Geometry of Calogero-Moser spaces

Cedric Bonnafe

CNRS (UMR 5149) - Universite de Montpellier

Poitiers, November 2016

Set-up

Set-up

dimC V = n <∞

W < GLC(V ), |W | <∞.

Ref(W ) = s ∈ W | codimCVs = 1

Set-up

dimC V = n <∞

W < GLC(V ), |W | <∞.

Ref(W ) = s ∈ W | codimCVs = 1

Hypothesis. W = 〈Ref(W )〉

Set-up

dimC V = n <∞

W < GLC(V ), |W | <∞.

Ref(W ) = s ∈ W | codimCVs = 1

Hypothesis. W = 〈Ref(W )〉(i.e. V /W ≃ Cn)

Set-up

dimC V = n <∞

W < GLC(V ), |W | <∞.

Ref(W ) = s ∈ W | codimCVs = 1

Hypothesis. W = 〈Ref(W )〉(i.e. V /W ≃ Cn)

C = c : Ref(W )/∼ −→ C

We fix c ∈ C

Cherednik algebra at t = 0

Cherednik algebra at t = 0

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

∀y ∈ V , ∀x ∈ V ∗, [y , x ] =∑

s∈Ref(W )

cs〈y , s(x) − x〉s

Cherednik algebra at t = 0

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

∀y ∈ V , ∀x ∈ V ∗, [y , x ] =∑

s∈Ref(W )

cs〈y , s(x) − x〉s

Cherednik algebra at t = 0

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

∀y ∈ V , ∀x ∈ V ∗, [y , x ] =∑

s∈Ref(W )

cs〈y , s(x) − x〉s

LetZc = Z(Hc)

Cherednik algebra at t = 0

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

∀y ∈ V , ∀x ∈ V ∗, [y , x ] =∑

s∈Ref(W )

cs〈y , s(x) − x〉s

LetZc = Z(Hc)

Easy fact.

C[V ]W , C[V ∗]W ⊂ Zc .

LetP = C[V ]W ⊗ C[V ∗]W ⊂ Zc

DefinitionThe Calogero-Moser space associated with the datum (W , c) is theaffine variety

Zc = Spec(Zc).

DefinitionThe Calogero-Moser space associated with the datum (W , c) is theaffine variety

Zc = Spec(Zc).

It is endowed with a morphism

ϕc : Zc −→ Spec(P) = V /W × V ∗/W

DefinitionThe Calogero-Moser space associated with the datum (W , c) is theaffine variety

Zc = Spec(Zc).

It is endowed with a morphism

ϕc : Zc −→ Spec(P) = V /W × V ∗/W ≃ C2n.

DefinitionThe Calogero-Moser space associated with the datum (W , c) is theaffine variety

Zc = Spec(Zc).

It is endowed with a morphism

ϕc : Zc −→ 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 |.

DefinitionThe Calogero-Moser space associated with the datum (W , c) is theaffine variety

Zc = Spec(Zc).

It is endowed with a morphism

ϕc : Zc −→ 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).

DefinitionThe Calogero-Moser space associated with the datum (W , c) is theaffine variety

Zc = Spec(Zc).

It is endowed with a morphism

ϕc : Zc −→ 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). ThenH0 = C[V × V ∗]⋊W ,

DefinitionThe Calogero-Moser space associated with the datum (W , c) is theaffine variety

Zc = Spec(Zc).

It is endowed with a morphism

ϕc : Zc −→ 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). ThenH0 = C[V × V ∗]⋊W ,

Z0 = C[V × V ∗]W ,

DefinitionThe Calogero-Moser space associated with the datum (W , c) is theaffine variety

Zc = Spec(Zc).

It is endowed with a morphism

ϕc : Zc −→ 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). ThenH0 = C[V × V ∗]⋊W ,

Z0 = C[V × V ∗]W ,

Z0 = (V × V ∗)/W .

Two extra-structures

Two extra-structures

There is a C×-action on Hc (i.e. a Z-grading):

Two extra-structures

There is a C×-action on Hc (i.e. a Z-grading): deg(V ) = −1 deg(V ∗) = 1 deg(W ) = 0

Two extra-structures

There is a C×-action on Hc (i.e. a Z-grading): deg(V ) = −1 deg(V ∗) = 1 deg(W ) = 0

So there is a C×-action on Zc and on Zc .

Two extra-structures

There is a C×-action on Hc (i.e. a Z-grading): deg(V ) = −1 deg(V ∗) = 1 deg(W ) = 0

So there is a C×-action on Zc and on Zc .

Poisson bracket:

, : Zc × Zc −→ Zc

(z , z ′) 7−→ limt→0[z , z ′]Ht,c

t

GLn(Fq), q = p?, ℓ 6= p.

GLn(Fq), q = p?, ℓ 6= p.

Steinberg (1952), Lusztig (1976):

GLn(Fq), q = p?, ℓ 6= p.

Steinberg (1952), Lusztig (1976):

partitions of n∼←→ Unip. char. of GLn(Fq)

GLn(Fq), q = p?, ℓ 6= p.

Steinberg (1952), Lusztig (1976):

partitions of n∼←→ Unip. char. of GLn(Fq)

λ 7−→ ρλ

GLn(Fq), q = p?, ℓ 6= p.

Steinberg (1952), Lusztig (1976):

partitions of n∼←→ Unip. char. of GLn(Fq)

λ 7−→ ρλ

Hypothesis: order(q mod ℓ) = d

GLn(Fq), q = p?, ℓ 6= p.

Steinberg (1952), Lusztig (1976):

partitions of n∼←→ Unip. char. of GLn(Fq)

λ 7−→ ρλ

Hypothesis: order(q mod ℓ) = d

Fact (Fong-Srinivasan, 1980’s):

GLn(Fq), q = p?, ℓ 6= p.

Steinberg (1952), Lusztig (1976):

partitions of n∼←→ Unip. char. of GLn(Fq)

λ 7−→ ρλ

Hypothesis: order(q mod ℓ) = d

Fact (Fong-Srinivasan, 1980’s):ρλ and ρµ are in the same ℓ-block

m♥d(λ) = ♥d(µ)

GLn(Fq), q = p?, ℓ 6= p.

Steinberg (1952), Lusztig (1976):

partitions of n∼←→ Unip. char. of GLn(Fq)

λ 7−→ ρλ

Hypothesis: order(q mod ℓ) = d

Fact (Fong-Srinivasan, 1980’s):ρλ and ρµ are in the same ℓ-block

m♥d(λ) = ♥d(µ)

Conjecture (Broue-Malle-Michel, 1993)

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

ρλ | H•c (X♥(r),Qℓ)⇐⇒ ♥d(λ) = γ.

EndQℓGLn(Fq)

(

H•c (Xγ(r))

)

≃ Heckeparams(G (d , 1, r))

GLn(Fq), q = p?, ℓ 6= p.

Steinberg (1952), Lusztig (1976):

partitions of n∼←→ Unip. char. of GLn(Fq)

λ 7−→ ρλ

Hypothesis: order(q mod ℓ) = d

Fact (Fong-Srinivasan, 1980’s):ρλ and ρµ are in the same ℓ-block

m♥d(λ) = ♥d(µ)

Conjecture (Broue-Malle-Michel, 1993)

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

ρλ | H•c (X♥(r),Qℓ)⇐⇒ ♥d(λ) = γ.

EndQℓGLn(Fq)

(

H•c (Xγ(r))

)

≃ Heckeparams(G (d , 1, r))

Z1(Sn), smooth, C×-action, ζ ∈ C×

Steinberg (1952), Lusztig (1976):

partitions of n∼←→ Unip. char. of GLn(Fq)

λ 7−→ ρλ

Hypothesis: order(q mod ℓ) = d

Fact (Fong-Srinivasan, 1980’s):ρλ and ρµ are in the same ℓ-block

m♥d(λ) = ♥d(µ)

Conjecture (Broue-Malle-Michel, 1993)

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

ρλ | H•c (X♥(r),Qℓ)⇐⇒ ♥d(λ) = γ.

EndQℓGLn(Fq)

(

H•c (Xγ(r))

)

≃ Heckeparams(G (d , 1, r))

Z1(Sn), smooth, C×-action, ζ ∈ C×

Gordon (2003):

partitions of n∼←→ Unip. char. of GLn(Fq)

λ 7−→ ρλ

Hypothesis: order(q mod ℓ) = d

Fact (Fong-Srinivasan, 1980’s):ρλ and ρµ are in the same ℓ-block

m♥d(λ) = ♥d(µ)

Conjecture (Broue-Malle-Michel, 1993)

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

ρλ | H•c (X♥(r),Qℓ)⇐⇒ ♥d(λ) = γ.

EndQℓGLn(Fq)

(

H•c (Xγ(r))

)

≃ Heckeparams(G (d , 1, r))

Z1(Sn), smooth, C×-action, ζ ∈ C×

Gordon (2003):partitions of n

∼←→ Z1(Sn)C×

λ 7−→ ρλ

Hypothesis: order(q mod ℓ) = d

Fact (Fong-Srinivasan, 1980’s):ρλ and ρµ are in the same ℓ-block

m♥d(λ) = ♥d(µ)

Conjecture (Broue-Malle-Michel, 1993)

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

ρλ | H•c (X♥(r),Qℓ)⇐⇒ ♥d(λ) = γ.

EndQℓGLn(Fq)

(

H•c (Xγ(r))

)

≃ Heckeparams(G (d , 1, r))

Z1(Sn), smooth, C×-action, ζ ∈ C×

Gordon (2003):partitions of n

∼←→ Z1(Sn)C×

λ 7−→ zλ

Hypothesis: order(q mod ℓ) = d

Fact (Fong-Srinivasan, 1980’s):ρλ and ρµ are in the same ℓ-block

m♥d(λ) = ♥d(µ)

Conjecture (Broue-Malle-Michel, 1993)

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

ρλ | H•c (X♥(r),Qℓ)⇐⇒ ♥d(λ) = γ.

EndQℓGLn(Fq)

(

H•c (Xγ(r))

)

≃ Heckeparams(G (d , 1, r))

Z1(Sn), smooth, C×-action, ζ ∈ C×

Gordon (2003):partitions of n

∼←→ Z1(Sn)C×

λ 7−→ zλ

Hypothesis: order(ζ) = d

Fact (Fong-Srinivasan, 1980’s):ρλ and ρµ are in the same ℓ-block

m♥d(λ) = ♥d(µ)

Conjecture (Broue-Malle-Michel, 1993)

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

ρλ | H•c (X♥(r),Qℓ)⇐⇒ ♥d(λ) = γ.

EndQℓGLn(Fq)

(

H•c (Xγ(r))

)

≃ Heckeparams(G (d , 1, r))

Z1(Sn), smooth, C×-action, ζ ∈ C×

Gordon (2003):partitions of n

∼←→ Z1(Sn)C×

λ 7−→ zλ

Hypothesis: order(ζ) = d

Fact (Haiman, 2000):ρλ and ρµ are in the same ℓ-block

m♥d(λ) = ♥d(µ)

Conjecture (Broue-Malle-Michel, 1993)

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

ρλ | H•c (X♥(r),Qℓ)⇐⇒ ♥d(λ) = γ.

EndQℓGLn(Fq)

(

H•c (Xγ(r))

)

≃ Heckeparams(G (d , 1, r))

Z1(Sn), smooth, C×-action, ζ ∈ C×

Gordon (2003):partitions of n

∼←→ Z1(Sn)C×

λ 7−→ zλ

Hypothesis: order(ζ) = d

Fact (Haiman, 2000):zλ and zµ are in the same irr. comp. of Z1(Sn)

ζ

m♥d(λ) = ♥d(µ)

Conjecture (Broue-Malle-Michel, 1993)

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

ρλ | H•c (X♥(r),Qℓ)⇐⇒ ♥d(λ) = γ.

EndQℓGLn(Fq)

(

H•c (Xγ(r))

)

≃ Heckeparams(G (d , 1, r))

Z1(Sn), smooth, C×-action, ζ ∈ C×

Gordon (2003):partitions of n

∼←→ Z1(Sn)C×

λ 7−→ zλ

Hypothesis: order(ζ) = d

Fact (Haiman, 2000):zλ and zµ are in the same irr. comp. of Z1(Sn)

ζ

m♥d(λ) = ♥d(µ)

Theorem (Haiman, ∼ 2000)

If γ is a d -core and n = |γ|+ dr , then there exists an irreduciblecomponent Z1(Sn)

ζγ of Z1(Sn)

ζ such that:

zλ ∈ Z1(Sn)ζγ ⇐⇒ ♥d(λ) = γ.

Z1(Sn)ζγ is diffeo. (conj. isom.) to Zparams(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 (Zc)smooth is a symplectic leaf;

Symplectic singularities

Theorem (Brown-Gordon, 2003)

(a) The symplectic leaves are obtained as follows (Zc)smooth is a symplectic leaf; ((Zc)sing)

Symplectic singularities

Theorem (Brown-Gordon, 2003)

(a) The symplectic leaves are obtained as follows (Zc)smooth is a symplectic leaf; ((Zc)sing)smooth

Symplectic singularities

Theorem (Brown-Gordon, 2003)

(a) The symplectic leaves are obtained as follows (Zc)smooth is a symplectic leaf; ((Zc)sing)smooth is a symplectic leaf;

Symplectic singularities

Theorem (Brown-Gordon, 2003)

(a) The symplectic leaves are obtained as follows (Zc)smooth is a symplectic leaf; ((Zc)sing)smooth is a symplectic leaf; (((Zc)sing)sing)smooth is a symplectic leaf; . . .

Symplectic singularities

Theorem (Brown-Gordon, 2003)

(a) The symplectic leaves are obtained as follows (Zc)smooth is a symplectic leaf; ((Zc)sing)smooth is a symplectic leaf; (((Zc)sing)sing)smooth is a symplectic leaf; . . .

(b) Zc 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 ifthere exists c ∈ C such that Zc is smooth.

Symplectic resolutions

Theorem (Ginzburg-Kaledin 2004, Namikawa 2007)

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

Theorem (Brown-Gordon, 2003)

Zc is smooth if and only if all the simple Hc-modules havedimension |W |.

Symplectic resolutions

Theorem (Ginzburg-Kaledin 2004, Namikawa 2007)

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

Theorem (Brown-Gordon, 2003)

Zc is smooth if and only if all the simple Hc-modules havedimension |W |.

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

Assume that W is irreducible.Then Z0 = (V × V ∗)/W admits a symplectic resolution if and onlyif W = G (d , 1, n) = Sn ⋉ (µd)

n ⊂ GLn(C) or W = G4 ⊂ GL2(C).

Cohomology

Cohomology

Theorem (Ginzburg-Kaledin, 2004)

Assume that Z0 → Z0 is a symplectic resolution.

Cohomology

Theorem (Ginzburg-Kaledin, 2004)

Assume that Z0 → Z0 is a symplectic resolution.

(1) H2i+1(Z0,C) = 0

Cohomology

Theorem (Ginzburg-Kaledin, 2004)

Assume that Z0 → Z0 is a symplectic resolution.

(1) H2i+1(Z0) = 0;

Cohomology

Theorem (Ginzburg-Kaledin, 2004)

Assume that Z0 → Z0 is a symplectic resolution.

(1) H2i+1(Z0) = 0;

(2) H2•(Z0) ≃ grF(Z(CW )).

Cohomology

Theorem (Ginzburg-Kaledin, 2004)

Assume that Z0 → Z0 is a symplectic resolution.

(1) H2i+1(Z0) = 0;

(2) H2•(Z0) ≃ grF(Z(CW )).

Conjecture EC (Ginzburg-Kaledin, 2004)

Assume that Z0 → Z0 is a symplectic resolution.

Cohomology

Theorem (Ginzburg-Kaledin, 2004)

Assume that Z0 → Z0 is a symplectic resolution.

(1) H2i+1(Z0) = 0;

(2) H2•(Z0) ≃ grF(Z(CW )).

Conjecture EC (Ginzburg-Kaledin, 2004)

Assume that Z0 → Z0 is a symplectic resolution.

(EC1) H2i+1C× (Z0) = 0;

Cohomology

Theorem (Ginzburg-Kaledin, 2004)

Assume that Z0 → Z0 is a symplectic resolution.

(1) H2i+1(Z0) = 0;

(2) H2•(Z0) ≃ grF(Z(CW )).

Conjecture EC (Ginzburg-Kaledin, 2004)

Assume that Z0 → Z0 is a symplectic resolution.

(EC1) H2i+1C× (Z0) = 0;

(EC2) H2•C×(Z0) ≃ ReesF(Z(CW )).

Cohomology

Theorem (Ginzburg-Kaledin, 2004)

Assume that Z0 → Z0 is a symplectic resolution.

(1) H2i+1(Z0) = 0;

(2) H2•(Z0) ≃ grF(Z(CW )).

Conjecture EC (Ginzburg-Kaledin, 2004)

Assume that Z0 → Z0 is a symplectic resolution.

(EC1) H2i+1C× (Z0) = 0;

(EC2) H2•C×(Z0) ≃ ReesF(Z(CW )).

Theorem (Vasserot, 2001)

(EC) holds if W = Sn (Z0 = Hilbn(C2)).

Theorem (Vasserot, 2001)

(EC) holds if W = Sn (Z0 = Hilbn(C2)).

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

Theorem (Vasserot, 2001)

(EC) holds if W = Sn (Z0 = Hilbn(C2)).

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

Question. What about the general case?

Cohomology

Theorem (Ginzburg-Kaledin, 2004)

Assume that Z0 → Z0 is a symplectic resolution.

(1) H2i+1(Z0) = 0;

(2) H2•(Z0) ≃ grF(Z(CW )).

Conjecture EC (Ginzburg-Kaledin, 2004)

Assume that Z0 → Z0 is a symplectic resolution.

(EC1) H2i+1C× (Z0) = 0;

(EC2) H2•C×(Z0) ≃ ReesF(Z(CW )).

Cohomology (smooth case)

Theorem (Ginzburg-Kaledin, 2004)

Assume that Zc is smooth.

(1) H2i+1(Zc) = 0;

(2) H2•(Zc) ≃ grF(Z(CW )).

Conjecture EC (Ginzburg-Kaledin, 2004)

Assume that Zc is smooth.

(EC1) H2i+1C× (Zc) = 0;

(EC2) H2•C×(Zc) ≃ ReesF(Z(CW )).

Cohomology (general case)

Conjecture C (Rouquier-B.)

(C1) H2i+1(Zc) = 0;

(C2) H2•(Zc) ≃ grF(ImΩc).

Conjecture EC (Rouquier-B.)

(EC1) H2i+1C× (Zc) = 0;

(EC2) H2•C×(Zc) ≃ ReesF(ImΩc).

Cohomology (general case)

Conjecture C (Rouquier-B.)

(C1) H2i+1(Zc) = 0;

(C2) H2•(Zc) ≃ grF(ImΩc).

Conjecture EC (Rouquier-B.)

(EC1) H2i+1C× (Zc) = 0;

(EC2) H2•C×(Zc) ≃ ReesF(ImΩc).

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

Example of a symplectic singularity

Example of a symplectic singularity

W = W (B2) = 〈s, t |s2 = t2 = (st)4 = 1〉

Example of a symplectic singularity

W = W (B2) = 〈s, t |s2 = t2 = (st)4 = 1〉

Let a = cs and b = ct .

Example of a symplectic singularity

W = W (B2) = 〈s, t |s2 = t2 = (st)4 = 1〉

Let a = cs and b = ct .

Minimal presentation of Zc : 8 generators, 9 equations

Example of a symplectic singularity

W = W (B2) = 〈s, t |s2 = t2 = (st)4 = 1〉

Let a = cs and b = ct .

Minimal presentation of Zc : 8 generators, 9 equationsZc −→ C8, dimZc = 4

Example of a symplectic singularity

W = W (B2) = 〈s, t |s2 = t2 = (st)4 = 1〉

Let a = cs and b = ct .

Minimal presentation of Zc : 8 generators, 9 equationsZc −→ C8, dimZc = 4

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

Example of a symplectic singularity

W = W (B2) = 〈s, t |s2 = t2 = (st)4 = 1〉

Let a = cs and b = ct .

Minimal presentation of Zc : 8 generators, 9 equationsZc −→ C8, dimZc = 4

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

The interesting case: assume from now on that a = b 6= 0.

Example of a symplectic singularity

W = W (B2) = 〈s, t |s2 = t2 = (st)4 = 1〉

Let a = cs and b = ct .

Minimal presentation of Zc : 8 generators, 9 equationsZc −→ C8, dimZc = 4

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

The interesting case: assume from now on that a = b 6= 0.⇒ Zc has only one singular point, named 0

Example of a symplectic singularity

W = W (B2) = 〈s, t |s2 = t2 = (st)4 = 1〉

Let a = cs and b = ct .

Minimal presentation of Zc : 8 generators, 9 equationsZc −→ C8, dimZc = 4

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

The interesting case: assume from now on that a = b 6= 0.⇒ Zc has only one singular point, named 0

Easy fact: dimC T0(Zc) = 8.

Example of a symplectic singularity

W = W (B2) = 〈s, t |s2 = t2 = (st)4 = 1〉

Let a = cs and b = ct .

Minimal presentation of Zc : 8 generators, 9 equationsZc −→ C8, dimZc = 4

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

The interesting case: assume from now on that a = b 6= 0.⇒ Zc has only one singular point, named 0

Easy fact: dimC T0(Zc) = 8.

Let m0 denote the maximal ideal of Zc corresponding to 0.

Example of a symplectic singularity

W = W (B2) = 〈s, t |s2 = t2 = (st)4 = 1〉

Let a = cs and b = ct .

Minimal presentation of Zc : 8 generators, 9 equationsZc −→ C8, dimZc = 4

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

The interesting case: assume from now on that a = b 6= 0.⇒ Zc has only one singular point, named 0

Easy fact: dimC T0(Zc) = 8.

Let m0 denote the maximal ideal of Zc corresponding to 0. Thenm0,m0 ⊂ m0 because 0 is a symplectic leaf.

Example of a symplectic singularity

W = W (B2) = 〈s, t |s2 = t2 = (st)4 = 1〉

Let a = cs and b = ct .

Minimal presentation of Zc : 8 generators, 9 equationsZc −→ C8, dimZc = 4

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

The interesting case: assume from now on that a = b 6= 0.⇒ Zc has only one singular point, named 0

Easy fact: dimC T0(Zc) = 8.

Let m0 denote the maximal ideal of Zc corresponding to 0. Thenm0,m0 ⊂ m0 because 0 is a symplectic leaf.

⇒ T0(Zc)∗ = m0/m

20 inherits from , a structure of Lie algebra!

Example (continued)

Example (continued)

W = W (B2) = 〈s, t |s2 = t2 = (st)4 = 1〉

a = cs = ct ,

Zc −→ C8 = T0(Zc)

⇒ T0(Zc)∗ is a Lie algebra for ,

Example (continued)

W = W (B2) = 〈s, t |s2 = t2 = (st)4 = 1〉

a = cs = ct ,

Zc −→ C8 = T0(Zc)

⇒ T0(Zc)∗ is a Lie algebra for ,

Computation (thanks to MAGMA): T0(Zc)∗ ≃ sl3(C) (!)

Example (continued)

W = W (B2) = 〈s, t |s2 = t2 = (st)4 = 1〉

a = cs = ct ,

Zc −→ C8 = T0(Zc)

⇒ T0(Zc)∗ is a Lie algebra for ,

Computation (thanks to MAGMA): T0(Zc)∗ ≃ sl3(C) (!)

So Zc −→ sl3(C)∗ ≃ sl3(C) (trace form).

Example (continued)

W = W (B2) = 〈s, t |s2 = t2 = (st)4 = 1〉

a = cs = ct ,

Zc −→ C8 = T0(Zc)

⇒ T0(Zc)∗ is a Lie algebra for ,

Computation (thanks to MAGMA): T0(Zc)∗ ≃ sl3(C) (!)

So Zc −→ sl3(C)∗ ≃ sl3(C) (trace form).

Computation (MAGMA):TC0(Zc) = M ∈ sl3(C) | M

2 = 0

Example (continued)

W = W (B2) = 〈s, t |s2 = t2 = (st)4 = 1〉

a = cs = ct ,

Zc −→ C8 = T0(Zc)

⇒ T0(Zc)∗ is a Lie algebra for ,

Computation (thanks to MAGMA): T0(Zc)∗ ≃ sl3(C) (!)

So Zc −→ sl3(C)∗ ≃ sl3(C) (trace form).

Computation (MAGMA):TC0(Zc) = M ∈ sl3(C) | M

2 = 0 = Omin.

Example (continued)

W = W (B2) = 〈s, t |s2 = t2 = (st)4 = 1〉

a = cs = ct ,

Zc −→ C8 = T0(Zc)

⇒ T0(Zc)∗ is a Lie algebra for ,

Computation (thanks to MAGMA): T0(Zc)∗ ≃ sl3(C) (!)

So Zc −→ sl3(C)∗ ≃ sl3(C) (trace form).

Computation (MAGMA):TC0(Zc) = M ∈ sl3(C) | M

2 = 0 = Omin.

So PTC0(Zc) = Omin/C× is smooth

Example (continued)

W = W (B2) = 〈s, t |s2 = t2 = (st)4 = 1〉

a = cs = ct ,

Zc −→ C8 = T0(Zc)

⇒ T0(Zc)∗ is a Lie algebra for ,

Computation (thanks to MAGMA): T0(Zc)∗ ≃ sl3(C) (!)

So Zc −→ sl3(C)∗ ≃ sl3(C) (trace form).

Computation (MAGMA):TC0(Zc) = M ∈ sl3(C) | M

2 = 0 = Omin.

So PTC0(Zc) = Omin/C× is smooth so Beauville classification

theorem applies:

Example (continued)

W = W (B2) = 〈s, t |s2 = t2 = (st)4 = 1〉

a = cs = ct ,

Zc −→ C8 = T0(Zc)

⇒ T0(Zc)∗ is a Lie algebra for ,

Computation (thanks to MAGMA): T0(Zc)∗ ≃ sl3(C) (!)

So Zc −→ sl3(C)∗ ≃ sl3(C) (trace form).

Computation (MAGMA):TC0(Zc) = M ∈ sl3(C) | M

2 = 0 = Omin.

So PTC0(Zc) = Omin/C× is smooth so Beauville classification

theorem applies:

Conclusion (Juteau-B.)

The symplectic singularities (Zc , 0) and (Omin, 0) are equivalent.

Example (continued)

W = W (B2) = 〈s, t |s2 = t2 = (st)4 = 1〉

a = cs = ct ,

Zc −→ C8 = T0(Zc)

⇒ T0(Zc)∗ is a Lie algebra for ,

Computation (thanks to MAGMA): T0(Zc)∗ ≃ sl3(C) (!)

So Zc −→ sl3(C)∗ ≃ sl3(C) (trace form).

Computation (MAGMA):TC0(Zc) = M ∈ sl3(C) | M

2 = 0 = Omin.

So PTC0(Zc) = Omin/C× is smooth so Beauville classification

theorem applies:

Conclusion (Juteau-B.)

The symplectic singularities (Zc , 0) and (Omin, 0) are equivalent. Inparticular, Zc is not rationally smooth.