Blowups, deformations to normal cones and Lie groupoids (joint work - - PowerPoint PPT Presentation
Blowups, deformations to normal cones and Lie groupoids (joint work with Claire Debord) Claire Debord & GS: Blowup constructions for Lie groupoids and a Boutet de Monvel type calculus (In preparation) Georges Skandalis Universit e
Universit´ e Paris-Diderot Paris 7 Institut de Math´ ematiques de Jussieu Paris Rive Gauche
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 1 / 24
1 Deformation to the normal cone (DNC) 2 Blowup (Blup).
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 2 / 24
V . The deformation to the normal cone is
V ⇥ {0}.
+ t N M V ⇥ {0}.
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 3 / 24
fV ✏
fM
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 4 / 24
r,s
i
Γ is a groupoid over N G(0) Γ(0) denoted N G Γ ◆ N G(0) Γ(0) .
Γ ⇥ {0} ◆ (G(0) ⇥ R⇤) t N G(0) Γ(0) ⇥ {0}
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 5 / 24
1 Tangent groupoid of Alain Connes
2 V submanifold of G(0), saturated for G, DNC(G, GV
V ) normal groupoid of
V ,
3 K maximal compact subgroup of a Lie group G, DNC(G, K) used by
4 Double deformation: G1 ⇢ G2 ⇢ G3.
M E ⇢ E ⇥ E:
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 6 / 24
V :
λX, 0) for t = 0
V ).
+ = M \ V t S(N M V ).
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 7 / 24
fV ✏
fM
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 8 / 24
r,s
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 9 / 24
V )) and G0 = SBlupr,s(G, V ).
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 10 / 24
rΓ(x) and sE : Ex ! E(0) sΓ(x),
+.
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 11 / 24
1 Unique linear groupoid structure on E: E ◆ F with source s, range r and
2 E is the action groupoid E ors E/F.
1 This construction can be done with any field. 2 If r 6= s, every orbit meets F \ {0}: the restriction ˚
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 12 / 24
1 Unique linear groupoid structure on E: E ◆ F with source s, range r and
2 E is the action groupoid F ors E/F.
1 This construction can be done with any field. 2 If r 6= s, every orbit meets F \ {0}: the restriction ˚
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 12 / 24
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 13 / 24
r,s
V ! V , F = N M V
V ! N M V .
V ◆ N M V , P(N G V ) ◆ P(N M V ) and S(N G V ) ◆ S(N M V ).
V ⇥ {0} ◆ M ⇥ R⇤ t N M V ⇥ {0}
V ) ◆ M \ V t P(N M V ).
V ) ◆ M \ V t S(N M V ).
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 14 / 24
r,s
Γ restriction of the groupoid N G Γ ◆ N M V
V \ V .
+ t N G Γ ⇥ {0} ◆ M ⇥ R⇤ + t N M V
+ = G ˚ M ˚ M t SN G Γ ◆ ˚
V )
+)
Γ )
M ˚ M)
Γ )
M ˚ M ⇥ R⇤ +)
Γ )
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 15 / 24
+)
Γ )
M ˚ M × R∗ +)
Γ )
DNC+
M ˚ M)
Γ )
Γ ), C∗(G × R∗ +)),
DNC+ ∈ KK1(C∗( ˚
Γ ), C∗(G ˚ M ˚ M × R∗ +)) and
Γ ), C∗(G ˚ M ˚ M)).
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 16 / 24
+)
Γ )
M ˚ M × R∗ +)
˚ β
β
Γ ) β∂
DNC+
M ˚ M)
Γ )
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 16 / 24
+)
Γ )
M ˚ M × R∗ +)
˚ β
j
β
Γ ) β∂
DNC+
M ˚ M)
Γ )
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 16 / 24
+)
Γ )
M ˚ M × R∗ +)
˚ β
j
β
Γ ) β∂
DNC+
M ˚ M)
Γ )
1 ∂ ^
DNC+ ⊗ [˚
Γ ), C∗(G)).
2 ∂SBlup ⊗ ˚
DNC+ ∈ KK1(C∗(SN G Γ ), C∗(˚
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 16 / 24
+)
Γ )
M ˚ M × R∗ +)
˚ β
j
β
Γ ) β∂
DNC+
M ˚ M)
Γ )
Γ ), C∗(G)).
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 16 / 24
+)
Γ )
M ˚ M ⇥ R⇤ +)
˚ β
j
β
Γ ) β∂
DNC+
M ˚ M)
Γ )
Γ ), C⇤(G)).
V )x is nonzero for every x 2 V , then ˚
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 16 / 24
+)
M ˚ M ⇥ R⇤ +)
˚ β
j
βΨ
DNC+ βΣ
DNC+
M ˚ M)
V )x is nonzero for every x 2 V , the j’s are K-equivalences.
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 17 / 24
V is
V )⇤), C⇤(N G V )).
+)
Ind
Ind
V )⇤)
' Ind
+)
Γ )
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 18 / 24
V is a submanifold.
V ⇥ (R ⇥ R), V ) ⇠ ⇢ SBlupr,s(G, V ).
Mt(V ⇥R), V ⇥ ({0, 1} ⇥ {0, 1}))
V ⇥ (R ⇥ R), V )
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 19 / 24
V ⇥ (R ⇥ R), V ) ◆ SBlup(V ⇥ R, V ) ' V ⇥ (R+ t R), restricted
V )ga of GV V :
+.
+) ! C⇤ +(DNC(H, H(0))) ! C0(A⇤H) ! 0
+:
+ ! 0
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 20 / 24
+ ! 0
E 0
P T (G,V )
EP T (G,V )
V ))
E 00
P T (G,V )
V )
V )
V )
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BM(G, V ) = K
V ).
˚ M ` V ˚ M ` V ) ! C⇤ BM(G, V ) rC⇤
V
bound(G, V ) ! 0, where
bound(G, V ) = algebra of Boutet de Monvel type boundary symbols:
V ), q 2 C(S⇤AGV V ), p, t⇤ 2 E 00 P T .
V : zero order symbol map of the Boutet de Monvel type calculus.
V
V (K)
V ;
V , rV , rV : restrictions to the boundary.
Claire Debord and GS Blowps, DNC and Lie groupoids Toulouse 29/05/17 22 / 24
BM(G, V ): algebra of matrices R =
V ).
V
BM(G, V ) ! ΣΨ⇤ bound(G, V ) defined by
V (Φ)
V : Ψ⇤(SBlupr,s(G, V )) ! Ψ⇤(SN G V ) is the restriction.
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