the analytically factorial singular surface S, and set U = S \ {0}, the smooth locus of S . Using the cover of U by the two open subsets S x ∪ S y ,, the cohomology classes of the Čech cocycles x -m y -n z k , 0 ≤ k ≤ 4 constitute a basis for H 1 (U, O U ). Consider the affine threefolds Z m,n,k ⊂ A 5 defined by

x m v -y n u -z k = 0, x 2 + y 3 + z 5 = 0, 0 ≤ k ≤ 4 for 0 ≤ k ≤ 4 . With π m,n,k : Z m,n,k → S induced by the ring inclusion A ⊂ C[Z m,n,k ],
the cocycle x -m y -n z k corresponds to the principal G abundle over U with total space E m,n,k = π -1 (U ) and bundle projection π. Then E m,n,k is smooth, factorial, affine threefold admitting a locally trivial G a -action with quotient isomorphic to U .

Smoothness of E m,n,k is clear from the defnition and factorialty follows from the fact that π is a Zariski fibration with base and fiber having trivial Picard group [START_REF] Magid | The Picard sequences of a fibration[END_REF]. In fact E m,n,k is the quotient of a principal G a -bundle over A 2 * by the action of a finite group (the binary icosahedral group). We will see in the next section that all nontrivial G a -bundles over A 2 * have affine total spaces and deduce that the E m,n,k are affine by Chevalley's theorem.

The counterexamples to cancellation in [START_REF] Finston | The automorphism group of certain factorial threefolds and a cancellation problem[END_REF] are constructed analogously to the E m,n,k , i.e. as principal G a -bundles over surfaces S a,b,c :

x a + y b + z c = 0, 1 a + 1 b + 1 c < 1
punctured at the singular point. The numerical criterion 1 a + 1 b + 1 c < 1 enabled the distinction of the total spaces as affine varieties. We do not have analogous results for the E m,n,k .

2. Principal G a -bundles over A 2 * = A 2 \ {0} We start with the observation that the total spaces of nontrivial principal G a -bundles over A 2 * are always affine schemes. Indeed, a nontrivial Čech cocycle with value in O A 2 * for the covering of A 2 * by the open subsets {x = 0} and {y = 0} can be written as g = p(x, y)x -m y -n ∈ C[x ±1 , y ±1 ] with deg x p < m, deg y p < n and denote by X(m, n, p) the total space for the so determined bundle over

A 2 * . Set A = C[x, y, u, v]/(x n v -y m u -p(x, y))
so that X(m, n, p) is isomorphic to Spec(A) \V (I) where I = (x, y)A.

If p(0, 0) = 0 then X(m, n, p) is the zero locus of x m v-y n u-p(x, y) in A 4 . Assume then that p(0, 0) = 0 and note that I is a height one prime ideal in A.

We show that IT

I (A) = T I (A) to conclude that X(m, n, p) = Spec(A)\V (I) is affine. Write p(x, y) = k i=0 p i (x)y i
Case 1 : p 0 (x) = 0. Let a ≥ 1 be the multiplicity of 0 as a root of p and write p 0 (x) = x a q 0 (x). From p(x, y) = x a q 0 (x) + y

k i=1 p i (x)y i-1 we obtain x m-a -q 0 y = y n-1 u + k i=1 p i (x)y i-1 x a ∈ T I (A).
Thus q 0 (x) ∈ IT I (A), but q 0 (0) = 0 implies that IT I (A) = T I (A). Case 2 : p 0 (x) = 0. Write p(x, y) = y b k-b i=0 q i (x)y i with q 0 (x) = 0. From

x m v = y b [y n-b u + k-b i=0 q i (x)y i ] we obtain w = v y b = y n-b u + k-b i=0 q i (x)y i ] x m ∈ T I (A).
Then v = y b w so that in T I (A),

x m y b w -y n u = y b k-b i=0 q i (x)y i x m w -y n-b u = k-b i=0 q i (x)y i . Replace A with A ′ = C[x, y, u, v]/(x m w -y n-b u -k-b i=0 q i (x)y i ) to re- duce to Case 1.
The topological universal cover of the E m,n,k of the previous section can be viewed as a principal G a -bundle over A 2 * and therefore as a "linear combination" of X m,n in the sense of Čech 1-cocycles. Indeed, for G the binary icosahedral group embedded in GL 2 (C) it is well known that A 2 * /G ∼ = U and that the étale mapping π :

A 2 * → A 2 * /G is the universal topological covering. Thus, given a nontrivial G a -bundle E m,n,k over U, we obtain by base extension E m,n,k = A 2 * × U E m,n,k a nontrivial principal G a -bundle over A 2
* whose total space yields the universal covering space of E m,n,k . Moreover, E m,n,k is recovered as E m,n,k /G where the finite group G acts freely on the first factor. It follows that each E m,n,k is affine, smooth, and factorial. Finally, since E m,n,k → E m,n,k ∼ = E m,n,k /G is a finite morphism, E m,n,k is affine (and smooth, factorial) by Chevalley's theorem.

The varieties X m,n (resp. E m,n,k ) in the preceding examples are total spaces for principal G a -bundles over the quasiaffine A 2 * (resp. U ). The base extension X m,n × A 2 * X p,q is therefore a principal G a -bundle over both X m,n and X p,q for any m, n, p, q. Since they are affine, we obtain X m,n × A 1 ∼ = X p,q × A 1 and the same holds true for the cylinders over the E m,n,k . The well known Danielewski surfaces D(n) given as the zero loci of the polynomials y 2 -2x n z -1, n > 0 in A 3 provide smooth but nonfactorial counterexamples to the generalized cancellation problem. In fact for n = m, D(n) and D(m) are not even homeomorphic in the Euclidean topology. In contrast, all X m,n (resp. E m,n,k ) are homeomorphic to A 2 * × A 1 (resp. U × A 1 since, as principal bundles, the Euclidean topology of the base has a countable basis and the fiber is solid [START_REF] Steenrod | Fibre Bundles[END_REF].

These considerations apply to any of the affine surfaces in A 3 with a rational double point (quotient singularity) at the origin. Principal G abundles over the quasiaffine surface obtained by puncturing at the singular point will again produce potential counterexamples to the generalized affine cancellation problem. The surface defined by x 2 + y 3 + z 5 = 0 was chosen for its historical significance and the fact that its singularity is the unique one that is analytically factorial (hence "closest" to the affine plane A 2 ). But we do not know whether or not the total spaces are isomorphic as varieties On the other hand, the construction does provide cancellation counterexamples for other quotient singularities. The following example relies on the Makar Limanov invariant of an affine domain A : M L(A) = ∩{ker δ : δ is a locally nilpotent derivation of A} Example 3. Let U = S \ {0} where S ⊂ A 3 is defined by

x 2 + y 2 z + z c = 0, c ≥ 3.
It is well known that S ∼ = A 2 /G where G is the binary dihedral group of order 4c. Define X m,n (G) ⊂ S × A 2 by x m vy n u -1 = 0 where m, n are positive integers. Then

(1) X m,n (G) is not factorial, but M L(C[X m,n (G)]) = C[S].
To see this, let δ be locally nilpotent with kernel R, and K = Frac(R). Then the extension of δ to K ⊗ R C[X m,n (G)] has a slice s and

K ⊗ R C[X m,n (G)] ∼ = K[s], δ(s) = 1
and K is the kernel of the extended derivation. If deg s (z) > 0, then z(s) divides both x(s), y(s) contradicting x m v-y n u = 1. Thus z ∈ R from which it follows that both x, y ∈ R. (2) X m,n (G) ∼ = X p,q (G) if and only if (m, n) = (p, q). This follows from arguments analogous to those in [START_REF] Finston | The automorphism group of certain factorial threefolds and a cancellation problem[END_REF] and the fact that any automorphism of C[Y ] is of the form (x, y, z) → (µ 1 x, µ 2 y, µ 3 z) for certain roots of unity µ i [START_REF] Masuda | Etale endomorphsims of algeraic surfaces[END_REF].

(3) A 2 * × U X m,n (G) → X m,n (
G) is an étale covering. Note that the total spaces A 2 * × U X m,n (G) of these principal G a -bundles over A 2 * are not isomorphic as G-varieties, but it is possible that there is some abstract isomorphism between them.

It is a classical result going back to Klein that the ring of invariants C[x, y]

G for the natural action of a finite subgroup G of SL 2 (C) on A 2 corresponding to the quotient surface singularities can be generated by 3 homogeneous polynomials. As a consequence, affine threefolds of the type X m,n (G) for such G can be interpreted as quasihomogeneous varieties with respect to a grading determined by the generators of C[x, y] G . Properties of such quasihomogeneous threefolds will be addressed in future work.

Isomorphy Classes of X m,n

Recall the hypersurfaces X m,n ⊂ A 4 = Spec(C[x, y, u, v]) defined by

x m v -y n u = 1
with mn = 0. The main result of this section shows that at least for some distinct pairs {m, n} = {p, q} we have X m,n ∼ = X p,q .

In his investigation of complex surfaces with G a -actions [START_REF] Tom Dieck | On the topology of complex surfaces with an action of the additive group of complex numbers[END_REF], tom Dieck observed that X m,m admits a free action of a semidirect product structure G 2m on G m × G a for which X m,m /G 2m ∼ = P 1 and X m,m → P 1 is a principal G 2m -bundle. Moreover, the induced mapping X m,m /G m → P 1 is diffeomorphic to the complex line bundle O P 1 (-2m). The quotient X m,m /G m turns out to be isomorphic to the Danielewski surface D(m) : y 2 -2x m z = 1 and the latter can thus be distinguished topologically for distinct values of m. A related construction enables us to prove the existence of an algebraic isomorphism X m,n ∼ = X p,q if m + n = p + q albeit with a different analysis of the structure.

For a given (m, n) we have actions of G m and G a on X m,n given respectively by

G m × X m,n → X m,n (λ, (x, y, u, v)) -→ (λx, λy, λ -n u, λ -m v). and G a × X m,n → X m,n (t, (x, y, u, v)) -→ (x, y, u + tx m , v + ty n )
The G a -action is generated by the derivation δ : u → x m → 0, v → y n → 0 and the so determined principal G a -bundle π :

X m,n → A 2 * is trivialized over the open covering of A 2
* by the open subsets

U x = {x = 0} and U y = {y = 0} Writing θ : (G m × G a ) × X m,n → X m,n for the composite action, note that θ((λ, t), (x, y, u, v)) = (λx, λy, λ -n u + tλ -n x m , λ -m v + tλ -m y n ) = (λx, λy, λ -n u + tλ -(m+n) (λx) m , λ -m v + tλ -(m+n) (λy) n ) = θ((λ, λ -(m+n) t), (x, y, u, v)) Set d = m + n.
From the last equation it is clear that X m,n is equipped with a free action of the semidirect product

G d = G m ⋉ d G a with multiplication (λ, t) (λ ′ , t ′ ) = (λλ ′ , t + λ d t ′ ).
The open covering π -1 (U x )∪ π -1 (U x ) that trivializes the G a -action on X m,n is clearly G d -stable as well. We let

V x = π -1 (U x ) ∼ = Spec(C[x ±1 , y, u, v]/(x m v -y n u -1)) ∼ = Spec(C[x ±1 , y, u]) V y = π -1 (U y ) ∼ = Spec(C[x, y ±1 , u, v]/(x m v -y n u -1)) ∼ = Spec(C[x, y ±1 , v]). Setting (u 1 , T 1 , L 1 ) = (yx -1 , x n u, x) and (u 2 , T 2 , L 2 ) = (xy -1 , y m v, y)
and observing that u i are G d -invariant, T i are G -invariant and translated by the G a -action (vice versa for L i ), we obtain G d -equivariant trivializations

V x ∼ = Spec(C[x ±1 , y, u]) ∼ = Spec(C[u 1 ][T 1 , L 1 , L -1 1 ]) ∼ = A 1 × G d V y ∼ = Spec(C[x, y ±1 , v ]) ∼ = Spec(C[u 2 ][T 2 , L 2 , L -1 2 ]) ∼ = A 1 × G d . Gluing V x /G d ∼ = Spec(C[u 1 ]) and V y /G d ∼ = Spec(C[u 2 ]) over their inter- section Spec(C[u 1 , u -1 1 ]) via u 1 → u -1 2 we obtain a quotient π m,n : X m,n → C m,n ∼ = P 1
which is a Zariski locally trivial principal G d -bundle. Transition isomorphisms are given by L 2 = y = u 1 L 1 and

T 2 = y m v = ( y x ) m x m v = u m 1 (1 + y n u) = u m 1 (1 + u n 1 T 1 ) = u m 1 + u d 1 T 1 proving Proposition 1.
The threefold X m,n admits the structure of a principal G m+n -bundle over P 1 that is locally trivial for the Zariski topology. The transition isomorphisms for this bundle are

(L, T ) -→ (uL, u m+n T + u m ).
In contrast to the differential-topological context of [START_REF] Tom Dieck | On the topology of complex surfaces with an action of the additive group of complex numbers[END_REF] we cannot conclude that the induced mapping X m,n /G m → C m,n ∼ = P 1 is algebraically isomorphic to the line bundle O P 1 (-(m + n)) (the translation part of the transition isomorphism cannot be algebraically contracted away). Nevertheless, we can conclude from the discussion above with d = m + n, that Corollary 2.

(1) The quotient X m,n /G m is isomorphic to the affine surface S d,m which is the total space of the

O P 1 (-d)-torsor ρ : S d,m → P 1 with transition isomorphism T → u d T + u m . (2) Pic(S d,m ) ∼ = H 1 (S d,m , O * S d,m ) ∼ = Z and the isomorphism class of the G m -bundle X m,n → S d,m is a generator of Pic(S d,m ).
Proof. The second assertion follows from the fact that every A 1 -bundle over P 1 can be realized as the complement of a section C in a suitable P 1bundle over P 1 [18, Theorem 2.3] (S d,m is explicitly realized this way in the proof of Theorem 1 below). The latter have divisor class group isomorphic to Z 2 , generated by the classes of a fiber and a section that can be chosen to be precisely C. Thus the divisor class group of S d,m is isomorphic to Z, generated for instance by the class of a fiber of ρ.

The isomorphism class of X m,n as a variety is therefore determined by a generator of Pic(S d,m ), and moreover is independent of the choice of generator. Indeed, choosing the other generator yields the G m -bundle Xm,n over S d,m with transition isomorphism (L, T ) -→ (u -1 L, u d T + u m ). The coordinate change L = L -1 on the fibers of the bundle yields an isomorphism Xm,n ∼ → X m,n . The structure of S d,m as an O P 1 (-d)-torsor (actually as an A 1 -bundle over P 1 ) enables the application of a famous theorem of Danilov-Gizatullin [START_REF] Danilov | Automorphisms of affine surfaces[END_REF] to determine its isomorphy class as an affine surface (for a recent proof of this theorem see [START_REF] Flenner | On the Danilov-Gizatullin isomorphism theorem[END_REF]).

The transition isomorphism T -→ u d T + u m determines an A 1 -bundle over P 1 which in turn can be completed to the P 1 -bundle P(E), where E is the rank 2 vector bundle over P 1 with transition matrix

M = u d u m 0 1
corresponding to a non trivial extension

0 → O P 1 (-d) → E → O P 1 → 0.
Since the goal is to show that X m,n ∼ = X p,q if p + q = m + n, and obviously X m,n ∼ = X n,m , we will assume that m ≥ n. The results and terminology about ruled surfaces in what follows can be found in [8, V.2].

Lemma 2. The total space of the P 1 -bundle P( E) → P 1 is isomorphic to the Nagata-Hirzebruch surface F 2m-d .

Proof. It suffices to show that E⊗O P 1 (m) is normalized, i.e. that H 0 (E ⊗ O P 1 (m)) = 0 and H 0 (E ⊗ O P 1 (m -1)) = 0. Indeed, if these conditions hold, then by counting degrees,

E ⊗ O P 1 (m) ∼ = O P 1 ⊕ O P 1 (2m -d) and, as is well known, P(E) ∼ =P(E ⊗ O P 1 (m)) ∼ = F 2m-d .
The vector bundle E ⊗ O P 1 (j) has transition matrix

M j = u d-j u m-j 0 u -j
and a nonzero global section for this bundle corresponds to a pair of nonzero vectors

g = g 1 g 2 ∈ C[u] × C[u] and h = h 1 h 2 ∈ C[u -1 ] × C[u -1 ]
with M j • g = h For j = m -1 this would mean

h 1 = u d-m+1 g 1 + ug 2 h 2 = u(u d-m g 1 + g 2 )
which is impossible as d > m. For j = m we have, for example,

g = 0 1 and h = 1 u -m .
Theorem 1. Let d = p + q = m + n. Then X m,n ∼ = X p,q as abstract varieties.

Proof. As noted above, since X m,n ∼ = X n,m , we may assume without loss of generality that m ≥ n, so that 2m ≥ d. With the notation of the corollary, X m,n is the total space of a G m -bundle over S d,m which in turn is the total space of an O P 1 (-d)-torsor over P 1 . Choosing homogeneous coordinates [u : v] on the fibers F of the P 1 -bundle P(E) ∼ = F 2m-d →P 1 , S d,m is obtained as the complement of the section v = 0. The associated divisor is C + mF where C is the special section with self-intersection d -2m. The self-intersection of C + mF is calculated as Since the class of X m,n (and Xm,n ) generates Pic(S d,m ) and X m,n ∼ = Xm,n we obtain the isomorphism X m,n ∼ = X p,q .

(C + mF ) 2 = C 2 + 2mC.F = -(2m -d) + 2m = d
While it appears to be rather difficult to construct an explicit isomorphism between even the first interesting examples, X 2,2 and X 3,1 , one can check that X 2,2 admits the structure of the principal G a -bundle over A 2 * corresponding to the Čech cocycle x -3 y.

Example 4. Set a(x, y, u, v) = x - 1 2 y b(x, y, u, v) = (6x -y) 8 v - (3y -2x) 2 u w(x, y, u, v) = 5 16 v 2 x + 5 2 vxu - 1 32 v 2 y - 5 4 vyu + u 2 x - 5 2 u 2 y The morphism π : X 2,2 → A 2 * ∼ = Spec(C[a, b]) \{0}. (x, y, u, v) → (a(x, y, u, v), b(x, y, u, v))
is a G a -bundle corresponding to the Čech cocycle a -3 b. In particular, one can check that the assignment

x → a 3 3 , y → a 3 , u → xb - 1 4 , v → 2yb -1 extends to a locally nilpotent C-derivation δ on A 2,2 = C[x, y, u, v]/(x 2 v - y 2 u -1)

A more general class of examples

A refinement of the arguments in the previous section enables a description of the isomorphy classes of total spaces of principal G a -bundles defined by arbitrary homogeneous bivariate polynomials in terms of the X m,n .

Let f, g ∈ C [x, y] be arbitrary homogeneous polynomials such that Spec (C [x, y] / (f, g)) is supported at the origin of A 2 and consider affine threefolds X f,g in A 4 = Spec (C [x, y, u, v]) defined by the equations f vgu -1 = 0. Again, these threefolds come naturally equiped with the structure π f,g : X f,g → A 2

* of G a -bundles over the complement of the origin in A 2 = Spec (C [x, y]) by restricting the projection p x,y : A 4 → A 2 to X f,g . The latter becomes trivial on the covering of A 2 * by means of the principal affine open subsets

U f = Spec C [x, y] f and U g = Spec C [x, y] g .
Suppose that f and g are homogeneous, say of degree m and n respectively. Similar to the case of the X m,n of the previous section, the G m -action λ • (x, y) = (λx, λy) on A 2 lifts to a free G m -action on X f,g defined by

λ • (x, y, u, v) = λx, λy, λ -m u, λ -n v which is compatible with the structural map π f,g : X f,g → A 2 * . It follows that π f,g descends to an A 1 -bundle ρ f,g : X f,g /G m → A 2 * /G m ≃ P 1 .
The latter is an O P 1 (-nm)-torsor which can be described more explicitly as follows. Recall that for every pair of integers m, n ≥ 1 , the quotient of

ρ : X f,g → X f,g /G m × {1} ≃ A 2 * × A 2 * \ D /G 2 m = F (m, n) \ ∆ f,g .
over the complement of the image ∆ f,g of D in F(mn).

The self-intersection of the curve ∆ f,g , which is a section of π, can determined as follows. The images by the quotient map of the G 2 m -invariant subsets {v = 0} and {u = 0} define two sections C v and C u of π such that C v • C u = 0. Letting L be a fiber of π, the divisors mL + C v and nL + C u are linearly equivalent since they differ by the divisor of the rational function h = f v/gu on F (m, n). Since ∆ f,g belongs to the complete linear system generated by these divisors, we conclude that

∆ 2 f,g = (mL + C v ) • (nL + C u ) = m + n.
The above description implies that X f,g /G m is isomorphic to the complement of a section with self-intersection m + n in F (m, n) ≃ F |m-n| , and so, the P 1 -bundle F |m-n| → P 1 restricts to an O P 1 (-mn)-torsor ρ f,g : X f,g /G m → A 2 * /G m ≃ P 1 .

Combined with the results of the previous section, this the leads to the following Proposition 2. With the notation above, X f,g is isomorphic to X m,n .

Proof. Recall that X m,n has the structure of a G m -bundle over an affine surface S d,m obtained as the complement of certain section ∆ m,n with self-intersection m + n in F |m-n| . By virtue of Gizatullin's classification [START_REF] Danilov | Automorphisms of affine surfaces[END_REF], F |m-n| \ ∆ f,g ≃ F |m-n| \ ∆ m,n = S d,m as abstract affine surfaces. It follows that Z = X m,n × S d,m X f,g is a G m -bundle over both X m,n and X f,g via the first and the second projection respectively. Since the Picard groups of X m,n and X f,g are both trivial, these G m -bundles are actually trivial, which yields an isomorphism X m,n × A 1 * ≃ X f,g × A 1 * as affine varieties. Since X m,n and X f,g are both irreducible and have no invertible functions except nonzero constants, the latter descends to an isomorphism X m,n ≃ X f,g .

Lemma 1 .

 1 Let (O, m) be a regular local ring of dimension n ≥ 2 and let A be a factorial, finitely generated O-domain with O ֒→ A. Let f : X → Y be the morphism induced by the ring inclusion, where X = Spec(A) and

  and thus depends only on the sum d = p + q = m + n. The theorem of Danilov-Gizatullin says exactly that S d,m and S d,p are isomorphic as affine surfaces. An isomorphism S d,m φ → S d,p carries a generator of Pic(S d,p ) to a generator of Pic(S d,m ) under the induced isomorphism Pic(S d,p ) φ * → Pic(S d,m ).

  with a, b ∈ ker(δ) andδ(y + a + ab) = a 3 , δ(w) = b. Since (a 3 , b)A 2,2 = A 2,2, we obtain a locally trivial G a -action on X 2,2 with quotient isomorphic to A 2 * corresponding to the Čech cocycle y + a + ab a 3

A 2 * × A 2 *

 22 by the free G 2 m -action given by (λ, µ) • (x, y, u, v) = λx, λy, µλ -n u, µλ -m v is the surface scrollπ : F (m, n) = P (O P 1 (m) ⊕ O P 1 (n)) → P 1 , with structural morphism π induced by the G 2 m -invariant morphism A 2 * × A 2 * → P 1 , (x, y, u, v) → [x : y]. Clearly, X f,g is contained in the complement of the G 2 m -invariant closed subset D of A 2 * × A 2 * with equation f vgu = 0. Furthermore, X f,g is invariantunder the action of the first factor of G 2 m and is a slice for the action of the second factor of G 2 m of A 2 * × A 2 * \ D. Thus the quotient map restricts to a G m = G m × {1}-bundle