Geometry of Calogero-Moser spaces

Geometry of Calogero-Moser spaces

C´edric Bonnaf´e

CNRS (UMR 5149) - Universit´e de Montpellier

Poitiers, November 2016

Set-up

Set-up

dimCV =n <∞

W <GL_C(V), |W|<∞. Ref(W) ={s ∈W | codim_CV^s =1}

Set-up

dimCV =n <∞

W <GL_C(V), |W|<∞. Ref(W) ={s ∈W | codim_CV^s =1}

Hypothesis. W =hRef(W)i

Set-up

dimCV =n <∞

W <GL_C(V), |W|<∞. Ref(W) ={s ∈W | codim_CV^s =1}

Hypothesis. W =hRef(W)i (i.e. V/W ≃Cⁿ)

Set-up

dimCV =n <∞

W <GL_C(V), |W|<∞. Ref(W) ={s ∈W | codim_CV^s =1}

Hypothesis. W =hRef(W)i (i.e. V/W ≃Cⁿ) C ={c :Ref(W)/∼ −→C}

We fix c ∈ C

Cherednik algebra at t = 0

Cherednik algebra at t = 0

H_c =C[V]⊗CW ⊗C[V^∗] (as a vector space)

∀y ∈V, ∀x ∈V^∗, [y,x] = X

s∈Ref(W)

cshy,s(x) −xis

Cherednik algebra at t = 0

H_c =C[V]⊗CW ⊗C[V^∗] (as a vector space)

∀y ∈V, ∀x ∈V^∗, [y,x] = X

s∈Ref(W)

cshy,s(x) −xis

Cherednik algebra at t = 0

H_c =C[V]⊗CW ⊗C[V^∗] (as a vector space)

∀y ∈V, ∀x ∈V^∗, [y,x] = X

s∈Ref(W)

cshy,s(x) −xis

Let

Zc =Z(H_c)

Cherednik algebra at t = 0

H_c =C[V]⊗CW ⊗C[V^∗] (as a vector space)

∀y ∈V, ∀x ∈V^∗, [y,x] = X

s∈Ref(W)

cshy,s(x) −xis

Let

Zc =Z(H_c)

Easy fact.

C[V]^W,C[V^∗]^W ⊂Zc. Let

P =C[V]^W ⊗C[V^∗]^W ⊂Zc

Definition

TheCalogero-Moser space associated with the datum (W,c) is the affine variety

Z_c =Spec(Zc).

Definition

TheCalogero-Moser space associated with the datum (W,c) is the affine variety

Z_c =Spec(Zc).

It is endowed with a morphism

ϕ_c :Z_c −→Spec(P) =V/W ×V^∗/W

Definition

TheCalogero-Moser space associated with the datum (W,c) is the affine variety

Z_c =Spec(Zc).

It is endowed with a morphism

ϕ_c :Z_c −→Spec(P) =V/W ×V^∗/W ≃C²ⁿ.

Definition

TheCalogero-Moser space associated with the datum (W,c) is the affine variety

Z_c =Spec(Zc).

It is endowed with a morphism

ϕ_c :Z_c −→Spec(P) =V/W ×V^∗/W ≃C²ⁿ.

Theorem (Etingof-Ginzburg, 2002)

Zc is an integrally closed domain, and is a free P-module of rank |W|.

Definition

TheCalogero-Moser space associated with the datum (W,c) is the affine variety

Z_c =Spec(Zc).

It is endowed with a morphism

ϕ_c :Z_c −→Spec(P) =V/W ×V^∗/W ≃C²ⁿ.

Theorem (Etingof-Ginzburg, 2002)

Zc is an integrally closed domain, and is a free P-module of rank |W|.

Example (the case where c =0).

Definition

TheCalogero-Moser space associated with the datum (W,c) is the affine variety

Z_c =Spec(Zc).

It is endowed with a morphism

ϕ_c :Z_c −→Spec(P) =V/W ×V^∗/W ≃C²ⁿ.

Theorem (Etingof-Ginzburg, 2002)

Zc is an integrally closed domain, and is a free P-module of rank |W|.

Example (the case where c =0). Then H₀ =C[V ×V^∗]⋊W,

Definition

TheCalogero-Moser space associated with the datum (W,c) is the affine variety

Z_c =Spec(Zc).

It is endowed with a morphism

ϕ_c :Z_c −→Spec(P) =V/W ×V^∗/W ≃C²ⁿ.

Theorem (Etingof-Ginzburg, 2002)

Zc is an integrally closed domain, and is a free P-module of rank |W|.

Example (the case where c =0). Then H₀ =C[V ×V^∗]⋊W,

Z0 =C[V ×V^∗]^W,

Definition

TheCalogero-Moser space associated with the datum (W,c) is the affine variety

Z_c =Spec(Zc).

It is endowed with a morphism

ϕ_c :Z_c −→Spec(P) =V/W ×V^∗/W ≃C²ⁿ.

Theorem (Etingof-Ginzburg, 2002)

Zc is an integrally closed domain, and is a free P-module of rank |W|.

Example (the case where c =0). Then H₀ =C[V ×V^∗]⋊W,

Z0 =C[V ×V^∗]^W, Z₀ = (V ×V^∗)/W.

Two extra-structures

Two extra-structures

There is a C^×-action onH_c (i.e. a Z-grading):

Two extra-structures

There is a C^×-action onH_c (i.e. a Z-grading):

◮ deg(V) = −1

◮ deg(V^∗) =1

◮ deg(W) =0

Two extra-structures

There is a C^×-action onH_c (i.e. a Z-grading):

◮ deg(V) = −1

◮ deg(V^∗) =1

◮ deg(W) =0

So there is a C^×-action onZc and on Z_c.

Two extra-structures

There is a C^×-action onH_c (i.e. a Z-grading):

◮ deg(V) = −1

◮ deg(V^∗) =1

◮ deg(W) =0

So there is a C^×-action onZc and on Z_c. Poisson bracket:

{,}: Zc×Zc −→ Zc

(z,z^′) 7−→ limt→0[z,z^′]H_t,c

t

GL_n(F_q),q =p^?,ℓ6=p.

GL_n(F_q),q =p^?,ℓ6=p.

Steinberg (1952), Lusztig (1976):

GL_n(F_q),q =p^?,ℓ6=p.

Steinberg (1952), Lusztig (1976):

{partitions of n}←→^∼ {Unip. char. of GL_n(Fq)}

GL_n(F_q),q =p^?,ℓ6=p.

Steinberg (1952), Lusztig (1976):

{partitions of n}←→^∼ {Unip. char. of GL_n(Fq)}

λ 7−→ ρ_λ

GL_n(F_q),q =p^?,ℓ6=p.

Steinberg (1952), Lusztig (1976):

{partitions of n}←→^∼ {Unip. char. of GL_n(Fq)}

λ 7−→ ρ_λ Hypothesis: order(q mod ℓ) =d

GL_n(F_q),q =p^?,ℓ6=p.

Steinberg (1952), Lusztig (1976):

{partitions of n}←→^∼ {Unip. char. of GL_n(Fq)}

λ 7−→ ρ_λ Hypothesis: order(q mod ℓ) =d Fact (Fong-Srinivasan, 1980’s):

GL_n(F_q),q =p^?,ℓ6=p.

Steinberg (1952), Lusztig (1976):

{partitions of n}←→^∼ {Unip. char. of GL_n(Fq)}

λ 7−→ ρ_λ

Hypothesis: order(q mod ℓ) =d Fact (Fong-Srinivasan, 1980’s):

ρ_λ and ρ_µ are in the same ℓ-block m

♥d(λ) =♥d(µ)

GL_n(F_q),q =p^?,ℓ6=p.

Steinberg (1952), Lusztig (1976):

{partitions of n}←→^∼ {Unip. char. of GL_n(Fq)}

λ 7−→ ρ_λ

Hypothesis: order(q mod ℓ) =d Fact (Fong-Srinivasan, 1980’s):

ρ_λ and ρ_µ are in the same ℓ-block m

♥d(λ) =♥d(µ)

Conjecture (Brou´e-Malle-Michel, 1993)

Ifγ is a d-core and n=|γ|+dr, then there exists a Deligne-Lusztig varietyX_γ(r) for G such that:

ρ_λ |H_c^•(X_♥(r),Q_ℓ)⇐⇒♥d(λ) =γ.

End_Q

ℓGL_n(Fq) H_c^•(X_γ(r))

≃Hecke_params(G(d,1,r))

GL_n(F_q),q =p^?,ℓ6=p.

Steinberg (1952), Lusztig (1976):

{partitions of n}←→^∼ {Unip. char. of GL_n(Fq)}

λ 7−→ ρ_λ

Hypothesis: order(q mod ℓ) =d Fact (Fong-Srinivasan, 1980’s):

ρ_λ and ρ_µ are in the same ℓ-block m

♥d(λ) =♥d(µ)

Conjecture (Brou´e-Malle-Michel, 1993)

Ifγ is a d-core and n=|γ|+dr, then there exists a Deligne-Lusztig varietyX_γ(r) for G such that:

ρ_λ |H_c^•(X_♥(r),Q_ℓ)⇐⇒♥d(λ) =γ.

End_Q

ℓGL_n(Fq) H_c^•(X_γ(r))

≃Hecke_params(G(d,1,r))

Z₁(S_n), smooth, C^×-action, ζ ∈C^× Steinberg (1952), Lusztig (1976):

{partitions of n}←→^∼ {Unip. char. of GL_n(Fq)}

λ 7−→ ρ_λ

Hypothesis: order(q mod ℓ) =d Fact (Fong-Srinivasan, 1980’s):

ρ_λ and ρ_µ are in the same ℓ-block m

♥d(λ) =♥d(µ)

Conjecture (Brou´e-Malle-Michel, 1993)

Ifγ is a d-core and n=|γ|+dr, then there exists a Deligne-Lusztig varietyX_γ(r) for G such that:

ρ_λ |H_c^•(X_♥(r),Q_ℓ)⇐⇒♥d(λ) =γ.

End_Q

ℓGL_n(Fq) H_c^•(X_γ(r))

≃Hecke_params(G(d,1,r))

Z₁(S_n), smooth, C^×-action, ζ ∈C^× Gordon (2003):

{partitions of n}←→^∼ {Unip. char. of GL_n(Fq)}

λ 7−→ ρ_λ

Hypothesis: order(q mod ℓ) =d Fact (Fong-Srinivasan, 1980’s):

ρ_λ and ρ_µ are in the same ℓ-block m

♥d(λ) =♥d(µ)

Conjecture (Brou´e-Malle-Michel, 1993)

Ifγ is a d-core and n=|γ|+dr, then there exists a Deligne-Lusztig varietyX_γ(r) for G such that:

ρ_λ |H_c^•(X_♥(r),Q_ℓ)⇐⇒♥d(λ) =γ.

End_Q

ℓGL_n(Fq) H_c^•(X_γ(r))

≃Hecke_params(G(d,1,r))

Z₁(S_n), smooth, C^×-action, ζ ∈C^× Gordon (2003):

{partitions of n}←→^∼ Z₁(S_n)^C× λ 7−→ ρ_λ

Hypothesis: order(q mod ℓ) =d Fact (Fong-Srinivasan, 1980’s):

ρ_λ and ρ_µ are in the same ℓ-block m

♥d(λ) =♥d(µ)

Conjecture (Brou´e-Malle-Michel, 1993)

Ifγ is a d-core and n=|γ|+dr, then there exists a Deligne-Lusztig varietyX_γ(r) for G such that:

ρ_λ |H_c^•(X_♥(r),Q_ℓ)⇐⇒♥d(λ) =γ. End_Q

ℓGL_n(Fq) H_c^•(X_γ(r))

≃Hecke_params(G(d,1,r))

Z₁(S_n), smooth, C^×-action, ζ ∈C^× Gordon (2003):

{partitions of n}←→^∼ Z₁(S_n)^C× λ 7−→ zλ

Hypothesis: order(q mod ℓ) =d Fact (Fong-Srinivasan, 1980’s):

ρ_λ and ρ_µ are in the same ℓ-block m

♥d(λ) =♥d(µ)

Conjecture (Brou´e-Malle-Michel, 1993)

Ifγ is a d-core and n=|γ|+dr, then there exists a Deligne-Lusztig varietyX_γ(r) for G such that:

ρ_λ |H_c^•(X_♥(r),Q_ℓ)⇐⇒♥d(λ) =γ. End_Q

ℓGL_n(Fq) H_c^•(X_γ(r))

≃Hecke_params(G(d,1,r))

Z₁(S_n), smooth, C^×-action, ζ ∈C^× Gordon (2003):

{partitions of n}←→^∼ Z₁(S_n)^C× λ 7−→ zλ

Hypothesis: order(ζ) =d Fact (Fong-Srinivasan, 1980’s):

ρ_λ and ρ_µ are in the same ℓ-block m

♥d(λ) =♥d(µ)

Conjecture (Brou´e-Malle-Michel, 1993)

Ifγ is a d-core and n=|γ|+dr, then there exists a Deligne-Lusztig varietyX_γ(r) for G such that:

ρ_λ |H_c^•(X_♥(r),Q_ℓ)⇐⇒♥d(λ) =γ. End_Q

ℓGL_n(Fq) H_c^•(X_γ(r))

≃Hecke_params(G(d,1,r))

Z₁(S_n), smooth, C^×-action, ζ ∈C^× Gordon (2003):

{partitions of n}←→^∼ Z₁(S_n)^C× λ 7−→ zλ

Hypothesis: order(ζ) =d Fact (Haiman, 2000):

ρ_λ and ρ_µ are in the same ℓ-block m

♥d(λ) =♥d(µ)

Conjecture (Brou´e-Malle-Michel, 1993)

Ifγ is a d-core and n=|γ|+dr, then there exists a Deligne-Lusztig varietyX_γ(r) for G such that:

ρ_λ |H_c^•(X_♥(r),Q_ℓ)⇐⇒♥d(λ) =γ. End_Q

ℓGL_n(Fq) H_c^•(X_γ(r))

≃Hecke_params(G(d,1,r))

Z₁(S_n), smooth, C^×-action, ζ ∈C^× Gordon (2003):

{partitions of n}←→^∼ Z₁(S_n)^C× λ 7−→ zλ

Hypothesis: order(ζ) =d Fact (Haiman, 2000):

zλ and zµ are in the same irr. comp. of Z₁(S_n)^ζ m

♥d(λ) =♥d(µ)

Conjecture (Brou´e-Malle-Michel, 1993)

Ifγ is a d-core and n=|γ|+dr, then there exists a Deligne-Lusztig varietyX_γ(r) for G such that:

ρ_λ |H_c^•(X_♥(r),Q_ℓ)⇐⇒♥d(λ) =γ. End_Q

ℓGL_n(F_q) H_c^•(X_γ(r))

≃Hecke_params(G(d,1,r))

Z₁(S_n), smooth, C^×-action, ζ ∈C^× Gordon (2003):

{partitions of n}←→^∼ Z₁(S_n)^C× λ 7−→ zλ

Hypothesis: order(ζ) =d Fact (Haiman, 2000):

zλ and zµ are in the same irr. comp. of Z₁(S_n)^ζ m

♥d(λ) =♥d(µ)

Theorem (Haiman, ∼ 2000)

Ifγ is a d-core and n=|γ|+dr, then there exists an irreducible componentZ₁(S_n)^ζ_γ of Z₁(S_n)^ζ such that:

zλ ∈Z₁(S_n)^ζ_γ ⇐⇒♥d(λ) =γ.

Z₁(S_n)^ζ_γ is diffeo. (conj. isom.) to Z_params(G(d,1,r)).

Symplectic singularities

Symplectic singularities

Theorem (Brown-Gordon, 2003)

(a) The symplectic leaves are obtained as follows

Symplectic singularities

Theorem (Brown-Gordon, 2003)

(a) The symplectic leaves are obtained as follows

◮ (Z_c)_smooth is a symplectic leaf;

Symplectic singularities

Theorem (Brown-Gordon, 2003)

(a) The symplectic leaves are obtained as follows

◮ (Z_c)_smooth is a symplectic leaf;

◮ ((Z_c)_sing)

Symplectic singularities

Theorem (Brown-Gordon, 2003)

(a) The symplectic leaves are obtained as follows

◮ (Z_c)_smooth is a symplectic leaf;

◮ ((Z_c)_sing)_smooth

Symplectic singularities

Theorem (Brown-Gordon, 2003)

(a) The symplectic leaves are obtained as follows

◮ (Z_c)_smooth is a symplectic leaf;

◮ ((Z_c)_sing)_smooth is a symplectic leaf;

Symplectic singularities

Theorem (Brown-Gordon, 2003)

(a) The symplectic leaves are obtained as follows

◮ (Z_c)_smooth is a symplectic leaf;

◮ ((Z_c)_sing)_smooth is a symplectic leaf;

◮ (((Z_c)_sing)_sing)_smooth is a symplectic leaf;

◮ . . .

Symplectic singularities

Theorem (Brown-Gordon, 2003)

(a) The symplectic leaves are obtained as follows

◮ (Z_c)_smooth is a symplectic leaf;

◮ ((Z_c)_sing)_smooth is a symplectic leaf;

◮ (((Z_c)_sing)_sing)_smooth is a symplectic leaf;

◮ . . .

(b) Z_c is a symplectic singularity (as defined by Beauville).

Symplectic resolutions

Symplectic resolutions

Theorem (Ginzburg-Kaledin 2004, Namikawa 2007)

Z₀ = (V ×V^∗)/W admits a symplectic resolution if and only if there exists c ∈C such thatZ_c is smooth.

Symplectic resolutions

Theorem (Ginzburg-Kaledin 2004, Namikawa 2007)

Z₀ = (V ×V^∗)/W admits a symplectic resolution if and only if there exists c ∈C such thatZ_c is smooth.

Theorem (Brown-Gordon, 2003)

Z_c is smooth if and only if all the simple H_c-modules have dimension |W|.

Symplectic resolutions

Theorem (Ginzburg-Kaledin 2004, Namikawa 2007)

Z₀ = (V ×V^∗)/W admits a symplectic resolution if and only if there exists c ∈C such thatZ_c is smooth.

Theorem (Brown-Gordon, 2003)

Z_c is smooth if and only if all the simple H_c-modules have dimension |W|.

Corollary (G.-K., B.-G., Bellamy 2008)

Assume that W is irreducible.

ThenZ₀ = (V ×V^∗)/W admits a symplectic resolution if and only if W =G(d,1,n) = Sn⋉(µ_d)ⁿ ⊂GL_n(C) or W =G4 ⊂GL₂(C).

Cohomology

Cohomology

Theorem (Ginzburg-Kaledin, 2004)

Assume thatZ˜₀ →Z₀ is a symplectic resolution.

Cohomology

Theorem (Ginzburg-Kaledin, 2004)

Assume thatZ˜₀ →Z₀ is a symplectic resolution.

(1) H²ⁱ⁺¹(Z˜₀,C) =0

Cohomology

Theorem (Ginzburg-Kaledin, 2004)

Assume thatZ˜₀ →Z₀ is a symplectic resolution.

(1) H²ⁱ⁺¹(Z˜₀) = 0;

Cohomology

Theorem (Ginzburg-Kaledin, 2004)

Assume thatZ˜₀ →Z₀ is a symplectic resolution.

(1) H²ⁱ⁺¹(Z˜₀) = 0;

(2) H^2•(Z˜₀)≃gr_F(Z(CW)).

Cohomology

Theorem (Ginzburg-Kaledin, 2004)

Assume thatZ˜₀ →Z₀ is a symplectic resolution.

(1) H²ⁱ⁺¹(Z˜₀) = 0;

(2) H^2•(Z˜₀)≃gr_F(Z(CW)).

Conjecture EC (Ginzburg-Kaledin, 2004)

Assume thatZ˜₀ →Z₀ is a symplectic resolution.

Cohomology

Theorem (Ginzburg-Kaledin, 2004)

Assume thatZ˜₀ →Z₀ is a symplectic resolution.

(1) H²ⁱ⁺¹(Z˜₀) = 0;

(2) H^2•(Z˜₀)≃gr_F(Z(CW)).

Conjecture EC (Ginzburg-Kaledin, 2004)

Assume thatZ˜₀ →Z₀ is a symplectic resolution.

(EC1) H_C²ⁱ⁺¹× (Z˜₀) =0;

Cohomology

Theorem (Ginzburg-Kaledin, 2004)

Assume thatZ˜₀ →Z₀ is a symplectic resolution.

(1) H²ⁱ⁺¹(Z˜₀) = 0;

(2) H^2•(Z˜₀)≃gr_F(Z(CW)).

Conjecture EC (Ginzburg-Kaledin, 2004)

Assume thatZ˜₀ →Z₀ is a symplectic resolution.

(EC1) H_C²ⁱ⁺¹× (Z˜₀) =0;

(EC2) H_C^2•×(Z˜₀)≃ReesF(Z(CW)).

Cohomology

Theorem (Ginzburg-Kaledin, 2004)

Assume thatZ˜₀ →Z₀ is a symplectic resolution.

(1) H²ⁱ⁺¹(Z˜₀) = 0;

(2) H^2•(Z˜₀)≃gr_F(Z(CW)).

Conjecture EC (Ginzburg-Kaledin, 2004)

Assume thatZ˜₀ →Z₀ is a symplectic resolution.

(EC1) H_C²ⁱ⁺¹× (Z˜₀) =0;

(EC2) H_C^2•×(Z˜₀)≃ReesF(Z(CW)).

Theorem (Vasserot, 2001)

(EC) holds if W = S_n (Z˜₀ =Hilb_n(C²)).

Theorem (Vasserot, 2001)

(EC) holds if W = S_n (Z˜₀ =Hilb_n(C²)).

Other cases. W =W(B2) or G4 (Shan-B. 2016).

Theorem (Vasserot, 2001)

(EC) holds if W = S_n (Z˜₀ =Hilb_n(C²)).

Other cases. W =W(B2) or G4 (Shan-B. 2016).

Question. What about the general case?

Cohomology

Theorem (Ginzburg-Kaledin, 2004)

Assume thatZ˜₀ →Z₀ is a symplectic resolution.

(1) H²ⁱ⁺¹(Z˜₀) = 0;

(2) H^2•(Z˜₀)≃gr_F(Z(CW)).

Conjecture EC (Ginzburg-Kaledin, 2004)

Assume thatZ˜₀ →Z₀ is a symplectic resolution.

(EC1) H_C²ⁱ⁺¹× (Z˜₀) =0;

(EC2) H_C^2•×(Z˜₀)≃ReesF(Z(CW)).

Cohomology (smooth case)

Theorem (Ginzburg-Kaledin, 2004)

Assume thatZ_c is smooth.

(1) H²ⁱ⁺¹(Z_c) =0;

(2) H^2•(Z_c)≃gr_F(Z(CW)).

Conjecture EC (Ginzburg-Kaledin, 2004)

Assume thatZ_c is smooth.

(EC1) H_C²ⁱ⁺¹× (Z_c) =0;

(EC2) H_C^2•×(Z_c)≃ReesF(Z(CW)).

Cohomology (general case)

Conjecture C (Rouquier-B.)

(C1) H²ⁱ⁺¹(Z_c) =0;

(C2) H^2•(Z_c)≃gr_F(ImΩ_c).

Conjecture EC (Rouquier-B.)

(EC1) H_C^2i+1× (Z_c) =0;

(EC2) H_C^2•×(Z_c)≃ReesF(ImΩ_c).

Cohomology (general case)

Conjecture C (Rouquier-B.)

(C1) H²ⁱ⁺¹(Z_c) =0;

(C2) H^2•(Z_c)≃gr_F(ImΩ_c).

Conjecture EC (Rouquier-B.)

(EC1) H_C^2i+1× (Z_c) =0;

(EC2) H_C^2•×(Z_c)≃ReesF(ImΩ_c).

Example (B.). (C) and (EC) are true if dimC(V) =1.

Example of a symplectic singularity

Example of a symplectic singularity

W =W(B2) =hs,t|s² =t² = (st)⁴ =1i

Example of a symplectic singularity

W =W(B2) =hs,t|s² =t² = (st)⁴ =1i Leta =cs and b=ct.

Example of a symplectic singularity

W =W(B2) =hs,t|s² =t² = (st)⁴ =1i Leta =cs and b=ct.

Minimal presentation ofZ_c: 8 generators, 9 equations

Example of a symplectic singularity

W =W(B2) =hs,t|s² =t² = (st)⁴ =1i Leta =cs and b=ct.

Minimal presentation ofZ_c: 8 generators, 9 equations Z_c ֒−→C⁸, dimZ_c =4

Example of a symplectic singularity

W =W(B2) =hs,t|s² =t² = (st)⁴ =1i Leta =cs and b=ct.

Minimal presentation ofZ_c: 8 generators, 9 equations Z_c ֒−→C⁸, dimZ_c =4

Easy fact: Z_c is smooth if and only ifab(a²−b²)6=0.

Example of a symplectic singularity

W =W(B2) =hs,t|s² =t² = (st)⁴ =1i Leta =cs and b=ct.

Minimal presentation ofZ_c: 8 generators, 9 equations Z_c ֒−→C⁸, dimZ_c =4

Easy fact: Z_c is smooth if and only ifab(a²−b²)6=0.

The interesting case: assume from now on that a=b6=0.

Example of a symplectic singularity

W =W(B2) =hs,t|s² =t² = (st)⁴ =1i Leta =cs and b=ct.

Minimal presentation ofZ_c: 8 generators, 9 equations Z_c ֒−→C⁸, dimZ_c =4

Easy fact: Z_c is smooth if and only ifab(a²−b²)6=0.

The interesting case: assume from now on that a=b6=0.

⇒Z_c has only one singular point, named 0

Example of a symplectic singularity

W =W(B2) =hs,t|s² =t² = (st)⁴ =1i Leta =cs and b=ct.

Minimal presentation ofZ_c: 8 generators, 9 equations Z_c ֒−→C⁸, dimZ_c =4

Easy fact: Z_c is smooth if and only ifab(a²−b²)6=0.

The interesting case: assume from now on that a=b6=0.

⇒Z_c has only one singular point, named 0 Easy fact: dimCT0(Z_c) =8.

Example of a symplectic singularity

W =W(B2) =hs,t|s² =t² = (st)⁴ =1i Leta =cs and b=ct.

Minimal presentation ofZ_c: 8 generators, 9 equations Z_c ֒−→C⁸, dimZ_c =4

Easy fact: Z_c is smooth if and only ifab(a²−b²)6=0.

The interesting case: assume from now on that a=b6=0.

⇒Z_c has only one singular point, named 0 Easy fact: dimCT0(Z_c) =8.

Letm₀ denote the maximal ideal of Zc corresponding to 0.

Example of a symplectic singularity

W =W(B2) =hs,t|s² =t² = (st)⁴ =1i Leta =cs and b=ct.

Minimal presentation ofZ_c: 8 generators, 9 equations Z_c ֒−→C⁸, dimZ_c =4

Easy fact: Z_c is smooth if and only ifab(a²−b²)6=0.

The interesting case: assume from now on that a=b6=0.

⇒Z_c has only one singular point, named 0 Easy fact: dimCT0(Z_c) =8.

Letm₀ denote the maximal ideal of Zc corresponding to 0. Then {m₀,m₀}⊂m₀ because{0}is a symplectic leaf.

Example of a symplectic singularity

W =W(B2) =hs,t|s² =t² = (st)⁴ =1i Leta =cs and b=ct.

Minimal presentation ofZ_c: 8 generators, 9 equations Z_c ֒−→C⁸, dimZ_c =4

Easy fact: Z_c is smooth if and only ifab(a²−b²)6=0.

The interesting case: assume from now on that a=b6=0.

⇒Z_c has only one singular point, named 0 Easy fact: dimCT0(Z_c) =8.

Letm₀ denote the maximal ideal of Zc corresponding to 0. Then {m₀,m₀}⊂m₀ because{0}is a symplectic leaf.

⇒T₀(Z_c)^∗ = m₀/m²₀ inherits from {,} a structure of Lie algebra!

Example (continued)

Example (continued)

W =W(B2) =hs,t|s² =t² = (st)⁴ =1i a=cs =ct,

Z_c ֒−→C⁸ =T₀(Z_c)

⇒T0(Z_c)^∗ is a Lie algebra for {,}

Example (continued)

W =W(B2) =hs,t|s² =t² = (st)⁴ =1i a=cs =ct,

Z_c ֒−→C⁸ =T₀(Z_c)

⇒T0(Z_c)^∗ is a Lie algebra for {,}

Computation (thanks toMAGMA): T0(Z_c)^∗ ≃sl₃(C) (!)

Example (continued)

W =W(B2) =hs,t|s² =t² = (st)⁴ =1i a=cs =ct,

Z_c ֒−→C⁸ =T₀(Z_c)

⇒T0(Z_c)^∗ is a Lie algebra for {,}

Computation (thanks toMAGMA): T0(Z_c)^∗ ≃sl₃(C) (!) So Z_c ֒−→sl₃(C)^∗ ≃sl₃(C) (trace form).

Example (continued)

W =W(B2) =hs,t|s² =t² = (st)⁴ =1i a=cs =ct,

Z_c ֒−→C⁸ =T₀(Z_c)

⇒T0(Z_c)^∗ is a Lie algebra for {,}

Computation (thanks toMAGMA): T0(Z_c)^∗ ≃sl₃(C) (!) So Z_c ֒−→sl₃(C)^∗ ≃sl₃(C) (trace form).

Computation (MAGMA):

TC0(Z_c) ={M ∈sl₃(C) | M² =0}

Example (continued)

W =W(B2) =hs,t|s² =t² = (st)⁴ =1i a=cs =ct,

Z_c ֒−→C⁸ =T₀(Z_c)

⇒T0(Z_c)^∗ is a Lie algebra for {,}

Computation (thanks toMAGMA): T0(Z_c)^∗ ≃sl₃(C) (!) So Z_c ֒−→sl₃(C)^∗ ≃sl₃(C) (trace form).

Computation (MAGMA):

TC0(Z_c) ={M ∈sl₃(C) | M² =0}=Omin.

Example (continued)

W =W(B2) =hs,t|s² =t² = (st)⁴ =1i a=cs =ct,

Z_c ֒−→C⁸ =T₀(Z_c)

⇒T0(Z_c)^∗ is a Lie algebra for {,}

Computation (thanks toMAGMA): T0(Z_c)^∗ ≃sl₃(C) (!) So Z_c ֒−→sl₃(C)^∗ ≃sl₃(C) (trace form).

Computation (MAGMA):

TC0(Z_c) ={M ∈sl₃(C) | M² =0}=Omin. So PTC0(Z_c) =Omin/C^× is smooth

Example (continued)

W =W(B2) =hs,t|s² =t² = (st)⁴ =1i a=cs =ct,

Z_c ֒−→C⁸ =T₀(Z_c)

⇒T0(Z_c)^∗ is a Lie algebra for {,}

Computation (thanks toMAGMA): T0(Z_c)^∗ ≃sl₃(C) (!) So Z_c ֒−→sl₃(C)^∗ ≃sl₃(C) (trace form).

Computation (MAGMA):

TC0(Z_c) ={M ∈sl₃(C) | M² =0}=Omin.

So PTC0(Z_c) =Omin/C^× is smooth so Beauville classification theorem applies:

Example (continued)

W =W(B2) =hs,t|s² =t² = (st)⁴ =1i a=cs =ct,

Z_c ֒−→C⁸ =T₀(Z_c)

⇒T0(Z_c)^∗ is a Lie algebra for {,}

Computation (thanks toMAGMA): T0(Z_c)^∗ ≃sl₃(C) (!) So Z_c ֒−→sl₃(C)^∗ ≃sl₃(C) (trace form).

Computation (MAGMA):

TC0(Z_c) ={M ∈sl₃(C) | M² =0}=Omin.

So PTC0(Z_c) =Omin/C^× is smooth so Beauville classification theorem applies:

Conclusion (Juteau-B.)

The symplectic singularities(Z_c,0)and (O_min,0)are equivalent.

Example (continued)

W =W(B2) =hs,t|s² =t² = (st)⁴ =1i a=cs =ct,

Z_c ֒−→C⁸ =T₀(Z_c)

⇒T0(Z_c)^∗ is a Lie algebra for {,}

Computation (thanks toMAGMA): T0(Z_c)^∗ ≃sl₃(C) (!) So Z_c ֒−→sl₃(C)^∗ ≃sl₃(C) (trace form).

Computation (MAGMA):

TC0(Z_c) ={M ∈sl₃(C) | M² =0}=Omin.

So PTC0(Z_c) =Omin/C^× is smooth so Beauville classification theorem applies:

Conclusion (Juteau-B.)

The symplectic singularities(Z_c,0)and (O_min,0)are equivalent. In particular,Z_c is not rationally smooth.