mini-course on fano foliations - cirm-math.commini-course on fano foliations joint with st ephane...
Post on 18-Feb-2021
0 Views
Preview:
TRANSCRIPT
-
Mini-course on Fano Foliations
Carolina Araujo (IMPA)
Lecture 1: Definition, examples and first properties
-
Mini-course on Fano Foliations
Joint with Stéphane Druel (CNRS/Université Claude Bernard Lyon 1)
Lecture 0: Algebraicity of smooth formal schemes and applications tofoliations
Lecture 1: Definition, examples and first properties
Lecture 2: Adjunction formula and applications
Lecture 3: Classification of Fano foliations of large index
-
Motivation from the MMP
KX > 0 KX = 0 KX < 0
Fano varieties
Special geometric properties
Fano manifolds are rationally connected (RC)
Classification
Classification of Fano manifolds with large index
-
Fano foliations
X normal complex projective variety
Foliation F ( TX on X
saturated ( TX/F torsion-free )integrable ( [F ,F ] ⊂ F )
Sing(F) ⊂ X degeneracy locus of the map F ↪→ TX
The canonical class of F : KF ∈ C`(X )
OX (KF ) ∼=(
det(F))∨
F ↪→ TX ΩrX → OX (KF )
ISing(F) is the image of the induced map(ΩrX (−KF )
)∨∨ → OX
-
Fano foliations
X normal complex projective variety
Foliation F ( TX on X
saturated ( TX/F torsion-free )integrable ( [F ,F ] ⊂ F )
Sing(F) ⊂ X degeneracy locus of the map F ↪→ TX
The canonical class of F : KF ∈ C`(X )
OX (KF ) ∼=(
det(F))∨
Definition
F is Fano if −KF is Q-Cartier and ample
-
Early examples: foliations on Pn
F ( TPn foliation of rank r on Pn
d = deg(F) degree of F
L = Pn−r ⊂ Pn general linear subspace
DF ⊂ L = Pn−r tangency hypersurface
d = deg(DF ) ≥ 0
Early problem: Classification of (codim 1) foliations of low degree on Pn
-
Foliations of degree 0 on Pn
F : r -planes on Pn containing a fixed L0 = Pr−1
Pn−r ⊂ Pn general linear subspace - everywhere transvere to Fdeg(F) = 0F is induced by the linear projection π : Pn 99K Pn−r from L0F = ker(dπ) ∼= O(1)⊕r and −KF = rH
Theorem (Jouanolou 1979, Déserti-Cerveau 2005)
These are the only foliations of degree 0 on Pn
-
The index of a Fano foliations
Definition
The index of a Fano foliation F on complex projective manifold X is
i(F) := max{m ∈ Z
∣∣ − KF ∼Z mA, A ample }Example
F ∼= O(1)⊕r foliation of degree 0 on Pn =⇒ i(F) = r
F ↪→ TX OX (−KF ) → (ΩrX )∨ ∼= Ωn−rX (−KX )
0 6= ω ∈ H0(X ,Ωn−rX (−KX + KF )
)Example (Fano foliations on Pn)F ( TPn Fano foliation on Pn of rank r and index i
ω ∈ H0(Pn,Ωn−rPn (n + 1− i)
)
-
The index of a Fano foliations
Definition
The index of a Fano foliation F on complex projective manifold X is
i(F) := max{m ∈ Z
∣∣ − KF ∼Z mA, A ample }Example
F ∼= O(1)⊕r foliation of degree 0 on Pn =⇒ i(F) = r
Example (Fano foliations on Pn)F ( TPn Fano foliation of rank r and index i
ω ∈ H0(Pn,Ωn−rPn (n + 1− i)
)L = Pn−r ⊂ Pn general linear subspaceω|L ∈ H0
(Pn−r ,Ωn−rPn−r (n + 1− i)
)= H0
(Pn−r ,OPn−r (r − i)
)i = r − d
-
The index of a Fano foliations
Definition
The index of a Fano foliation F on complex projective manifold X is
i(F) := max{m ∈ Z
∣∣ − KF ∼Z mA, A ample }Example (Fano foliations on Pn)F ( TPn Fano foliation of rank r , index i , and degree d
i = r − d ≤ r
Theorem (A.- Druel - Kovács 2008)
F ( TX Fano foliation of rank r on a complex projective manifold Xi(F) ≤ ri(F) = r =⇒ X ∼= Pn
-
The index of a Fano foliations
Definition
The index of a Fano foliation F on complex projective manifold X is
i(F) := max{m ∈ Z
∣∣ − KF ∼Z mA, A ample }Example (Fano foliations on Pn)F ( TPn Fano foliation of rank r , index i , and degree d
i = r − d ≤ r
Theorem (A.- Druel 2014, Höring 2014)
F ( TX Fano foliation of rank r on a normal complex projective variety Xi(F) ≤ ri(F) = r =⇒ X is a generalized cone
-
Foliations of degree 1 on Pn (index r − 1)
Construction (Algebraically integrable foliation)
ϕ : X 99K Y dominant rational map with connected fibers
ϕ◦ : X ◦ → Y ◦ equidimensional morphism
X ◦ ⊂ X open subset with codimX (X \ X ◦) ≥ 2
F ⊂ TX saturation of ker(dϕ◦) in TX
KF = KX/Y − R(ϕ)
R(ϕ) ramification divisor of ϕ
Example 1:ϕ : Pn 99K P(1n−r , 2)
(x0 : · · · : xn) 7−→(L1 : · · · : Ln−r : Q
)−KF = (r − 1)H
-
Foliations of degree 1 on Pn (index r − 1)
Construction (Pullback foliations)
ϕ : X 99K Y dominant rational map with connected fibers
ϕ◦ : X ◦ → Y ◦ equidimensional morphism
X ◦ ⊂ X open subset with codimX (X \ X ◦) ≥ 2
G ⊂ TY foliation on Y
Pullback foliation F = ϕ∗G
F is the saturation of (dϕ◦)−1(G|Y ◦) in TX
KF = KX/Y + ϕ∗KG − R(ϕ)G
-
Foliations of degree 1 on Pn (index r − 1)Construction (Pullback foliations)
ϕ : X 99K Y dominant rational map with connected fibers
ϕ◦ : X ◦ → Y ◦ equidimensional morphism
X ◦ ⊂ X open subset with codimX (X ◦) ≥ 2
G ⊂ TY foliation on Y F = ϕ∗G
KF = KX/Y + ϕ∗KG − R(ϕ)G
Example 2: π : Pn 99K Pn−r+1 linear projectionC ⊂ TPn−r+1 foliation induced by a global vector field (KC = 0 ) F = π∗C ⊂ TPn
−KF = (r − 1)H
If C is general, then C and F have transcendental leaves
-
Foliations of degree 1 on Pn (index r − 1)
Theorem (Jouanolou 1979, Loray-Pereira-Touzet 2018)
There are 2 types of foliations of degree 1 on Pn (index r − 1) :
F is induced by Pn 99K P(1n−r , 2)
∃ ϕ : Pn 99K Pn−r+1 and such that F = ϕ∗C for C ⊂ TPn−r+1foliation of rank 1 induced by a global vector field
Theorem (Cerveau - Lins Neto 1996)
There are 6 types of codimension 1 foliations of degree 2 on Pn
( rank r = n − 1 and index i = r − 2 )
-
More examples: foliations on hypersurfaces
Construction (Restrictions of foliations)
F foliation of codimension q ≥ 1 on smooth projective variety X
Y ⊂ X smooth subvariety generically transverse to F
F restrics to a foliation FY of codimension q on Y
F ⊂ TX ! ω ∈ H0(X ,ΩqX (−KX + KF )
)FY ⊂ TY ! ωY ∈ H0
(Y ,ΩqY (−KY + KFY )
)(−KX + KF )|Y = −KY + KFY + B ( B ≥ 0 )
KFY = c1(NY /X
)+ (KF )|Y − B
-
Fano foliations on hypersurfaces
Construction (Restrictions of foliations)
F foliation of codimension q ≥ 1 on smooth projective variety X
Y ⊂ X smooth subvariety generically transverse to F
F restrics to a foliation FY of codimension q on Y
KFY = c1(NY /X
)+ (KF )|Y − B ( B ≥ 0 )
Example (X = Pn, Y ⊂ Pn hypersurface of degree d ≥ 2)
F ⊂ TPn Fano foliation of index i ≤ r
−KFY ≥ (i − d) H|Y
-
More examples: foliations on hypersurfaces
Construction (Restrictions of foliations)
F foliation of codimension q ≥ 1 on smooth projective variety X
Y ⊂ X smooth subvariety generically transverse to F
F restrics to a foliation FY of codimension q on Y
KFY = c1(NY /X
)+ (KF )|Y − B ( B ≥ 0 )
Example (X = Pn, Y ⊂ Pn hypersurface of degree d ≥ 2)
F ⊂ TPn Fano foliation of index i = r
−KFY = (i − d) H|Y( h0
(Y ,ΩqY (q + 1− d)
)= 0 )
-
Algebraicity properties of Fano foliations
Theorem (Campana - Păun 2019)
X normal projective Q-factorial variety, α ∈ N1(X )R movable curve class,G ⊂ TX foliation on X with µminα (G) > 0.
Then G has algebraic and RC leaves.
µα(•) = det(•)·αrank(•)µminα (G) = inf
{µα(Q) | Q 6= 0 is a torsion-free quotient of G
}Corollary
X normal projective Q-factorial variety, F ⊂ TX Fano foliation.
Then ∃ subfoliation G ⊂ F with algebraic and RC leaves.
-
Proof of corollary
X normal projective Q-factorial variety, α ∈ N1(X )R movable curve class,F torsion free sheaf on X
The Harder-Narasimhan filtration:
0 = F0 ( F1 ( · · · ( Fk = F
with Qi = Fi/Fi−1 µα-semistable, and
µα(Q1) > µα(Q2) > · · · > µα(Qk)
µminα (Fi ) = µα(Qi )
F foliation =⇒ Fi foliation whenever µα(Qi ) ≥ 0
F Fano foliation =⇒ µα(F) > 0 =⇒ µα(Q1) > 0
-
Proof of corollary
The Harder-Narasimhan filtration:
0 = F0 ( F1 ( · · · ( Fk = F
with Qi = Fi/Fi−1 µα-semistable, and
µα(Q1) > µα(Q2) > · · · > µα(Qk)
µminα (Fi ) = µα(Qi )
F foliation =⇒ Fi foliation whenever µα(Qi ) ≥ 0F Fano foliation =⇒ µα(F) > 0 =⇒ µα(Q1) > 0
Theorem (Campana - Păun 2019)
X normal projective Q-factorial variety, α ∈ N1(X )R movable curve class,F1 ⊂ TX foliation on X with µminα (F1) > 0.
Then F1 has algebraic and RC leaves.
-
Proof of corollary
The Harder-Narasimhan filtration:
0 = F0 ( F1 ( · · · ( Fk = F
with Qi = Fi/Fi−1 µα-semistable, and
µα(Q1) > µα(Q2) > · · · > µα(Qk)
µminα (Fi ) = µα(Qi )
F foliation =⇒ Fi foliation whenever µα(Qi ) ≥ 0F Fano foliation =⇒ µα(F) > 0 =⇒ µα(Q1) > 0
Remark
s := max{
1 ≤ i ≤ k |µα(Qi ) > 0}≥ 1
Then F1, . . . ,Fs have algebraic and RC leaves.
-
The algebraic rank of a foliation
X normal complex projective variety
F ⊂ TX foliation on X
∃ ϕ : X 99K Y dominant rational map with connected fibers
∃ G purely transcendental foliation on Y
F = ϕ∗G
Definition (algebraic rank)
rkalg (F) := dim(X )− dim(Y )
-
Bounding the algebraic rank
Theorem (A.-Druel 2019)
X complex projective manifold, F ( TX Fano foliation of index i(F)
rkalg (F) ≥ i(F)
rkalg (F) = i(F) =⇒ X ∼= Pn
∃ ϕ : Pn 99K Pm and G purely transcendental foliation on Pm suchthat
KG ≡ 0 and F = ϕ∗G
-
del Pezzo foliations
Theorem (A.- Druel - Kovács 2008)
F ( TX Fano foliation of rank r on a complex projective manifold Xi(F) ≤ ri(F) = r =⇒ X ∼= Pn
Definition
A Fano foliation F ( TX of rank r on a complex projective manifold X isa del Pezzo foliation if i(F) = r − 1.
-
The algebraic rank of del Pezzo foliations
Definition
A Fano foliation F ( TX of rank r on a complex projective manifold X isa del Pezzo foliation if i(F) = r − 1.
Theorem (A.-Druel 2019)
X complex projective manifold, F ( TX Fano foliation of index i(F)rkalg (F) ≥ i(F)rkalg (F) = i(F) =⇒ X ∼= Pn
Corollary (A.- Druel 2013)
A del Pezzo foliation F on a complex projective manifold X 6∼= Pn isalgebraically integrable.
-
del Pezzo foliations
Definition
A Fano foliation F ( TX of rank r on a complex projective manifold X isa del Pezzo foliation if i(F) = r − 1.
Corollary (A.- Druel 2013)
A del Pezzo foliation F on a complex projective manifold X 6∼= Pn isalgebraically integrable.
Problem
Classification of del Pezzo foliations
-
Thank you!
top related