Olivier Biquard

Heather Macbeth

Steady Kähler-Ricci solitons on crepant resolutions of finite quotients of C n

We prove the existence of steady Kähler-Ricci solitons on equivariant crepant resolutions of C n /G, where G is a finite subgroup of SU (n) acting freely on C n .

Our construction proceeds by taking Joyce's well-known family [Joy00] of Ricci-flat Kähler metrics on crepant resolutions of orbifolds C n /G (which automatically have c 1 (M ) = 0), and modifying their metrics near infinity by gluing them to a G-quotient of Cao's soliton on C n .

Introduction

Overview

The 'fixed points' of the Ricci flow are steady Ricci solitons. Such objects, natural generalizations of Ricci-flat metrics, are pairs (g, X) of a Riemannian metric and a vector field, satisfying the elliptic partial differential equation Ric(g) + 1 2 L X g = 0. They arise in the study of the Ricci flow as models of singularities [DdP07,GZ08,DS10,AIK15], and as backward limits of ancient solutions. They are also critical points, in a suitable sense [START_REF] Haslhofer | A renormalized Perelman-functional and a lower bound for the ADMmass[END_REF], of the Perelman F -functional, and thus can be considered canonical among all such pairs (g, X).

Steady Ricci solitons which are not Ricci-flat must be noncompact [START_REF] Ivey | Ricci solitons on compact three-manifolds[END_REF]. The few known examples include several which are Kähler, with holomorphic vector field: Hamilton's cigar soliton [START_REF] Hamilton | The Ricci flow on surfaces[END_REF] on C, H.-D. Cao's generalizations [START_REF] Cao | Existence of gradient Kähler-Ricci solitons[END_REF] on C n and K CP n-1 , and further generalizations by Dancer-M. Wang [START_REF] Dancer | On Ricci solitons of cohomogeneity one[END_REF] and B. Yang [START_REF] Yang | A characterization of noncompact Koiso-type solitons[END_REF]. Non-Kähler examples include the well-known constructions of Bryant and of Ivey. All these examples -and, we believe, all known examplesare highly symmetric, and the soliton metric is given either explicitly or by solving an ODE.

In this article we use PDE methods to construct new steady Kähler-Ricci solitons (M, ω, X), in all complex dimensions n ≥ 2 (real dimensions 2n ≥ 4), of infinitely many topological types in dimensions 2 and 3 at least. Like all steady Kähler-Ricci solitons, they have first Chern class

c 1 (M ) = [Ric(ω)] = [-1 2 L X ω] = 0.
Theorem 1.1. Let G be a finite subgroup of SU (n), which acts freely on C n \ {0}. Let M G be a crepant resolution of C n /G, which is equivariant with respect to the action of C * . Let k be a Kähler class on M G which contains an asymptotically locally Euclidean Kähler metric. Then for all ϵ sufficiently small, there exists a steady Kähler-Ricci soliton on M G in the cohomology class ϵk, whose drift vector field is the extension to M G of the radial vector field -2r∂/∂r (this extension exists by the equivariance of the resolution).

Moreover, outside a compact set these solitons have the form ω 0 + i 2 ∂∂Ψ, where ω 0 is Cao's steady Kähler-Ricci soliton metric on C n (descending to C n /G), and, for any k and any λ < n,

|∇ k Ψ| ω 0 = O(e -λt ),
where t is the distance to a fixed point in M G .

Since crepant resolutions have trivial 1-cohomology, the soliton has to be a gradient solution, that is the drift vector field is the gradient of a function (the soliton potential): given the asymptoics obtained in the theorem, this function is asymptotic to the soliton potential of the model Cao soliton given in Section 2.

Such gluing constructions have been performed before in other geometric settings. Early uses of the method include Taubes' construction [START_REF] Henry | The existence of anti-self-dual conformal structures[END_REF] of anti-self-dual metrics, and Kapouleas' [START_REF] Kapouleas | Complete constant mean curvature surfaces in Euclidean three-space[END_REF][START_REF] Kapouleas | Compact constant mean curvature surfaces in Euclidean three-space[END_REF] of minimal surfaces. More recently the method has been used [START_REF] Hien Nguyen | Translating tridents[END_REF][START_REF] Hien Nguyen | Complete embedded self-translating surfaces under mean curvature flow[END_REF] to construct self-translators, the mean curvature flow analogue of steady Ricci solitons. Joyce himself [START_REF] Joyce | Compact manifolds with special holonomy[END_REF] used his Calabi-Yau's as building blocks for gluing constructions of manifolds of exceptional holonomy.

In the Kähler setting, Biquard-Minerbe [START_REF] Biquard | A Kummer construction for gravitational instantons[END_REF] used gluing methods to construct noncompact Calabi-Yau manifolds of complex dimension 2, of several different asymptotic behaviours. Gluing techniques have also been used by various authors to construct constant scalar curvature and extremal Kähler metrics, see for example [START_REF] Rollin | Non-minimal scalar-flat Kähler surfaces and parabolic stability[END_REF][START_REF] Arezzo | Blowing up and desingularizing constant scalar curvature Kähler manifolds[END_REF][START_REF] Arezzo | Extremal metrics on blowups[END_REF][START_REF] Székelyhidi | On blowing up extremal Kähler manifolds[END_REF][START_REF] Biquard | Smoothing singular constant scalar curvature Kähler surfaces and minimal Lagrangians[END_REF].

We also prove a general uniqueness statement. We show that a steady Kähler-Ricci soliton is unique for its cohomology class, asymptotics (as previously mentioned, nontrivial steady solitons are noncompact), and drift vector field, in the following sense: Proposition 1.2. Let (M, ω, X) be a steady Kähler-Ricci soliton. Let ω + i 2 ∂∂Ψ be another steady Kähler-Ricci soliton for the same drift vector field X, such that at infinity we have 1. Ψ → 0; 2. XΨ → 0;

∂∂Ψ → 0 (by comparison with ω).

Then Ψ = 0.

Related work

The form of Theorem 1.1 suggests generalization in two directions, and after this article was posted to the arXiv, Conlon-Deruelle [START_REF] Conlon | Steady gradient Kähler-Ricci solitons on crepant resolutions of Calabi-Yau cones[END_REF] succeeded in carrying out both these extensions.

First, the orbifolds C n /G are only the simplest examples of Calabi-Yau cones. It turns out that equivariant crepant resolutions of arbitrary Calabi-Yau cones admit steady Kähler-Ricci solitons modeled on the Cao soliton, with a subtlety in the asymptotics of the soliton metric depending on whether the Kähler class is or is not compactly supported (in the case of resolutions of C n /G all Kähler classes are compactly supported).

Secondly, a steady Kähler-Ricci soliton should exist on the manifolds M G , associated to the radial vector field X, in every Kähler class, not just small Kähler classes as constructed in this article. Rather than using gluing machinery, the argument in [START_REF] Conlon | Steady gradient Kähler-Ricci solitons on crepant resolutions of Calabi-Yau cones[END_REF] solves the PDE by a continuity method, with the C 0 estimate relying on, among other things, a weighted energy estimate and an adapted Nash-Moser iteration, as well as ideas from pluripotential theory. The alternate construction presented here remains useful in providing the asymptotics of the construction as the Kähler class tends to zero.

Outline

In Sections 2-5, we first review the construction of the Cao soliton, and then introduce the required analytical tools to handle the analysis in weighted Hölder spaces on the Cao soliton, with an exponential weight at infinity. Finally, for finite quotients of the Cao soliton, we combine this analysis with the weighted analysis at the singular point of the quotient.

In Section 6 we review the material needed on crepant resolutions of C n /G, and on the canonical Kähler, Ricci-flat, asymptotically locally Euclidean metrics on these spaces. We also discuss some specific examples of equivariant crepant resolutions.

The proof of uniqueness (Proposition 1.2) is given in Section 7. Finally, in Sections 8-13, we proceed to the gluing construction, using in particular a blowup analysis (Section 11) to bound the inverse of the linearization of the problem in suitable weighted spaces. Theorem 1.1 is proved in Section 13.

U (n)-invariant Kähler metrics and the soliton equation

We begin by recalling Cao's construction [START_REF] Cao | Existence of gradient Kähler-Ricci solitons[END_REF] of a steady soliton on C n and a one-parameter family of steady solitons on the bundle O(-n) over CP n-1 . We habitually identify the complement of the zero section of O(-n) with (C n \ {0})/Z n , to which it is biholomorphic. All Cao's solitons are invariant under the natural action of U (n) on finite quotients of C n : therefore we start by reviewing this ansatz, see [START_REF] Feldman | Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons[END_REF].

From Section 4 onwards, only the soliton on C n will be used. The family of solitons on O(-n) is reviewed because it is the motivating example for the solitons on desingularizations of (C n \ {0})/G constructed in the rest of this article (see Remark 3.2).

Let Φ : R → R, and denote its derivative by φ(t) = Φ t (t). Throughout this article we use interchangeably the variables z ∈

C n \ {0}, r ∈ R + , t ∈ R, with |z| 2 = r 2 = e t .
We introduce real symmetric 2-tensors on C n \ {0}:

g F S = g S 2n-1 -η 2 , g cyl = dr 2 r 2 + η 2 = 1 4 dt 2 + η 2
where g S 2n-1 is the round sphere metric, the 1-form η is the connection 1-form on the Hopf bundle S 2n-1 → CP n-1 , and g F S is the pull-back of the standard Fubini-Study metric of CP n-1 with holomorphic sectional curvature -4. We denote by ω F S and ω cyl the associated (1,1)-forms.

The Cao U (n)-invariant Kähler forms on C n \ {0} will be of the form

ω = i 2 ∂∂Φ(t) = φ(t)ω F S + φ t (t)ω cyl . (2.1)
This formula defines a Kähler metric outside the origin if and only if both φ and φ t are everywhere positive. We henceforth assume that they are. The Ricci form is then

ρ(ω) = i∂∂ log ω n vol C n = i∂∂ log φ n-1 φ t e -nt = i∂∂ log(φ n-1 φ t ) -nt . (2.2) Fix µ ∈ R.
Let X be the radial vector field -2µr∂/∂r = -4µ∂/∂t. The pair (ω, X) is a steady Ricci soliton if Ric(g) + 1 2 L X g = 0, which by (2.2) can be rewritten

φ n-1 φ t e µφ = cst. e nt .
By a translation on t (that is, an homothety of C n ), one can take the constant to be 1, and the final equation is therefore

φ n-1 φ t e µφ = e nt .
(2.3) Such Ricci solitons are gradient Ricci solitons: with the gradient function f (z) = -µφ(t),

Ric(g Φ ) + (∇ g Φ ) 2 f = 0.
By multiplying φ by a constant, one can restrict to the case µ = 1, which we will henceforth do.

The equation (2.3) is solved in the following way. Let F (s) be the degree-(n -1) polynomial

F (s) = n-1 r=0 (-1) n-r-1 (n-1)! r! s r . Then one can check that d ds [F (s)e s ] = s n-1 e s ,
and for all s > s 0 ≥ 0, 0 < F (s)e s -F (s 0 )e s 0 < s n-1 e s .

(2.4)

It follows that the soliton equation (2.3) can be rewritten as

F (φ)e φ = e nt n + cst.
We can therefore define a family of solitons parametrized by a nonnegative real number a ≥ 0 by taking φ a = φ to be implicitly defined by the equation

F (φ(t))e φ(t) = e nt n + F (a)e a .
(2.5) From (2.4) the function φ is well defined, with φ(t) > a for all t; again from (2.4), one has φ n-1 e φ > e nt n so equation (2.3) implies that 0 < φ t < n.

(2.6)

Thus Φ := φdt is the Kähler potential of a steady Ricci soliton with associated holomorphic vector field X = -2r∂/∂r = -4∂/∂t.

Example 2.1 (Hamilton's cigar soliton). In dimension n = 1, the equation (2.5) reduces to e φa(t) = e t +e a , and then (φ a ) t = (1+e a e -t ) -1 by (2.3). The associated Kähler-Ricci soliton on C is (φ a ) t g cyl , so the dependence in a is just by translation, and we get a unique soliton (Hamilton's cigar soliton), (φ 0 ) t g cyl = dx 2 + dy 2 r 2 + 1 .

Asymptotics of the Cao potentials

In this section, we establish the asymptotics of the functions φ a (t) constructed in the previous section as t → -∞ and as t → ∞. The former is necessary to determine the topological structure of the metric completion of the solitons. The latter determines the asymptotics at infinity of the Cao solitons.

Lemma 3.1. As t → -∞, φ a (t) = a + 1 n a 1-n e -a e nt + O(e 2nt ), a > 0. e t -1 n+1 e 2t + O(e 3t ), a = 0.

Proof. Easy calculation starting from (2.5).

Since e t = r 2 , when a = 0, the metric associated to φ = φ 0 on C n \ {0} extends smoothly across the origin (φ 0 has a smooth development in powers of e t ). This is Cao's steady soliton (unique up to translation and rescaling) on C n .

When a > 0, then φ t (t) = a 1-n e -a e nt + O(e 2nt ) with e nt = r 2n (and again there is a smooth development in powers of e nt ), so formula (2.1) says that the metric associated to φ = φ a on the quotient (C n \ {0})/Z n ∼ = O(-n) \ CP n-1 extends smoothly across the zero section CP n-1 to give a smooth metric on O(-n). This is Cao's family of steady solitons (a one-parameter family, up to translation and rescaling) on O(-n).

Remark 3.2. As a → 0, the solitons on O(-n) converge to the Cao soliton on C n /Z n with an orbifold singularity at the origin. This is an explicit example of our construction in this paper. It is also interesting to see what 'bubble' occurs in this limit: an easy calculation gives

φ a (t + log a) a → (1 + e nt )
1 n (the translation in t corresponds to homotheties in C n ). The limit is the well-known Kähler Ricciflat metric on the total space of O(-n) constructed by Calabi [START_REF] Calabi | Métriques kählériennes et fibrés holomorphes[END_REF] (for n = 2 this is the Eguchi-Hanson metric).

We now study the behaviour at infinity. The following lemma is again a consequence of equation (2.5). For further terms in the expansion, down to an error of

O (log t) 2 t 2 , see [CD20, Proposition 3.1]. Lemma 3.3. As t → ∞, φ(t) = nt -(n -1) log t -n log n + O log t t . φ t (t) = n - n -1 t + O log t t 2 . Proof. Let φ 0 (t) := nt -(n -
φ 0 (t) + A 1 log t t < φ(t) < φ 0 (t) + A 2 log t t .
This establishes the first statement. For the second, substitute this expansion into equation (2.3).

Therefore the metric has the asymptotic behaviour when t → +∞

g ∼ n 1 4 dt 2 + η 2 + tg F S (3.1)
and in particular the volume of the ball of radius t is of order t n , that is half-dimensional; also the injectivity radius is bounded below. More generally:

Lemma 3.4. The Cao solitons have injectivity radius bounded below and curvature and all their covariant derivatives bounded.

Proof. It remains to be proved that the Riemannian curvature and its derivatives are bounded.

Given the explicit form of the metric, this reduces to the fact that all derivatives of φ are bounded, which is obvious (at infinity φ t → n and the higher derivatives go to zero).

Analysis on the Cao soliton

For the rest of this article, we need only the soliton obtained for a = 0; that is, the Cao soliton on

C n with Kähler form ω 0 = φω F S + φ t ω cyl . In this section we do some analysis on this soliton and on manifolds asymptotic to it. We first work generally on a Riemannian manifold (M, g) with a single end isometric to the portion {T 0 ≤ t} of the Cao soliton (C, ω 0 ), or to its quotient by some freely-acting G ⊆ SU (n). Fix δ > 0 and define a weight function

w(t) = e δφ .
So w ≥ 1 and at infinity w(t) ∼ e δnt . We use the Hölder weighted spaces

C k,α δ (M ) = w -1 C k,α (M )
, where C k,α (M ) are the usual Hölder spaces. Proposition 4.1 (Schauder estimate for Cao asymptotics). Let a be a symmetric positive-definite bivector field, and b a vector field, whose (unweighted) C α norms are finite. Suppose moreover that for some real λ > 0, the uniform global bound a ≥ λg -1 holds. Denote by L the second-order elliptic operator Lu = ⟨a, ∇ 2 u⟩ + ⟨b, du⟩.

Then there exists a constant C, such that for all t 0 , t 1 with

T 0 + 1 ≤ t 0 + 1 ≤ t 1 < ∞, for all u on {t < t 1 } ⊆ M , ∥u∥ C 2,α δ ({t<t 0 }) ≤ C ∥u∥ C 0 δ ({t<t 1 }) + ∥Lu∥ C α δ ({t<t 1 }) .
Proof. By Lemma 3.4, there exist s > 0 and Q > 0 such that for all x ∈ M , there exist (harmonic) co-ordinates on the ball B x (s) in which the metric tensor g is C 1,α controlled by Q. See, for example, [Heb96, Section 1.2]. Therefore on balls of radii ≤ s, the Schauder norms with respect to the metric g are uniformly equivalent to the Euclidean Schauder norms. Moreover, by assumption, the coefficients a, b of the operator are uniformly controlled on balls B x (s). Therefore the Euclidean Schauder estimate [GT01, Theorem 6.2] also holds, with uniform constant, on balls B x (s) with Riemannian Schauder norm.

The weight function w δ has the property that, for some uniform constant C, for all x and all y ∈ B x (s), C -1 w(x) ≤ w(y) ≤ Cw(x). We may therefore combine the uniform Schauder estimates on balls B x (s) to give the desired global weighted estimate.

We now specialize to the Cao manifold (C n , ω 0 ) itself, with the weight function w(t) = e δφ . itself. Denote by ∆ the Laplace-Beltrami operator of ω 0 , and by X (as in Section 2) the radial vector field -2r∂/∂r = -4∂/∂t. The linearization of the soliton equation on a Kähler potential is the operator ∆ -X. We have the following C 0 estimates for this operator:

Lemma 4.2. Let 0 < δ < 1. 1. For t 0 ∈ R, and u on {t ≤ t 0 } ⊆ C n , sup t≤t 0 w|u| ≤ max sup t=t 0 w|u|, 1 4δ(1 -δ)n sup t≤t 0 w|(∆ -X)u| . (4.1) 2. For r 0 ∈ [1, ∞], and u on {1 ≤ r ≤ r 0 } ⊆ C n such that u| {r=r 0 } = (if r 0 < ∞) or u ∈ C 2 δ ({1 ≤ r}) (if r 0 = ∞), sup 1≤r≤r 0 w|u| ≤ max sup r=1 w|u|, 1 4δ(1 -δ)n sup 1≤r≤r 0 w|(∆ -X)u| . (4.2) Proof. The Laplacian of a radial function is ∆f = 4 φ n-1 φ t ∂ ∂t φ n-1 ∂f ∂t .
Therefore, using equation (2.3) and

X = -4 ∂ ∂t , (∆ -X)e -δφ = - 4δ φ n-1 φ t ∂ ∂t e -δφ φ n-1 φ t + 4δφ t = - 4δ e nt-φ ∂ ∂t e nt-(1+δ)φ + 4δφ t = -4δ(n -δφ t )e -δφ .
Since 0 < φ t < n by (2.6), one deduces

4δ(1 -δ)n < -e δφ (∆ -X)e -δφ < 4δn.
This estimate enables the use of a barrier argument: if (∆ -X)f = g, then write f = e -δφ F ; then (∆ -X)F + F e δφ (∆ -X)e -δφ -2δ⟨dφ, dF ⟩ = e δφ g.

(4.3)
On a compact domain t ≤ t 0 , the maximum principle applied to (4.3) implies sup

{t≤t 0 } e δφ |f | = sup {t≤t 0 } |F | ≤ max sup {t=t 0 } |F | + 1 4δ(1 -δ)n sup {t≤t 0 } e δφ |g| .
This is (4.1). Similarly, if r 0 < ∞, then on the compact domain 1 ≤ r ≤ r 0 , if u| {r=r 0 } = 0, then we obtain (4.2). Finally, if r 0 = ∞, we may select a sequence of compactly-supported functions χ k , with 0 ≤

χ k ≤ 1, with χ k → 1 pointwise as k → ∞, such that for all u ∈ C 2 δ ({1 ≤ r}), lim k→∞ ∥(∆ -X)(χ k u)∥ C 0 δ ({1≤r}) = ∥(∆ -X)u∥ C 0 δ ({1≤r}) .
(Indeed, we may take χ k (t) = χ(t/k), for some fixed cutoff function χ.) We can then extract the noncompact (r 0 = ∞) case of (4.2) from the sequence of compact results.

Theorem 4.3 (Drift Laplacian on the Cao soliton). Let 0 < δ < 1. Then the operator

∆ -X : C 2,α δ (C n ) → C α δ (C n ) is an isomorphism.
Proof. To prove surjectivity: Let g ∈ C α δ (C n ). For each of a sequence of domains {t ≤ t i }, with t i → ∞, let f i be the solution to the Dirichlet problem (∆ -X)f i = g, f i | {t=t i } = 0. Such a solution exists since ∆ -X has no zero-th order term. By the estimate (4.1),

sup t≤t i w|f i | ≤ 1 4δ(1 -δ)n sup t≤t i w|(∆ -X)f i | = 1 4δ(1 -δ)n sup t≤t i w|g|.
So, by the Schauder estimate Proposition 4.1, for all t 0 and all i such that t i ≥ t 0 + 1,

∥f i ∥ C 2,α δ ({t≤t 0 }) ≤ C ∥f i ∥ C 0 δ ({t≤t i }) + ∥(∆ -X)f i ∥ C α δ ({t≤t i }) ≤ C∥g∥ C α δ ({t≤t i }) ,
where the constant C is independent both of t 0 and of i. By a diagonal argument, we can extract a subsequence, also denoted f i , which C 2 -converges on each compact subset. Let f denote the limit. On each compact subset {t ≤ t 0 }, we have that

∥f ∥ C 2,α δ ({t≤t 0 }) ≤ lim sup i→∞ ∥f i ∥ C 2,α δ ({t≤t 0 }) ≤ C∥g∥ C α δ . So ∥f ∥ C 2,α δ ≤ c∥g∥ C α δ .
This proves the surjectivity of ∆ -X. To prove injectivity: suppose that (∆ -X)f = 0 with f = O(e -δφ ). Let δ ′ be such that 0 < δ ′ < δ. Applying the estimate (4.1) with the parameter δ ′ , sup

t≤t 0 e δ ′ φ |f | ≤ sup t=t 0 e δ ′ φ |f | = e (δ ′ -δ)φ(t 0 ) sup t=t 0 e δφ |f |.
The right hand side tends to 0 as t 0 → ∞, so f = 0.

Analysis on finite quotients of the Cao soliton

For the rest of this article, fix a finite subgroup G ⊂ SU (n) acting freely on C n \ {0}. We now get an orbifold soliton (C G = (C n \ {0})/G, ω 0 ) with an isolated singularity at the origin.

It is possible to construct weighted orbifold Hölder spaces on C G , consisting of the functions on C G whose lifts to C n \ {0} extend to functions in the spaces C k,γ δ (C n ) of the previous section. By Theorem 4.3, the operator ∆ -X is an isomorphism between appropriate such weighted-at-infinity orbifold Hölder spaces. However, this result is not sufficient for our gluing procedure.

We need to consider Hölder spaces which are also nontrivially weighted at the origin, with weight functions polynomial in the distance to the origin. We take as the weight function the positive radial function w 0 (t) := e γt/2 , t < 0, e δφ(t) , t > 0, (we can smooth this function at t = 0 but this is not really needed). So w 0 (t) ∼ e δnt at infinity (the weights of the previous section) and w 0 (t) ∼ r γ at the origin. If it is necessary to specify the parameters, we will use the notation w 0;γ,δ for w 0 . Choose a positive function σ : C G → R such that at each point x, the injectivity radius at x is greater than σ(x). We can take σ = σ(t) to be a radial function, with

σ(t) =
cst. e t/2 = cst. r, t ≪ 0, cst, t ≫ 0.

(5.1)

We can now define the weighted Hölder space C k,α γ,δ (C G ) by the norm

∥f ∥ C k,α γ,δ = k i=0 sup σ(x) i w 0 (x)|∇ i f | + sup σ(x) k+α w 0 (x) sup y∈Bx(σ(x)) |P y,x ∇ k f (y) -∇ k f (x)| d(x, y) α , (5.2)
where P y,x is the parallel transport along the unique minimizing geodesic from y to x. (We use the Riemannian metric ω 0 for all geometric aspects of this construction:

∇, | • |, d(•, •), P x,y .
) This weighted Hölder norm is equivalent to the previous section's norm on C n , for functions supported on {t > 0}, and to the norm classically used on cones for functions supported on {t < 0}. Before we do this, we note, for future reference, some (essentially classical) facts about ∆ -X when considered as an operator on a bounded portion {t ≤ t 0 } of C G . Let C k,α γ (t ≤ t 0 ) denote the Hölder space of functions on that portion, with norm the analogue of the formula (5.2) restricting to x, y ∈ {t ≤ t 0 }. Then let Ĉ2,α γ (t ≤ t 0 ) denotes the set of functions f ∈ C 2,α γ (t ≤ t 0 ) with (Dirichlet) boundary condition f | t=t 0 = 0. Lemma 5.1. For each γ ∈ (0, 2n -2), the operator

∆ -X : Ĉ2,α γ (t ≤ t 0 ) -→ C α γ+2 (t ≤ t 0 )
is an isomorphism.

Proof. We note that the only critical weights of ∆ at the origin in the interval [0, 2n -2] are 0 (corresponding to the constants) and 2n -2 (corresponding to the Green function). This implies that the Laplacian ∆ has index zero as a map between the spaces Ĉ2,α γ (t ≤ t 0 ) and C α γ+2 (t ≤ t 0 ) (its adjoint is itself with the adjoint weight γ ′ = 2n -2 -γ ∈ (0, 2n -2), so the operator and its adjoint have the same kernel).

Since X = -2r ∂ ∂r is a compact operator between the spaces in question, this means that ∆ -X has index zero.

To show the injectivity of the operator, observe that the calculation of the critical weights implies that a solution of (∆ -X)f = 0 with f ∈ Ĉ2,α γ (t ≤ t 0 ) and 0 < γ < 2n -2 must actually be bounded and have a limit at the origin. By local elliptic regularity it must be smooth (in the orbifold sense of lifting to a smooth function on the G-prequotient B = B 0 (exp(t 0 /2)) ⊆ C n ). So we have a solution of (∆ -X)f = 0 in the space C 2,α (B), with Dirichlet boundary conditions, so f = 0.

Therefore the operator is an isomorphism.

We can now give the doubly-weighted isomorphism.

Theorem 5.2 (Drift Laplacian on finite quotients of the Cao soliton). Let 0 < δ < 1 and

0 < γ < 2n -2. Then the operator ∆ -X : C 2,α γ,δ (C G ) → C α γ+2,δ (C G ) is an isomorphism.
Proof. This theorem will be a consequence of Theorem 4.3 (which deals with the analysis at infinity) and of the local theory for polynomial weights in cone points (as in Lemma 5.1).

To show the injectivity of the operator, we follow the same argument as in Lemma 5.1. A solution of (∆ -X)f = 0 with f ∈ C 2,α γ,δ and 0 < γ < 2n -2 must actually be bounded and have a limit at the origin, and so lift to a solution of (∆ -X)f = 0 in the space C 2,α δ . We then find f = 0 by Theorem 4.3.

To show the surjectivity of the operator, take g ∈ C α γ+2 (C G ). By Lemma 5.1 we find f 0 ∈ C 2,α γ ({t ≤ 0}) defined for t ≤ 0, such that (∆ -X)f 0 = g. Take χ a cutoff function so that χ(t) = 0 for t ≥ 0 and χ(t) = 1 for t ≤ -1. Then, using the isomorphism of Theorem 4.3, and the fact that g -(∆ -X)(χf 0 ) has support included in {t ≥ -1}, we can find

f 1 ∈ C 2,α δ (C G ) ⊆ C 2,α γ,δ (C G ) such that (∆ -X)f 1 = g -(∆ -X)(χf 0 ). Then f = χf 0 + f 1 ∈ C 2,α γ,δ (C G )
is the required solution to (∆ -X)f = g. We also note the appropriate Schauder estimate on these doubly-weighted spaces: Proposition 5.3 (Schauder estimate on the Cao soliton quotient). There exists a constant C, such that for all t 0 ≤ ∞, for all u on {t ≤ t 0 }, with also

u| {t=t 0 } = 0 if t 0 < ∞, ∥u∥ C 2,α γ,δ ({t<t 0 }) ≤ C ∥u∥ C 0 γ,δ ({t<t 0 }) + ∥(∆ -X)u∥ C α γ+2,δ ({t<t 0 }) .
Proof. The proof is similar to that of Proposition 4.1. In this situation we have that there exists Q > 0 such that for all x ∈ M , there exist (harmonic) co-ordinates on the ball B x (σ(x)) in which the metric tensor g is C 1,α controlled by Q: the varying radius σ(x) (defined in (5.1)) is needed because the injectivity radius is small near the orbifold point at the origin. Therefore on balls of radii σ(x), the weighted Schauder norms with respect to the metric g are uniformly equivalent to the weighted Euclidean Schauder norms. So the weighted Euclidean Schauder boundary estimate [GT01, Lemma 6.4] also holds, with uniform constant, on balls B x (σ(x)) with weighted Riemannian Schauder norm.

Again, the weight functions w γ,δ have the property that, for some uniform constant C, for all x and all y ∈ B x (σ(x)), C -1 w(x) ≤ w(y) ≤ Cw(x). We may therefore combine the uniform Schauder estimates on balls B x (σ(x)) to give the desired global weighted estimate; the weight w γ+2,δ appears as the product σ 2 w γ,δ .

Joyce metrics on crepant resolutions

We continue to fix a finite subgroup G of SU (n) acting freely on C n \ {0}, as in Section 5. For the rest of this article we also fix an equivariant crepant resolution π : J G → C n /G of the quotient. In this context, equivariant means that the C * action on C n /G extends to an action on J G . Not all such quotients C n /G admit equivariant crepant resolutions, or even crepant resolutions. (By the argument of [CDS19, proof of Lemma 2.13], the existence of an unique crepant resolution implies that that resolution is equivariant.) We note some examples which do admit equivariant crepant resolutions: Example 6.2. In dimension n = 3, there are some ten infinite classes of such subgroups G. There exists a canonical equivariant crepant resolution of C 3 /G, the G-orbit Hilbert scheme of C 3 , whose existence was conjectured by Nakamura [START_REF] Nakamura | Hilbert schemes of abelian group orbits[END_REF] and proved by Bridgeland-King-Reid [START_REF] Bridgeland | The McKay correspondence as an equivalence of derived categories[END_REF]. There may also exist other crepant resolutions (perhaps not equvariant).

Example 6.3. Toric geometry [START_REF] Dais | All toric local complete intersection singularities admit projective crepant resolutions[END_REF][START_REF] Chiang | On hypersurface quotient singularities of dimension 4[END_REF] provides numerous examples, in arbitrary dimension, of orbifolds with toric, and therefore equivariant, crepant resolutions.

The equivariance property means that the radial vector field X = -2R ∂ ∂R on C n /G extends to J G , and we will continue to denote this extension by X; the vector field JX generates the action of S 1 ⊂ C * .

We will use the notation R for the radius function on J G , to distinguish it from the homothetic radius r on the Cao soliton.

As previously for the Cao soliton, we introduce Hölder weighted spaces adapted to the geometry of the manifolds J G . Consider a positive locally bounded weight w - γ , which for R ≥ 2 satisfies

w - γ = R γ .
If there is no ambiguity, we will denote w - γ by w -.

Take any reference metric on J G which coincides with the flat metric of C n /G near infinity. There is a positive function σ such that at each point σ(x) is smaller than the injectivity radius at x, and near infinity σ(x) = cst. R. So w - γ and σ γ are comparable (w - γ /σ γ and σ γ /w - γ are bounded). Then the analogue of formula (5.2), with this reference metric, this σ and this weight w - γ , defines a weighted Hölder space C k,α γ (J G ). We recall the Ricci-flat asymptotically locally Euclidean (ALE) Kähler metrics on J G , implicit in [START_REF] Tian | Complete Kähler manifolds with zero Ricci curvature[END_REF] and constructed directly in [START_REF] Joyce | Compact manifolds with special holonomy[END_REF]. These metrics are asymptotic to the flat metric on C n /G, up to a term in the weighted space C ∞ 2n (J G ). Theorem 6.4 (Joyce). Let k ∈ H 2 (J G , R) be a Kähler class on J G containing an ALE Kähler metric. Then k contains a unique ALE Ricci-flat metric ω -. Moreover, near infinity one has ω -= i 2 ∂∂Φ -, with

Φ -= R 2 + AR 2-2n + ψ, ψ ∈ C ∞ γ (J G ) for all γ < 2n -1.
Finally, if J G is an equivariant resolution, the metric ω -is S 1 -invariant.

The last statement follows from the uniqueness, since the S 1 action preserves the ALE behaviour at infinity.

Standard analysis on weighted spaces gives immediately: Proposition 6.5. For any γ ∈ (0, 2n -2), the Laplacian is an isomorphism

∆ : C 2,α γ (J G ) → C α γ+2 (J G ).

Proposition 6.6 (Schauder estimate on the Joyce manifold). There exists a constant C, such that for all u on

J G , ∥u∥ C 2,α γ (J G ) ≤ C ∥u∥ C 0 γ (J G ) + ∥∆u∥ C α γ+2 (J G ) . Since H 1 (J G , R) = 0, the S 1 -action generated by JX has a moment map µ -, that is a function such that dµ -= ι JX ω -. (6.1) It follows that L X ω -= dι X ω -= -dJdµ = -2i∂∂µ -.
Since near infinity ω -= i 2 ∂∂Φ -, we observe that dµ -= ι JX i 2 ∂∂Φ -= -1 4 dΦ -. Therefore, up to a constant,

µ -= - 1 4 XΦ -.

Generalities on steady Kähler-Ricci solitons

In this section, we digress to prove some general facts about steady Kähler-Ricci solitons, culminating in the uniqueness theorem stated in the Introduction. We work on an arbitrary noncompact complex manifold M .

Lemma 7.1. 1. Suppose that the Kähler forms ω and ω + i 2 ∂∂Ψ are both steady solitons with drift vector field X. Then

0 = i∂∂ -log (ω + i 2 ∂∂Ψ) n ω n + 1 4 XΨ .
2. Suppose that ω is Kähler, that µ is such that 1 2 L X ω = -i∂∂µ, and that Ω is a global holomorphic n-form. Let f = -log ω n Ω∧Ω -µ. Then ω + i 2 ∂∂Ψ is a steady soliton with drift vector field X, if and only if

0 = i∂∂ -log (ω + i 2 ∂∂Ψ) n ω n + 1 4 XΨ + f .
Proof. For the first part, we calculate

0 = Ric(ω + i 2 ∂∂Ψ) + 1 2 L X (ω + i 2 ∂∂Ψ) -Ric(ω) - 1 2 L X (ω) = -i∂∂ log (ω + i 2 ∂∂Ψ) n ω n + i∂∂( 1 4 XΨ).
For the second,

Ric(ω + i 2 ∂∂Ψ) + 1 2 L X (ω + i 2 ∂∂Ψ) = -i∂∂ log (ω + i 2 ∂∂Ψ) n Ω ∧ Ω -i∂∂(µ - 1 4 XΨ) = i∂∂ -log (ω + i 2 ∂∂Ψ) n ω n + 1 4 XΨ + f . (7.1)
We can now prove the uniqueness of the Kähler-Ricci soliton in its Kähler class, as stated in the Introduction. This relies on the well-known strong maximum principle of Hopf [GT01, Theorem 3.5].

Proof of Proposition 1.2. Suppose that the Kähler forms ω and ω + i 2 ∂∂Ψ are both steady solitons, with Ψ, XΨ and ∂∂Ψ going to 0 at infinity (the last in comparison with ω). Then

log (ω + i 2 ∂∂Ψ) n ω n - 1 4 XΨ
tends to 0 at infinity, and by Lemma 7.1

∆ ω -log (ω + i 2 ∂∂Ψ) n ω n + 1 4 XΨ = 0.
By the strong maximum principle, ∆ ω -harmonic functions on M either do not attain their extrema, or are constant. So, since the function tends to 0 at infinity, it vanishes. By the convexity of the logarithmic function,

0 = log (ω + i 2 ∂∂Ψ) n ω n - 1 4 XΨ ≤ 1 4 (∆ ω -X)Ψ.
Again by the strong maximum principle, (∆ ω -X)-subharmonic functions either do not attain their maximum, or are constant. Since Ψ → 0 at infinity, Ψ ≤ 0 everywhere. Inverting the roles of ω and ω + i 2 ∂∂Ψ gives the other inequality Ψ ≥ 0, so Ψ = 0.

The approximate solution and the Monge-Ampère equation

We now begin the gluing construction. For each small ϵ > 0, define a diffeomorphism p ϵ (z) = ϵ -1 z of C n /G, which also extends as a diffeomorphism of the equivariant resolution J G . Despite the fact that it is the same complex manifold, we wish to distinguish the desingularization of the Cao manifold, denoted M G , from the ALE space J G used for this desingularization. In this way, we see p ϵ as a 'blowup map'

p ϵ : M G -→ J G .
We will always consider J G equipped with the Joyce metric ω -(from Section 6), whereas we want to construct on M G a Kähler-Ricci soliton desingularizing the Cao soliton ω 0 on C G = (C n \ {0})/G (from Section 3). Let Φ 0 (on C G ) and Φ -(on the complement of a compact subset of J G ) be as in those sections, so that ω 0 = i 2 ∂∂Φ 0 and ω -= i ∂∂Φ -. For each small ϵ > 0, we define a reference Kähler metric ω ϵ on M G in the following way. Let χ be a cutoff function such that χ(r) = 0 for r ≤ 1 and χ(r) = 1 for r ≥ 2, and χ λ (r) = χ(r/λ). The gluing procedure will occur around the radius

r ϵ = ϵ n n+1 .
We construct ω ϵ in the following way:

• on r ≥ 2r ϵ , let ω ϵ = ω 0 ; • on r ≤ r ϵ , let ω ϵ = ϵ 2 p * ϵ ω -; • on r ϵ ≤ r ≤ 2r ϵ , let ω ϵ = i 2 ∂∂Φ ϵ , where Φ ϵ = χ rϵ (r)Φ 0 + (1 -χ rϵ (r))ϵ 2 p * ϵ Φ -. (8.1)
This is well-defined since in the intermediate region, ω ϵ coincides with ω 0 near r = 2r ϵ and with ϵ 2 p * ϵ ω -near r = r ϵ . It is of course S 1 -invariant. Observe that the function Φ ϵ defined in (8.1) is actually naturally defined on a larger region, say r ≥ ϵ (corresponding to R ≥ 1 in J G ), and that Φ ϵ is a potential for ω ϵ on this larger region.

To see that the forms ω ϵ are indeed Kähler for ϵ sufficiently small, note that we have Φ 0 -

r 2 = O(r 4 ) and Φ -(R) -R 2 = O(R 2-2n
), with control on the derivatives:

∇ k (Φ -(R) -R 2 ) = O(R 2-2n-k ). It follows that, for r ϵ ≤ r ≤ 2r ϵ , one has r 2 -ϵ 2 p * ϵ Φ -= O(ϵ 2n r 2-2n ) = O(r 4 ϵ ), ∇ 2 (r 2 -ϵ 2 p * ϵ Φ -) = O(ϵ 2n r -2n ) = O(r 2 ϵ ).
Therefore on r ϵ ≤ r ≤ 2r ϵ we have

ω ϵ - i 2 ∂∂r 2 = O(r 2 ϵ ),
so for ϵ small enough ω ϵ is positive, so is a Kähler form everywhere.

Since we also have bounds on all derivatives of Φ -, we more generally get on the same transition region r ϵ ≤ r ≤ 2r ϵ the estimates

∇ k (Φ ϵ -r 2 ) = O(r 4-k ϵ ), ∇ k (ω ϵ - i 2 ∂∂r 2 ) = O(r 2-k ϵ ). (8.2)
As in Section 6, there is a global moment map µ ϵ : M G → R, such that dµ ϵ = ι JX ω ϵ . We can fix the constant by deciding that, for r ≥ ϵ one has

µ ϵ = - 1 4 XΦ ϵ = -1 4 XΦ 0 , r ≥ 2r ϵ ; -1 4 X ϵ 2 p * ϵ Φ -= ϵ 2 p * ϵ µ -, ϵ ≤ r ≤ r ϵ . (8.3)
The latter implies that, in fact, µ ϵ = ϵ 2 p * ϵ µ -throughout the region r ≤ r ϵ . One again has

L X ω ϵ = -2i∂∂µ ϵ .
Let Ω be the global holomorphic n-form on M G which agrees, on C G = C n /G, with the Euclidean n-form. (Such an n-form exists since the resolution π :

M G → C G is crepant.) Let f ϵ = -log ω n ϵ Ω ∧ Ω -µ ϵ . (8.4)
and observe that by (2.3), f ϵ vanishes on r ≥ 2r ϵ . Motivated by Lemma 7.1, we now introduce the operator

T ϵ (Ψ) := (ω ϵ + i 2 ∂∂Ψ) n ω n ϵ -e 1 4 XΨ+fϵ . (8.5)
For the rest of the article we study the operators T ϵ .

Glued function spaces

We define weight functions w ϵ = w ϵ,γ,δ depending on the parameters γ ∈ (0, 2n -2) and δ ∈ (0, 1) (we will add the dependence in the parameters, for example w ϵ,γ to specify the dependence in γ, only in case of ambiguity) in the following way:

• for r ≥ r ϵ the weight w ϵ coincides with the weight defined on the Cao soliton: w ϵ = w 0 ; recall that w 0 (r) = r γ near r = 0;

• for r ≤ 2r ϵ the weight w ϵ coincides up to a constant with the weight for the Joyce metric: w ϵ = ϵ γ p * ϵ w -; since for R large one has w -(R) = R γ , this choices ensures that the two definitions coincide for r ϵ ≤ r ≤ 2r ϵ .

We define a weighted Hölder space C k,α ϵ,γ,δ (M G ), "gluing the Hölder spaces of the Cao manifold and of the Joyce manifold," by the analogue of formula (5.2) for the glued metrics ω ϵ , glued weights w ϵ,γ,δ , and similarly glued σ ϵ . For example, the C 0 norm is

∥f ∥ C 0 ϵ,γ,δ = sup w ϵ,γ,δ |f |.
A perhaps-clarifying remark is that, for cutoff functions χ λ as in the previous section,

∥f ∥ C k,α ϵ,γ,δ ∼ ∥χ rϵ (r)f ∥ C k,α γ,δ (C G ) + ϵ γ ∥(p ϵ ) * (1 -χ rϵ (r))f ∥ C k,α γ (J G )
. (The factor ϵ γ is exactly the factor required to make the two norms be equivalent on the transition region r ϵ ≤ r ≤ 2r ϵ .) Proposition 9.1 (Uniform Schauder estimate). There exists a constant C, such that for all ϵ > 0 sufficiently small, and all u on M G ,

∥u∥ C 2,α ϵ,γ,δ (M G ) ≤ C ∥u∥ C 0 ϵ,γ,δ (M G ) + ∥(∆ ϵ -X)u∥ C 0,α ϵ,γ+2,δ (M G ) . Proof.
As for Propositions 4.1 and 5.3.

Error term

We want to control f ϵ defined by (8.4). We know that f ϵ vanishes on r ≥ 2r ϵ , so the needed estimate is only on the Joyce manifold. We prove: Proposition 10.1. For all γ ≥ -4, δ, and k, there exists C such that for ϵ sufficiently small,

∥f ϵ ∥ C k ϵ,γ+2,δ ≤ Cr 4+γ ϵ .
Corollary 10.2. Under the same hypotheses, one has

∥T ϵ (0)∥ C k ϵ,γ+2,δ = ∥1 -e fϵ ∥ C k ϵ,γ+2,δ ≤ Cr 4+γ ϵ .
Proof of Corollary 10.2. Observe that the bound in Proposition 10.1 reads as

ϵ γ+2 sup R≤2ϵ -1 rϵ w - γ+2+k |∇ k f ϵ | ω -≤ C k ϵ n n+1 (4+γ) that is sup |∇ k f ϵ | ω -≤ C k (w - γ+2+k ) -1 ϵ 2n-2-γ n+1
for ϵ small enough. Expanding ∇ k (e fϵ -1) in terms of the derivatives of f ϵ now gives the same bounds on e fϵ -1.

Proof of Proposition 10.1. Since f ϵ vanishes for r ≥ 2r ϵ , we only need to bound f ϵ for r ≤ 2r ϵ . We will bound independently the two terms log ω n ϵ Ω∧Ω and µ ϵ . The first term is nonzero only on the transition region r ϵ ≤ r ≤ 2r ϵ . From the bounds (8.2), we get that on this region

∇ k ω n ϵ Ω ∧ Ω = O(r 2-k ϵ ),
and so sup rϵ≤r≤2rϵ

r 2+γ+k ∇ k log ω n ϵ Ω ∧ Ω ≤ c k r 4+γ ϵ
which is exactly the required bound. By (8.3), the second term µ ϵ coincides with -1 4 XΨ ϵ on the region where the potential Ψ ϵ exists, that is r ≥ ϵ; and with ϵ 2 p * ϵ µ -on r ≤ r ϵ . From the estimate (8.2) we see that on the transition region r ϵ ≤ r ≤ 2r ϵ we have

∇ k µ ϵ = O(r 2-k ϵ ),
and so sup rϵ≤r≤2rϵ

r 2+γ+k ∇ k µ ϵ ≤ c k r 4+γ ϵ ,
the desired estimate. Finally, on r ≤ r ϵ it is easier to write the required estimate in terms of norms on the manifold J G :

∥µ ϵ ∥ C k ϵ,γ+2,δ ({r≤rϵ}) = ϵ γ+2 ∥(p -1 ϵ ) * µ ϵ ∥ C k γ+2 ({R≤ϵ -1 rϵ}) = ϵ γ+4 ∥µ -∥ C k γ+2 ({R≤ϵ -1 rϵ})
≤ Cr 4+γ ϵ so long as γ ≥ -4.

Remark 10.3. The choice of r ϵ was made to get an optimal bound on f ϵ : the two errors coming from cutting off the Cao soliton, and from gluing the Joyce manifold (which is Ricci-flat, not a soliton), are of the same order around r ϵ . For example, for n = 2 we have r ϵ = ϵ 2 3 . A different choice (such as r ϵ = ϵ 1 2 , which means taking a bigger r ϵ ) would give a larger error term; it would then be necessary to correct the Joyce metric as in [BM11, § 2.4] before gluing, but we do not need to do this with our choice of r ϵ .

First-order term: Blowup argument

The linearization of the operator T ϵ defined in (8.5) is 1 4 (∆ ϵ -e fϵ X). We first study the simpler operator ∆ ϵ -X, then, in Corollary 11.4, use the control on e fϵ from Corollary 10.2 to extend the results to ∆ ϵ -e fϵ X.

Lemma 11.1. For each γ, δ, and each ϵ > 0, the operator

∆ ϵ -X : C 2,α ϵ,γ,δ (M G ) → C α ϵ,γ+2,δ (M G )
is continuous.

Proof. We have that (∆ ϵ -X)u = ⟨ω -1 ϵ , ∇ 2 u⟩ + ⟨-X, du⟩, where ⟨•, •⟩ denotes a full trace. Since both ω -1 ϵ and X are bounded with respect to the metric ω ϵ , the operator is continuous.

Proposition 11.2. For each 0 < δ < 1, each 0 < γ < 2n -2, and each 0 < α < 1, there exists a constant c, such that for all ϵ sufficiently small, for all r 0 ∈ [1, ∞], and for all u ∈ C 2 ϵ,γ,δ ({r ≤ r 0 }), with the Dirichlet boundary condition

u| r=r 0 = 0 if r 0 < ∞, ∥u∥ C 0 ϵ,γ,δ (r≤r 0 ) ≤ c∥(∆ ϵ -X)u∥ C 0 ϵ,γ+2,δ (r≤r 0 ) .
Proof. We prove this by contradiction. Suppose this false; then there exist a sequence ϵ i → 0, a sequence r i ∈ [1, ∞], and a sequence of functions u i defined on {r ≤ r i }, satisfying u i | r=r i = 0, with

∥(∆ -X)u i ∥ C 0 ϵ i ,γ+2,δ
→ 0, (11.1) but for all i,

∥u i ∥ C 0 ϵ i ,γ,δ = 1.
The first observation is that by (4.2), together with the fact that for r ≥ 1 the weight w used for the Cao metric coincides with the weights w ϵ,γ , w ϵ,γ+2 used to define the norms C 0 ϵ,γ,δ and C 0 ϵ,γ+2,δ , sup

1≤r≤r i w ϵ,γ |u i | ≤ max sup r=1 w ϵ,γ |u i |, 1 4δ(1 -δ)n sup 1≤r≤r i w ϵ,γ+2 |(∆ -X)u i | .
Since by hypothesis sup w ϵ,γ+2 |(∆ -X)u i | → 0, we see that there exist points z i such that:

|z i | ≤ 1 w ϵ,γ (z i )|u i (z i )| = 1.
We will now use a blowup argument at z i . Up to extracting a subsequence, we can suppose that r i → r ∞ ∈ [1, ∞]. We distinguish three cases:

Case 1: 0 < inf i |z i |.
We can suppose z i → z ∞ . From local elliptic regularity (including at the boundary r = r i if it is finite), as in [GT01, Theorems 8.32-3], we have local C 1,α ′ bounds on u i , so we can extract a subsequence C 1,α -converging on each compact subset of the domain {r < r ∞ } of the Cao manifold C G . Let u ∞ be its limit; since the weights w ϵ converge to the weight w 0 for C G , we get

(∆ -X)u ∞ = 0 (and u ∞ is smooth), sup w 0 |u ∞ | = w 0 (z ∞ )|u ∞ (z ∞ )| = 1, u ∞ | {r=r∞} = 0 if r ∞ < ∞.
Proposition 5.3, the Schauder estimate on the Cao manifold C G , gives that

∥u ∞ ∥ C 2,α γ,δ ({r≤r∞}) ≤ C ∥u ∞ ∥ C 0 γ,δ ({r≤r∞}) + ∥(∆ -X)u ∞ ∥ C α γ+2,δ ({r≤r∞}) = C sup {r≤r∞} w 0 |u ∞ | < ∞. So u ∞ ∈ C 2,α γ,δ ({r ≤ r ∞ }) is a nonzero element of the kernel of ∆ -X. If r ∞ = ∞, this contradicts Theorem 5.2. If r ∞ < ∞,
noting that u ∞ also satisfies the Dirichlet boundary condition, this contradicts Lemma 5.1.

Case 2:

sup i ϵ -1 i |z i | < ∞. Let ζ i := p ϵ i (z i ) ∈ J G ; we can suppose that ζ i → ζ ∞ ∈ J G . Let v i := ϵ γ i (p ϵ i ) * u i .
By definition the weight w ϵ,γ,δ is equal to the Joyce weight ϵ γ p * ϵ w - γ on the larger and larger domains {R ≤ ϵ -1 r ϵ } of J G , so sup

{R≤ϵ -1 rϵ} w - γ |v i | = w - γ (ζ i )|v i (ζ i )| = 1. (11.2)
On the other hand, since ϵ -2 ω ϵ = ω -for R ≤ ϵ -1 r ϵ , the control (11.1) translates into sup

{R≤ϵ -1 rϵ} w - γ+2 |(∆ ω --ϵ 2 X)v i | = sup {r≤rϵ} w ϵ,γ+2,δ |(∆ ωϵ -X)u i | -→ 0. (11.3)
The operators ∆ ω --ϵ 2 X are uniformly locally controlled, so by local elliptic regularity [GT01, Theorems 8.32-3], as in Case 1, we can extract a subsequence C 1,α -converging on each compact domain of J G . Let v ∞ be its limit. By (11.2) and (11.3),

∆ ω -v ∞ = 0, sup w - γ |v ∞ | = 1.
We conclude as before, using Proposition 6.6 (the Schauder estimate on the Joyce manifold) and Proposition 6.5 (the proof of isomorphism). Case 3: We may pass to a subsequence with z i → 0 and ϵ -1 i |z i | → ∞. In this case we perform a similar extraction on the cone C n /G itself, by scaling by

λ i = |z i |.
As in Case 2, we consider a sequence of homotheties which restrict the points to a compact region. In this case we take the homotheties p λ i , and introduce rescaled functions

v i := λ γ i (p λ i ) * u i . We can suppose ζ i = z i λ i → ζ ∞ ∈ C n /G. Finally, now λ -2 i (p λ i ) * ω ϵ i converges
(R) = R γ , in both instances because 1 ≫ λ i ≫ ϵ i .
The next part of the proof is unchanged from Case 2: the operators ∆ λ -2 i (p λ i ) * ωϵ i -λ 2 i X are uniformly locally controlled, and we get a nonzero solution v ∞ of ∆v ∞ = 0 on the whole flat cone

C n /G, with sup C n /G R γ |v ∞ | < ∞.
Finally, since γ ∈ (0, 2n -2) is not a critical weight of the Laplacian, ∆ :

C 2,α γ (C n /G) → C α γ+2 (C n /G
) is an isomorphism and such a v ∞ cannot exist.

Proposition 11.3. For each 0 < δ < 1, each 0 < γ < 2n -2, and each 0 < α < 1, there exists a constant c, such that for all ϵ sufficiently small, the operator

∆ ϵ -X : C 2,α ϵ,γ,δ (M G ) → C α ϵ,γ+2,δ (M G ) is an isomorphism, and for all u ∈ C 2,α ϵ,γ,δ (M G ), ∥u∥ C 2,α ϵ,γ,δ ≤ c∥(∆ ϵ -X)u∥ C 0,α ϵ,γ+2,δ . (11.4)
Proof. First observe that the estimate (11.4), and the injectivity of the operator, follow from the C 0 estimate in Proposition 11.2, combined with the uniform Schauder estimate Proposition 9.1. We now show the surjectivity of ∆ -X. This is the same argument as in Theorem 4.3. We want to solve (∆ -X)u = v. On each of a sequnce of bounded domains {t ≤ t i }, one can solve (∆ -X)u i = v with u i | r=r i = 0, since the operator ∆ -X, being a compact perturbation of ∆, has index zero for the Dirichlet problem, and is injective (by the maximum principle).

By Proposition 11.2, together with the Schauder estimate Proposition 4.1, we have that for all t 0 and all i such that t i ≥ t 0 + 1,

∥u i ∥ C 2,α ϵ,γ,δ ({t≤t 0 }) ≤ C∥v∥ C α ϵ,γ+2,δ (M G ) ,
where the constant C is independent both of t 0 and of i. Therefore when i → ∞ we can extract a

C 2 loc -limit u still satisfying ∥u∥ C 2,α ϵ,γ,δ ≤ c∥v∥ C 0 ϵ,γ+2,δ
, and solving (∆ -X)u = v.

Corollary 11.4. Proposition 11.3 is also valid for the family of operators ∆ ϵ -e fϵ X.

Proof. It suffices to check that the norm of the operator (e fϵ -1)X :

C 2,α ϵ,γ,δ (M G ) -→ C α ϵ,γ+2,δ (M G )
goes to zero, so that adding it to ∆ ϵ -X does not perturb the invertibility and the estimate on the inverse. This follows easily from Corollary 10.2.

12 Second-order term

Let us write

T ϵ (Ψ) = T ϵ (0) + L ϵ (Ψ) + Q ϵ (Ψ),
where L ϵ = 1 4 (∆ ϵ -e fϵ X) is the linearization of T ϵ at the origin.

Proposition 12.1. There exists constants c, c

′ > 0 such that if ∥Ψ∥ C 2,α ϵ,γ,δ , ∥Ψ ′ ∥ C 2,α ϵ,γ,δ ≤ cϵ 2+γ , then ∥Q ϵ (Ψ) -Q ϵ (Ψ ′ )∥ C α ϵ,γ+2,δ ≤ c ′ ϵ -2-γ ∥ψ -ψ ′ ∥ C 2,α ϵ,γ,δ (∥ψ∥ C 2,α ϵ,γ,δ + ∥ψ ′ ∥ C 2,α ϵ,γ,δ
).

Proof. Such an estimate is obvious on a ball, without weights, if the geometry of the metric is uniformly controlled: it comes down to saying that the second derivative D 2 T remains bounded on a ball. We can apply this remark on balls on the Cao part of the manifold (r ≥ 1). The weighted norm on balls is just the usual Hölder norm, multiplied by the factor w ϵ :

∥u∥ C k,α ϵ,γ,δ (B(x 0 ,1)) ∼ w ϵ (x 0 )∥u∥ C k,α (B(x 0 ,1)) .
Then, for ∥Ψ∥ C 2,α (B(x 0 ,1)) , ∥Ψ ′ ∥ C 2,α (B(x 0 ,1)) ≤ c we have the standard estimate ∥Q ϵ (Ψ) -Q ϵ (Ψ ′ )∥ C α (B(x 0 ,1)) ≤ c ′ ∥Ψ -Ψ ′ ∥ C 2,α (B(x 0 ,1)) (∥Ψ∥ C 2,α (B(x 0 ,1)) + ∥Ψ ′ ∥ C 2,α (B(x 0 ,1)) ).

Therefore,

∥Q ϵ (Ψ) -Q ϵ (Ψ ′ )∥ C α ϵ,γ+2,δ (B(x 0 ,1))
≤ c ′ w ϵ (x 0 ) -1 ∥Ψ -Ψ ′ ∥ C 2,α ϵ,γ,δ (B(x 0 ,1)) (∥Ψ∥ C 2,α ϵ,γ,δ (B(x 0 ,1)) + ∥Ψ ′ ∥ C 2,α ϵ,γ,δ (B(x 0 ,1)) )

which gives the required estimate since w ϵ (x 0 ) ≥ 1.

The situation is quite different on the part where we glued the Joyce manifold, because the injectivity radius and the weight become small. First observe that

T ϵ (Ψ) = ( ωϵ ϵ 2 + i 2 ∂∂ Ψ ϵ 2 ) n ( ωϵ ϵ 2 ) n -e fϵ+ ϵ 2 4 X Ψ ϵ 2 ,
and recall that on the Joyce region r ≤ r ϵ , ωϵ ϵ 2 = ω -, the metric on J G , and f ϵ = -ϵ 2 p * ϵ µ -. Therefore on that region

T ϵ (Ψ) = (ω -+ i 2 ∂∂ Ψ ϵ 2 ) n ω n - -e ϵ 2 [-µ -+ 1 4 X Ψ ϵ 2 ] .
On a large compact part K ⊂ J G , we have as above, for norms taken with respect to ω -,

∥Q ϵ (Ψ) -Q ϵ (Ψ ′ )∥ C α ≤ c ′ Ψ ϵ 2 - Ψ ′ ϵ 2 C 2,α Ψ ϵ 2 C 2,α + Ψ ′ ϵ 2 C 2,α for ∥ Ψ ϵ 2 ∥ C 2,α , ∥ Ψ ′ ϵ 2 ∥ C 2,
∥Q ϵ (Ψ) -Q ϵ (Ψ ′ )∥ C α ϵ,γ+2,δ ≤ c ′ ϵ -2-γ ∥Ψ -Ψ ′ ∥ C 2,α ϵ,γ,δ (∥Ψ∥ C 2,α ϵ,γ,δ + ∥Ψ ′ ∥ C 2,α ϵ,γ,δ
).

This is again the required estimate. The case of the conical region between K and {r ≥ 1} is similar and we will not write the details. The worst constants are obtained on the K part of the manifold, because this is the part of the manifold with smallest injectivity radius and weight. This explains the constants in the statement of the Proposition.

Inverse function theorem

The proof of Theorem 1.1 will now follow directly from the following quantitative version of the inverse function theorem, previously used in [START_REF] Biquard | A Kummer construction for gravitational instantons[END_REF]: Lemma 13.1. Let T : E → F be a smooth map between Banach spaces and define Q := T -T (0) -DT | 0 . Suppose that there are positive constants q, r 0 and c, such that 1. ∥Q(x) -Q(y)∥ ≤ q∥x -y∥(∥x∥ + ∥y∥) for every x and y in B E (0, r 0 ); 2. DT | 0 is an isomorphism with inverse bounded by c; 3. ∥T (0)∥ < 1 2c min(r 0 , 1 2qc ). Then the equation T (x) = 0 admits a unique solution x in B E (0, min(r 0 , 1 2qc )). Proof of Theorem 1.1. For fixed 0 < δ < 1 and 0 < γ < 2n -2, consider the family of operators

T ϵ : C 2,α
ϵ,γ,δ (M G ) → C 0,α ϵ,γ+2,δ (M G ). We will show that for ϵ sufficiently small, the operator T ϵ satisfies the hypotheses of Lemma 13.1. By Corollary 10.2, for ϵ sufficiently small ).

Thus if we take q := Cϵ -(γ+2) and r 0 := Cϵ γ+2 , then (1) and (2) are satisfied, and moreover 1 2c min(r 0 , 1 2qc ) ≥ C -1 ϵ γ+2 ≥ C -1 ϵ -2n-2-γ n+1 ∥T a (0)∥.

Since γ < 2n -2, for ϵ sufficiently small we then have that C -1 ϵ -2n-2-γ n+1 ≥ 1, and so (3) is satisfied for ϵ sufficiently small. Therefore by Lemma 13.1 there exists a solution Ψ ϵ to the equation T ϵ (Ψ ϵ ) = 0, i.e., (ω ϵ + i 2 ∂∂Ψ ϵ ) n = e (XΨϵ)/4+fϵ Ω ∧ Ω. The solution Ψ ϵ satisfies, ∥Ψ ϵ ∥ C 2,α ϵ,γ,δ (M G ) ≤ Cϵ 2+γ . Since Ψ ∈ C 2,α ϵ,γ,δ (M G ) is a solution to the equation T ϵ (Ψ) = 0, its differential dΨ is a weak solution to a quasilinear elliptic equation 0 = dT ϵ (Ψ) = ∆ ωϵ+ i 2 ∂∂Ψ dΨ + F (∇ 2 Ψ, dΨ).

Bootstrapping using the Schauder estimate Proposition 4.1, we conclude that Ψ ∈ C k,α ϵ,γ,δ (M G ) for all k. In particular, by Lemma 7.1, ω ϵ + i 2 ∂∂Ψ is a steady Kähler-Ricci soliton.

Example 6. 1 .

 1 In dimension n = 2, finite subgroups G ⊆ SU (2) which act freely are classified by Dynkin diagrams, falling into two infinite classes A k , D k together with some exceptional examples. The associated orbifolds C 2 /G are the Kleinian or Du Val singularities, and there is a unique minimal resolution of each such singularity, which is equivariant and crepant.

≤

  ∥T ϵ (0)∥ C 0,α ϵ,γ+2,δ (M G ) = ∥1 -e fϵ ∥ C 0,α ϵ,γ+2,δ (M G ) ≤ Cϵ (4+γ) n n+1 .By Corollary 11.4, for ϵ sufficiently smallL ϵ = 1 4 [∆ a -e fa X] : C 2,α ϵ,γ,δ (M G ) → C 0,α ϵ,γ+2,δ (M G) is an isomorphism, and its inverse is bounded by a constant independent of ϵ.By Proposition 12.1, for ϵ sufficiently small, for all Ψ, Ψ ′ with∥Ψ∥ C 2,α ϵ,γ,δ , ∥Ψ ′ ∥ C 2,α ϵ,γ,δ Cϵ γ+2 the map Q ϵ := T ϵ -T ϵ (0) -L ϵ satisfies, ∥Q ϵ (Ψ) -Q ϵ (Ψ ′ )∥ C 0,α ϵ,γ+2,δ ≤ Cϵ -(γ+2) ∥Ψ -Ψ ′ ∥ C 2,α ϵ,γ,δ (∥Ψ∥ C 2,α ϵ,γ,δ + ∥Ψ ′ ∥ C 2,α ϵ,γ,δ

  Therefore, for A sufficiently large and positive, say A 2 , this function is positive for all t sufficiently large, and for A sufficiently small, say A 1 , this function is negative for all t sufficiently large. That is, for all t sufficiently large,

	This function has limit			A -	(n -1) 2 n
	as t → ∞. F φ 0 (t) + A 1	log t t	e (φ0(t)+A1 log t t ) <	e nt n	+ F (a)e a < F φ 0 (t) + A 2	log t t	e (φ0(t)+A2 log t t ) .
	Since e nt n + F (a)e a = F (φ(t)) (by (2.5)), by the monotonicity of the function F (s)e s we conclude
	that for all t sufficiently large
			1) log t -n log n. For a fixed real number A, consider the function
	  	nF φ 0 (t) + A log t t	exp φ 0 (t) + A log t t e nt	-1 -	nF (a)e a e nt	  	t log t	.

  to the flat cone metric on C n /G, and the weights λ γ i p * λ i w ϵ i ,γ,δ converge to the cone weight w γ

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2016 semester.


𝑟 Savitri Gallego

𝑞 Chien-You Huang

𝑓 Chris Karwin

𝑓 Carolyn Kierans

𝑟 Jan Peter Lommler

Saurabh Mittal

𝑑 Michela Negro

𝑓 ,𝑙 Eliza Neights

𝑏 Sean N Pike

𝑢, 𝑓 Israel Martinez-Castellanos

𝑐 Yong Sheng

Thomas Siegert

Hiroki Yoneda

𝑎 Andreas Zoglauer

𝑎 John A Tomsick

𝑏,𝑎 Steven E Boggs

𝑐 Dieter Hartmann

𝑐 Marco Ajello

𝑑 Eric Burns

𝑒 Chris Fryer

𝑎 Alexander Lowell

𝑔 Julien Malzac

𝑏 Jarred Roberts

𝑎 Pascal Saint-Hilaire

𝑓 Albert Shih

𝑖 Clio Sleator

𝑗 Tadayuki Takahashi

𝑘 Fabrizio Tavecchio

𝑖 Eric Wulf

𝑎 Jacqueline Beechert

𝑎 Hannah Gulick

𝑎 Alyson Joens

𝑎 Hadar Lazar

Juan Carlos

𝑎 Martinez Oliveros

𝑗 Shigeki Matsumoto

𝑗 Tom Melia

𝑚 Mark Amman

𝑐 Dhruv Bal

𝑔 Peter Von Ballmoos

𝑐 Hugh Bates

𝑛 Markus Böttcher

𝑜 Andrea Bulgarelli

𝑝 Elisabetta Cavazzuti

𝑞 Hsiang-Kuang Chang

𝑎 Claire Chen

𝑞 Che-Yen Chu

Alex Ciabattoni

𝑝 𝑜 Luigi Costamante

𝑛 Lente Dreyer

𝑜 Valentina Fioretti

𝑝 Francesco Fenu

𝑘 Giancarlo Ghirlanda

𝑖 Eric Grove

𝑔 Pierre Jean

𝑐 Nikita Khatiya

𝑔 Jürgen Knödlseder

𝑠 Martin Krause

𝑐 Mark Leising

𝑓 ,𝑤 Tiffany R Lewis

Lea Marcotulli

𝑡 Samer

𝑎 Al Nussirat

𝑣 Kazuhiro Nakazawa

𝑟 Uwe Oberlack

𝑑 David Palmore

𝑜 Gabriele Panebianco

𝑜 Nicolo Parmiggiani

Tyler Parsotan

𝑓 Field

𝑛 Hester Schutte

𝑓 Alan P Smale

𝑖 Jacob Smith

𝑑 Aaron Trigg

𝑓 Tonia Venters

Yu Watanabe

Haocheng Zhang

The cosipy library: COSI's high-level analysis software Israel

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Overview

The Compton Spectrometer and Imager (COSI) [START_REF] Tomsick | The Compton Spectrometer and Imager[END_REF] is a compact Compton telescope sensitive to gamma rays between 200 keV and 5 MeV. COSI is a survey instrument with a large field of view -> 𝜋 sr-excellent energy resolution -<1.2% (<0.07%) at 511 keV (1.157 MeV)-capable of imaging with a good angular resolution -< 4.1 • (< 2.1 • ) at 511 keV (1.809 MeV)-and performing polarization measurements. COSI will reduce the so-called "MeV gap", the low-energy part of the gamma-ray spectrum that despite its great scientific potential is currently under explored. COSI will uncover the origin of the Galactic positrons, advance our understanding of how the elements were formed, and probe the physics of extreme environments and multi-messenger sources.

The COSI team is currently developing cosipy, a Python library that will be capable of performing all the high-level task expected for COSI: imaging, spectral analysis and polarimetry. On this contribution we first introduce, in Section 2, the basic principles of Compton data analysis. This is then followed by the design of cosipy and its current status. We also include our plans for the gradual improvement of cosipy and public data challenges, leading up to the expected launch of COSI in 2027.

Compton data analysis

In the MeV regime photons interact with the detector predominantly through Compton scattering. As shown in Fig. 1, the incoming photon results in a series of energy deposits in the detector until it is finally absorbed due to the photoelectric effect. Compact telescope like COSI cannot resolve the order of the interactions based on their timing, and therefore it must be derived based on the kinematics of Compton scattering. Once this is determined, the scattering angle 𝜙 and direction (l',b') of the outgoing photon in the first interaction define a circle in the sky that contains the direction of the incoming photon (l,b). The set of parameters {𝜙, l', b'} is usually called the Compton data space (CDS). Although the following interactions also contain useful information, the CDS is the minimum information needed to perform imaging and polarimetry.

Since the radius of the Compton circle is fully constrained by the scattering angle 𝜙, the combination over all possible values of 𝜙 define a cone in the three-dimensional CDS. The walls of this cones are broaden due to instrumental effects, such as the unknown kinetic energy of the electron that participated in the first interaction, the finite energy resolution, missed interactions due to the detection threshold, etc. The distribution of the broaden Compton cone is the point spread function (PSF) of the instrument, as shown in Fig. 2.

A source can be localized by combining the information from multiple events, as shown in Fig 3 . The imaging of the sky involves multiple sources and backgrounds events, not shown in Fig. 3a, that require the use of complex deconvolution algorithms. All these calculations are performed in the CDS where the Compton cones for each source can be distinguished, as exemplified in Fig. 3b.

Polarimetry is possible thanks to the azimuthal dependence of Compton scattering in response to a linearly polarized source. As shown in Fig. 4, the distribution of events in the Compton cone is modulated by the polarization angle -determining the phase-and the polarization fraction -proportional to the modulation factor.

The cosipy library

Although the basic principles of Compton data analysis can be applied to approximate the spectrum, location and polarization of source, real data presents significantly more challenges. The large number of events, high and variable backgrounds, instrument artifacts, and the continuous rotation of the spacecraft, all call for a more complex and robust high-analysis software. The timedependent spectra and polarization, and correlations between energy, polarization and direction, also complicate the analysis.

The cosipy library is a component of a series of software packages called cositools, and the last step of the COSI calibration and analysis pipeline, shown in 5. It builds upon the development during the COSI balloon campaigns. For a full description of the calibration and pipelines see [START_REF] Beechert | Calibrations of the Compton Spectrometer and Imager[END_REF][START_REF] Zoglauer | COSI: From Calibrations and Observations to All-sky Images[END_REF].

Current design and status

The cosipy library follows a likelihood-based approach. It uses a forward folding technique to compare the expected measurements to the data -energy 𝐸 𝑚 , scattering angle 𝜙, and outgoing photon direction (𝑙 ′ , 𝑏 ′ ), which together we will call the data space Ωgiven a sky model hypothesis -i.e. a distribution of sources, each with a given spectrum. The relationship between the expected measurements and physical photon parameters -incoming direction (𝑙, 𝑏), energy 𝐸 𝑖 and polarization angle 𝑃 𝐴, which we will call the photon parameter space Λis handled by the detector response, and encoded in a a multi-dimensional matrix.

The likelihood can be written as:

𝜆 𝑖 (s, b) = ∫ ΔΩ 𝑖 ∫ 𝐴 eff (Λ) 𝑃 (Ω|Λ) 𝐹 (s, Λ)𝑑Λ + Bkg(b) 𝑑Ω ,
where s and b are the signal and background parameters to fit; 𝑃(𝑛|𝜆) is the Poisson probability distribution; the term 𝐴 eff (Λ) 𝑃 (Ω|Λ) is the detector response, equal to the effective area times the probability of a detected event with physical parameters Λ to have the measured parameters Ω; and 𝐹 is the flux for a given location in the sky at a given energy and polarization. The detector response is obtained from event by event Monte Carlo simulations using MEGALib [START_REF] Zoglauer | MEGAlib The Medium Energy Gamma-ray Astronomy Library[END_REF]. All passive and active components are carefully modeled, and the detector effects simulated, in order to match benchmark calibration measurements.

The cosipy library is compatible with the Multi-Mission Maximum Likelihood framework (3ML) [START_REF] Vianello | The multi-mission maximum likelihood framework (3ml)[END_REF] which provides a common interface between multiple instruments. This is achieved by the use of plugins that receive a source model in a common format, and output the corresponding likelihood. This allows us to gain access to the maximum likelihood and Bayesian algorithms provided by 3ML in order to do spectral and polarization analysis.

Imaging deconvolution can also be performed by maximizing the likelihood in Eq. 1 over multiple hypothetical sources in a grid fully covering the sky. Due to the high dimensionality of the problem we use the Richardson-Lucy algorithm [START_REF] Lucy | An iterative technique for the rectification of observed distributions[END_REF][START_REF] Richardson | Bayesian-Based Iterative Method of Image Restoration[END_REF] which consists of an interactive method that converges to the maximum likelihood solution.

We have preliminary versions of the detectors response handling, likelihood calculation, imaging and spectral analysis. Selected results are shown in Fig. 6. During these early stages of development we are using a simplified detector response and coarse binning in order to test and profile the code. Currently we are working on optimizing the CPU and memory usage such that the computation can scale well to requirements of the flight version.

Data challenges and future plans

We will progressively add new capabilities and improvements to cosipy, leading up to the launch of COSI in 2027. Starting in 2024, we will organize public data challenges with a yearly cadence, consisting of simulated data sets for sources and science topics of interest. This will build upon the COSI balloon data challenge released in 2023 [START_REF] Karwin | The COSI Data Challenges and Simulations[END_REF]. We want to obtain feedback from the community to find issues and identify missing features required to satisfy all COSI science goals as well as to expand its portfolio. Some of the plans for the upcoming releases include:

• Simultaneous location, spectral and polarization fits.

• Time-dependent polarization analysis.

• Implementation of alternative imaging deconvolution algorithms, such as Multiresolution Regularized Expectation Maximization, and machine-learning-based image creation.

Conclusion

The cosipy library will perform all high-level analysis tasks to satisfy the scientific goals of the COSI mission, an MeV gamma-ray telescope launching in 2027. Currently under development, cosipy follows a modern likelihood-based approach, is open source, and compatible with the Multi-Mission Maximum Likelihood framework (3ML). Starting in 2024, we will make yearly releases accompanied by a public data challenge in order to obtain feedback from the community.

Figure 1 :

 1 Figure 1: COSI measures the location and the energy deposited by the different interactions of a gamma ray with the detector. During reconstruction, this path is deduced based on the kinematics of Compton scattering, constraining the direction of the incoming photon to a circle in the sky, centered in the direction of the first scattered gamma ray (l',b') and with a radius equal to the scatting angle of the first interaction 𝜙.

Figure 2 :

 2 Figure 2: Different slices of a simplified point spread function (PSF) in the Compton data space (CDS). The PSF was sliced along the 𝜙-axis (scattering angle) and shown in an ortographic projection of the sphere describing the direction of the outgoing scattered photon. The red dot shows the true location of the incoming photon.

Figure 3 :

 3 Figure 3: (a) A simplified example demonstrating the localization of a source (red dot) based on three detected events (white crosses, with Compton circles shown with dashed lines). (b,c) Events produced by two sources (s 1 and s 2 ) distributed in the Compton Data Space (CDS). Views from the top (b) and side (c). By keeping track of the scattered angle 𝜙 the cones can be discriminated and the image deconvolved.

Figure 4 :

 4 Figure 4: Left: simplified point spread function (PSF) for a linearly polarized sources. Right: projection onto the azimuth angle. The ratio between the black dotted lines is proportional to the polarization fraction. The phase is determined by the polarization angle (PA).

Figure 5 :

 5 Figure 5:The cosipy library will perform the High-level Data Analysis tasks. It is the last step of a series of software packages that compose the COSI calibration and analysis pipeline.

Figure 6 :

 6 Figure 6: Results from an early version of cosipy. (a) Simulated binned data, using a coarse grid, showing the location of a source (red dot) and the Compton circle. (b) Deconvolved image of the source using the Richardson-Lucy algorithm. (c) Fitted spectrum of the source using the 3ML framework.

The Compton Spectrometer and Imager (COSI) is a selected Small Explorer (SMEX) mission launching in 2027. It consists of a large field-of-view Compton telescope that will probe with increased sensitivity the under-explored MeV gamma-ray sky (0.2-5 MeV). We will present the current status of cosipy, a Python library that will perform spectral and polarization fits, image deconvolution, and all high-level analysis tasks required by COSI's broad science goals: uncovering the origin of the Galactic positrons, mapping the sites of Galactic nucleosynthesis, improving our models of the jet and emission mechanism of gamma-ray bursts (GRBs) and active galactic nuclei (AGNs), and detecting and localizing gravitational wave and neutrino sources. The cosipy library builds on the experience gained during the COSI balloon campaigns and will bring the analysis of data in the Compton regime to a modern open-source likelihood-based code, capable of performing coherent joint fits with other instruments using the Multi-Mission Maximum Likelihood framework (3ML). In this contribution, we will also discuss our plans to receive feedback from the community by having yearly software releases accompanied by publicly-available data challenges.38th International Cosmic Ray Conference (ICRC2023) 26 July -3 August, 2023 Nagoya, Japan