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!