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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.
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