Poitiers, November 2016bonnafe/Exposes/poitiers-2016.pdf · Poitiers, November 2016. Set-up. Set-up...
Transcript of Poitiers, November 2016bonnafe/Exposes/poitiers-2016.pdf · Poitiers, November 2016. Set-up. Set-up...
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Geometry of Calogero-Moser spaces
Cedric Bonnafe
CNRS (UMR 5149) - Universite de Montpellier
Poitiers, November 2016
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Set-up
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Set-up
dimC V = n <∞
W < GLC(V ), |W | <∞.
Ref(W ) = s ∈ W | codimCVs = 1
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Set-up
dimC V = n <∞
W < GLC(V ), |W | <∞.
Ref(W ) = s ∈ W | codimCVs = 1
Hypothesis. W = 〈Ref(W )〉
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Set-up
dimC V = n <∞
W < GLC(V ), |W | <∞.
Ref(W ) = s ∈ W | codimCVs = 1
Hypothesis. W = 〈Ref(W )〉(i.e. V /W ≃ Cn)
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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
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Cherednik algebra at t = 0
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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
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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
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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)
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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
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DefinitionThe Calogero-Moser space associated with the datum (W , c) is theaffine variety
Zc = Spec(Zc).
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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
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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.
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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 |.
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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).
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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 ,
![Page 18: Poitiers, November 2016bonnafe/Exposes/poitiers-2016.pdf · Poitiers, November 2016. Set-up. Set-up dimCV =n < ... 0 =C[V × V ∗]W, Definition The Calogero-Moser space associated](https://reader036.fdocuments.net/reader036/viewer/2022063013/5fce16c27a99a059d25effa1/html5/thumbnails/18.jpg)
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 ,
![Page 19: Poitiers, November 2016bonnafe/Exposes/poitiers-2016.pdf · Poitiers, November 2016. Set-up. Set-up dimCV =n < ... 0 =C[V × V ∗]W, Definition The Calogero-Moser space associated](https://reader036.fdocuments.net/reader036/viewer/2022063013/5fce16c27a99a059d25effa1/html5/thumbnails/19.jpg)
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 .
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Two extra-structures
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Two extra-structures
There is a C×-action on Hc (i.e. a Z-grading):
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Two extra-structures
There is a C×-action on Hc (i.e. a Z-grading): deg(V ) = −1 deg(V ∗) = 1 deg(W ) = 0
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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 .
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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
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GLn(Fq), q = p?, ℓ 6= p.
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GLn(Fq), q = p?, ℓ 6= p.
Steinberg (1952), Lusztig (1976):
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GLn(Fq), q = p?, ℓ 6= p.
Steinberg (1952), Lusztig (1976):
partitions of n∼←→ Unip. char. of GLn(Fq)
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GLn(Fq), q = p?, ℓ 6= p.
Steinberg (1952), Lusztig (1976):
partitions of n∼←→ Unip. char. of GLn(Fq)
λ 7−→ ρλ
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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
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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):
![Page 31: Poitiers, November 2016bonnafe/Exposes/poitiers-2016.pdf · Poitiers, November 2016. Set-up. Set-up dimCV =n < ... 0 =C[V × V ∗]W, Definition The Calogero-Moser space associated](https://reader036.fdocuments.net/reader036/viewer/2022063013/5fce16c27a99a059d25effa1/html5/thumbnails/31.jpg)
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)
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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))
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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))
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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))
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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))
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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))
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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))
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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))
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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))
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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))
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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)).
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Symplectic singularities
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Symplectic singularities
Theorem (Brown-Gordon, 2003)
(a) The symplectic leaves are obtained as follows
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Symplectic singularities
Theorem (Brown-Gordon, 2003)
(a) The symplectic leaves are obtained as follows (Zc)smooth is a symplectic leaf;
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Symplectic singularities
Theorem (Brown-Gordon, 2003)
(a) The symplectic leaves are obtained as follows (Zc)smooth is a symplectic leaf; ((Zc)sing)
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Symplectic singularities
Theorem (Brown-Gordon, 2003)
(a) The symplectic leaves are obtained as follows (Zc)smooth is a symplectic leaf; ((Zc)sing)smooth
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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;
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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; . . .
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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).
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Symplectic resolutions
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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.
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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 |.
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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).
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Cohomology
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Cohomology
Theorem (Ginzburg-Kaledin, 2004)
Assume that Z0 → Z0 is a symplectic resolution.
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Cohomology
Theorem (Ginzburg-Kaledin, 2004)
Assume that Z0 → Z0 is a symplectic resolution.
(1) H2i+1(Z0,C) = 0
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Cohomology
Theorem (Ginzburg-Kaledin, 2004)
Assume that Z0 → Z0 is a symplectic resolution.
(1) H2i+1(Z0) = 0;
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Cohomology
Theorem (Ginzburg-Kaledin, 2004)
Assume that Z0 → Z0 is a symplectic resolution.
(1) H2i+1(Z0) = 0;
(2) H2•(Z0) ≃ grF(Z(CW )).
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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.
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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;
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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 )).
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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 )).
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Theorem (Vasserot, 2001)
(EC) holds if W = Sn (Z0 = Hilbn(C2)).
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Theorem (Vasserot, 2001)
(EC) holds if W = Sn (Z0 = Hilbn(C2)).
Other cases. W = W (B2) or G4 (Shan-B. 2016).
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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?
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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 )).
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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 )).
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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).
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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.
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Example of a symplectic singularity
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Example of a symplectic singularity
W = W (B2) = 〈s, t |s2 = t2 = (st)4 = 1〉
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Example of a symplectic singularity
W = W (B2) = 〈s, t |s2 = t2 = (st)4 = 1〉
Let a = cs and b = ct .
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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
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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
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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.
![Page 76: Poitiers, November 2016bonnafe/Exposes/poitiers-2016.pdf · Poitiers, November 2016. Set-up. Set-up dimCV =n < ... 0 =C[V × V ∗]W, Definition The Calogero-Moser space associated](https://reader036.fdocuments.net/reader036/viewer/2022063013/5fce16c27a99a059d25effa1/html5/thumbnails/76.jpg)
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.
![Page 77: Poitiers, November 2016bonnafe/Exposes/poitiers-2016.pdf · Poitiers, November 2016. Set-up. Set-up dimCV =n < ... 0 =C[V × V ∗]W, Definition The Calogero-Moser space associated](https://reader036.fdocuments.net/reader036/viewer/2022063013/5fce16c27a99a059d25effa1/html5/thumbnails/77.jpg)
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
![Page 78: Poitiers, November 2016bonnafe/Exposes/poitiers-2016.pdf · Poitiers, November 2016. Set-up. Set-up dimCV =n < ... 0 =C[V × V ∗]W, Definition The Calogero-Moser space associated](https://reader036.fdocuments.net/reader036/viewer/2022063013/5fce16c27a99a059d25effa1/html5/thumbnails/78.jpg)
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.
![Page 79: Poitiers, November 2016bonnafe/Exposes/poitiers-2016.pdf · Poitiers, November 2016. Set-up. Set-up dimCV =n < ... 0 =C[V × V ∗]W, Definition The Calogero-Moser space associated](https://reader036.fdocuments.net/reader036/viewer/2022063013/5fce16c27a99a059d25effa1/html5/thumbnails/79.jpg)
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.
![Page 80: Poitiers, November 2016bonnafe/Exposes/poitiers-2016.pdf · Poitiers, November 2016. Set-up. Set-up dimCV =n < ... 0 =C[V × V ∗]W, Definition The Calogero-Moser space associated](https://reader036.fdocuments.net/reader036/viewer/2022063013/5fce16c27a99a059d25effa1/html5/thumbnails/80.jpg)
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.
![Page 81: Poitiers, November 2016bonnafe/Exposes/poitiers-2016.pdf · Poitiers, November 2016. Set-up. Set-up dimCV =n < ... 0 =C[V × V ∗]W, Definition The Calogero-Moser space associated](https://reader036.fdocuments.net/reader036/viewer/2022063013/5fce16c27a99a059d25effa1/html5/thumbnails/81.jpg)
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!
![Page 82: Poitiers, November 2016bonnafe/Exposes/poitiers-2016.pdf · Poitiers, November 2016. Set-up. Set-up dimCV =n < ... 0 =C[V × V ∗]W, Definition The Calogero-Moser space associated](https://reader036.fdocuments.net/reader036/viewer/2022063013/5fce16c27a99a059d25effa1/html5/thumbnails/82.jpg)
Example (continued)
![Page 83: Poitiers, November 2016bonnafe/Exposes/poitiers-2016.pdf · Poitiers, November 2016. Set-up. Set-up dimCV =n < ... 0 =C[V × V ∗]W, Definition The Calogero-Moser space associated](https://reader036.fdocuments.net/reader036/viewer/2022063013/5fce16c27a99a059d25effa1/html5/thumbnails/83.jpg)
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 ,
![Page 84: Poitiers, November 2016bonnafe/Exposes/poitiers-2016.pdf · Poitiers, November 2016. Set-up. Set-up dimCV =n < ... 0 =C[V × V ∗]W, Definition The Calogero-Moser space associated](https://reader036.fdocuments.net/reader036/viewer/2022063013/5fce16c27a99a059d25effa1/html5/thumbnails/84.jpg)
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) (!)
![Page 85: Poitiers, November 2016bonnafe/Exposes/poitiers-2016.pdf · Poitiers, November 2016. Set-up. Set-up dimCV =n < ... 0 =C[V × V ∗]W, Definition The Calogero-Moser space associated](https://reader036.fdocuments.net/reader036/viewer/2022063013/5fce16c27a99a059d25effa1/html5/thumbnails/85.jpg)
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).
![Page 86: Poitiers, November 2016bonnafe/Exposes/poitiers-2016.pdf · Poitiers, November 2016. Set-up. Set-up dimCV =n < ... 0 =C[V × V ∗]W, Definition The Calogero-Moser space associated](https://reader036.fdocuments.net/reader036/viewer/2022063013/5fce16c27a99a059d25effa1/html5/thumbnails/86.jpg)
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
![Page 87: Poitiers, November 2016bonnafe/Exposes/poitiers-2016.pdf · Poitiers, November 2016. Set-up. Set-up dimCV =n < ... 0 =C[V × V ∗]W, Definition The Calogero-Moser space associated](https://reader036.fdocuments.net/reader036/viewer/2022063013/5fce16c27a99a059d25effa1/html5/thumbnails/87.jpg)
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.
![Page 88: Poitiers, November 2016bonnafe/Exposes/poitiers-2016.pdf · Poitiers, November 2016. Set-up. Set-up dimCV =n < ... 0 =C[V × V ∗]W, Definition The Calogero-Moser space associated](https://reader036.fdocuments.net/reader036/viewer/2022063013/5fce16c27a99a059d25effa1/html5/thumbnails/88.jpg)
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
![Page 89: Poitiers, November 2016bonnafe/Exposes/poitiers-2016.pdf · Poitiers, November 2016. Set-up. Set-up dimCV =n < ... 0 =C[V × V ∗]W, Definition The Calogero-Moser space associated](https://reader036.fdocuments.net/reader036/viewer/2022063013/5fce16c27a99a059d25effa1/html5/thumbnails/89.jpg)
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:
![Page 90: Poitiers, November 2016bonnafe/Exposes/poitiers-2016.pdf · Poitiers, November 2016. Set-up. Set-up dimCV =n < ... 0 =C[V × V ∗]W, Definition The Calogero-Moser space associated](https://reader036.fdocuments.net/reader036/viewer/2022063013/5fce16c27a99a059d25effa1/html5/thumbnails/90.jpg)
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.
![Page 91: Poitiers, November 2016bonnafe/Exposes/poitiers-2016.pdf · Poitiers, November 2016. Set-up. Set-up dimCV =n < ... 0 =C[V × V ∗]W, Definition The Calogero-Moser space associated](https://reader036.fdocuments.net/reader036/viewer/2022063013/5fce16c27a99a059d25effa1/html5/thumbnails/91.jpg)
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.