On some Fano fourfolds

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Cubic fourfolds Prime Fano fourfolds of degree 10 and index 2

On some Fano fourfolds

Olivier DEBARRE (joint with Atanas ILIEV and Laurent MANIVEL)

´Ecole normale sup´erieure de Paris, France

Yeosu, February 2013

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Unirationality and rationality Periods

Special cubics

Associated irreducible symplectic variety

Unirationality

All smooth cubic fourfolds are unirational

X ⊂P⁵ smooth (complex) cubic fourfold. Pick a line `⊂X.

Define a mapπ :P(TX|_`)99KX by sending a line tangent to X atx ∈`to the third intersection point withX.

π is dominant of degree 2: forx ∈X general, h`,xi ∩X is the union of`and a conic; the two intersection points are π⁻¹(x). No smooth cubic fourfold has yet been proven to be irrational

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Special cubics

Unirationality

All smooth cubic fourfolds are unirational X ⊂P⁵ smooth (complex) cubic fourfold.

Pick a line `⊂X.

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Special cubics

Unirationality

Pick a line `⊂X.

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Special cubics

Unirationality

Pick a line `⊂X.

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Special cubics

Unirationality

Pick a line `⊂X.

π is dominant of degree 2: for x∈X general, h`,xi ∩X is the union of`and a conic; the two intersection points are π⁻¹(x).

No smooth cubic fourfold has yet been proven to be irrational

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Special cubics

Unirationality

Pick a line `⊂X.

π is dominant of degree 2: for x∈X general, h`,xi ∩X is the union of`and a conic; the two intersection points are π⁻¹(x).

No smooth cubic fourfold has yet been proven to be irrational

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Special cubics

Pfaffian cubics

Some smooth cubic fourfolds are rational

Pf(C⁶) :={η ∈P(∧²C^6∗)|η degenerate} ⊂P¹⁴

is a cubic hypersurface with singular locus of codimension 6 inP¹⁴. A general linear sectionX∩P⁵ is a (smooth) Pfaffian cubic. In

P(C⁶)^oo ^p¹ {(v, η)∈P(C⁶)×Pf(C⁶)|v ∈kerη} ^p² ^//Pf(C⁶), p₁ is a P⁹-bundle, p₂ is generically aP¹-bundle. It restricts to

P(C⁶)^oo ^1:1 {(v, η)∈P(C⁶)×X |v ∈kerη} ^P¹^-bundle ^//X. IfH⊂P(C⁶) hyperplane,

H ^oo ^1:1 {(v, η)∈H×X |v ∈kerη} ^1:1 ^//X.

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Special cubics

Pfaffian cubics

is a cubic hypersurface with singular locus of codimension 6 inP¹⁴. A general linear sectionX∩P⁵ is a (smooth) Pfaffian cubic.

In P(C⁶)^oo ^p¹ {(v, η)∈P(C⁶)×Pf(C⁶)|v ∈kerη} ^p² ^//Pf(C⁶), p₁ is a P⁹-bundle, p₂ is generically aP¹-bundle. It restricts to

H ^oo ^1:1 {(v, η)∈H×X |v ∈kerη} ^1:1 ^//X.

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Special cubics

Pfaffian cubics

P(C⁶)^oo ^p¹ {(v, η)∈P(C⁶)×Pf(C⁶)|v ∈kerη} ^p² ^//Pf(C⁶), p₁ is aP⁹-bundle, p₂ is generically aP¹-bundle.

It restricts to P(C⁶)^oo ^1:1 {(v, η)∈P(C⁶)×X |v ∈kerη} ^P¹^-bundle ^//X. IfH⊂P(C⁶) hyperplane,

H ^oo ^1:1 {(v, η)∈H×X |v ∈kerη} ^1:1 ^//X.

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Special cubics

Pfaffian cubics

P(C⁶)^oo ^p¹ {(v, η)∈P(C⁶)×Pf(C⁶)|v ∈kerη} ^p² ^//Pf(C⁶), p₁ is aP⁹-bundle, p₂ is generically aP¹-bundle. It restricts to

P(C⁶)^oo ^1:1 {(v, η)∈P(C⁶)×X |v ∈kerη} ^P¹^-bundle ^//X.

IfH⊂P(C⁶) hyperplane,

H ^oo ^1:1 {(v, η)∈H×X |v ∈kerη} ^1:1 ^//X.

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Special cubics

Pfaffian cubics

P(C⁶)^oo ^p¹ {(v, η)∈P(C⁶)×Pf(C⁶)|v ∈kerη} ^p² ^//Pf(C⁶), p₁ is aP⁹-bundle, p₂ is generically aP¹-bundle. It restricts to

H ^oo ^1:1 {(v, η)∈H×X |v ∈kerη} ^1:1 ^//X.

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Special cubics

Period domain

X ⊂P⁵ smooth cubic fourfold

Primitive Hodge structure is “of K3 type”

H⁴(X,C)0 = H^1,3(X) ⊕ H^2,2(X)0 ⊕ H^3,1(X)

dimensions 1 20 1

Lattices

H⁴(X,Z) ' I21,2

H⁴(X,Z)0 ' 2E8⊕2U⊕A2 =: Λ0

Local period domain (20-dimensional, bounded symmetric domain of type IV)

Q0 :={ω∈P(Λ₀⊗C)|ω·ω = 0, ω·ω <¯ 0}

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Special cubics

Period domain

H⁴(X,C)0 = H^1,3(X) ⊕ H^2,2(X)0 ⊕ H^3,1(X)

dimensions 1 20 1

Lattices

H⁴(X,Z) ' I21,2

H⁴(X,Z)0 ' 2E8⊕2U⊕A2 =: Λ0

Q0 :={ω∈P(Λ₀⊗C)|ω·ω = 0, ω·ω <¯ 0}

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Special cubics

Period domain

H⁴(X,C)0 = H^1,3(X) ⊕ H^2,2(X)0 ⊕ H^3,1(X)

dimensions 1 20 1

Lattices

H⁴(X,Z) ' I21,2

H⁴(X,Z)0 ' 2E8⊕2U⊕A2 =: Λ0

Q0:={ω ∈P(Λ₀⊗C)|ω·ω = 0, ω·ω <¯ 0}

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Special cubics

Period map

Moduli space is the 20-dimensional affine GIT quotient M :=P(H⁰(P⁵,OP⁵(3)))⁰//SL(C⁶)

Period map

p :M →D0 := Γ₀\Q0 where Γ₀ ⊂O(Λ₀) Period domainD0 is a quasi-projective variety with projective Baily-Borel compactificationD^BB₀

Theorem

p is ´etale (“local Torelli”; infinitesimal calculation); p is injective (Voisin);

image of p known (Looijenga, Laza).

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Special cubics

Period map

Moduli space is the 20-dimensional affine GIT quotient M :=P(H⁰(P⁵,OP⁵(3)))⁰//SL(C⁶) Period map

p :M →D0 := Γ₀\Q0 where Γ₀ ⊂O(Λ₀)

Period domainD0 is a quasi-projective variety with projective Baily-Borel compactificationD^BB₀

Theorem

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Special cubics

Period map

Theorem

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Special cubics

Period map

Theorem

p is ´etale (“local Torelli”; infinitesimal calculation);

p is injective (Voisin);

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Special cubics

Period map

Theorem

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Special cubics

Period map

Theorem

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Special cubics

Noether-Lefschetz locus

Dominance of period map implies

H⁴(X,Z)∩H^2,2(X,Z) =Zh² for X very general (1)

For each rank-2 saturated lattice

Zh² ⊂K ⊂I21,2

define (Hassett)

Q^K₀ :={ω∈Q0 |ω·K = 0}irreducible hypersurface in Q0, image D0^d ⊂D0: it depends only ond := disc(K).

The Noether-Lefschetz locus (where (1) does not hold) is p⁻¹ [

d∈Z

D0^d

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Special cubics

Noether-Lefschetz locus

H⁴(X,Z)∩H^2,2(X,Z) =Zh² for X very general (1) For each rank-2 saturated lattice

Zh² ⊂K ⊂I_21,2 define (Hassett)

d∈Z

D0^d

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Special cubics

Noether-Lefschetz locus

Q^K₀ :={ω∈Q0 |ω·K = 0}irreducible hypersurface in Q0,

image D0^d ⊂D0: it depends only ond := disc(K). The Noether-Lefschetz locus (where (1) does not hold) is

p⁻¹ [

d∈Z

D0^d

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Special cubics

Noether-Lefschetz locus

d∈Z

D0^d

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Special cubics

Noether-Lefschetz locus

d∈Z

D0^d

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Special cubics

Examples

1) IfX contains a plane P, one has h²·P = 1 andP²= 3: we are inQ^K₀ with K =

3 1 1 3

. The period is a general point of D₀⁸.

2) IfX is a Pfaffian cubic, it contains a quartic scrollT and T² = 10: we are inQ₀^K with K =

3 4 4 10

. The period is a general point ofD₀¹⁴.

So cubics whose period is a general point of D₀¹⁴ are rational. Infinitely many divisors in D₀⁸ correspond to rational cubics (Hassett), but one expects a general cubic containing a plane to be irrational.

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Special cubics

Examples

3 1 1 3

. The period is a general point of D₀⁸. 2) IfX is a Pfaffian cubic, it contains a quartic scrollT and T²= 10: we are in Q₀^K with K =

3 4 4 10

So cubics whose period is a general point of D₀¹⁴ are rational. Infinitely many divisors in D₀⁸ correspond to rational cubics (Hassett), but one expects a general cubic containing a plane to be irrational.

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Special cubics

Examples

3 1 1 3

3 4 4 10

So cubics whose period is a general point of D₀¹⁴ are rational.

Infinitely many divisors in D₀⁸ correspond to rational cubics (Hassett), but one expects a general cubic containing a plane to be irrational.

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Special cubics

Examples

3 1 1 3

3 4 4 10

So cubics whose period is a general point of D₀¹⁴ are rational.

Infinitely many divisors in D₀⁸ correspond to rational cubics (Hassett), but one expects a general cubic containing a plane to be irrational.

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Special cubics

Image of period map

One has:

D₀^d 6=∅iff d >0 andd ≡0,2 (mod 6) (Hassett),

image of p is D^BB₀ D²₀ D⁶₀ (Laza).

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Special cubics

Image of period map

One has:

D₀^d 6=∅iff d >0 andd ≡0,2 (mod 6) (Hassett), image of p is D^BB0 D²0 D⁶0 (Laza).

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Special cubics

Essential tool

F(X)⊂G(1,P⁵) smooth fourfold parametrizing lines inX

Theorem (Beauville-Donagi)

F(X) is an irreducible symplectic variety;

the incidence correspondance induces an isomorphism of polarized Hodge structures H⁴(X,Z)₀→^∼ H²(F(X),Z)₀(−1). Here, the groupH²(F(X),Z)0 is endowed with the Beauville quadratic form.

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Special cubics

Essential tool

F(X)⊂G(1,P⁵) smooth fourfold parametrizing lines inX Theorem (Beauville-Donagi)

the incidence correspondance induces an isomorphism of polarized Hodge structures H⁴(X,Z)₀→^∼ H²(F(X),Z)₀(−1). Here, the groupH²(F(X),Z)0 is endowed with the Beauville quadratic form.

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Special cubics

Essential tool

the incidence correspondance induces an isomorphism of polarized Hodge structures H⁴(X,Z)₀→^∼ H²(F(X),Z)₀(−1).

Here, the groupH²(F(X),Z)0 is endowed with the Beauville quadratic form.

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Special cubics

Essential tool

the incidence correspondance induces an isomorphism of polarized Hodge structures H⁴(X,Z)₀→^∼ H²(F(X),Z)₀(−1).

Here, the groupH²(F(X),Z)0 is endowed with the Beauville quadratic form.

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Special fourfolds Fibers of the period map

Unirationality

LetX be a smooth projective fourfold with Pic(X)'Z[H], where H is ample,H⁴ = 10, andKX ≡

lin−2H.

(In most cases,)X is the intersection ofG(2,V5)⊂P⁹ with a hyperplane and a quadric (Mukai).

Proposition

All (smooth) fourfolds of this type are unirational.

The construction of a double coverP⁴ 99KX is a bit involved but elementary.

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Unirationality

lin−2H.

Proposition

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Unirationality

lin−2H.

Proposition

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Unirationality

lin−2H.

Proposition

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Examples (Roth, Prokhorov)

1) IfX contains a plane P :=P(V1∧V4),

the projection from P X 99KP⁵

is birational onto its image, a smooth quadric: X is rational. 2) IfX contains aτ-quadric surface Σ (linear section ofG(2,V4)), the projection

X 99KP⁴ from Σ is birational: X is rational.

3) IfX contains a plane P :=G(2,V₃), the projection X 99KP⁵

fromP is birational onto its image, a smooth cubic containing a cubic scroll (expected to be irrational in general).

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Examples (Roth, Prokhorov)

1) IfX contains a plane P :=P(V1∧V4), the projection fromP X 99KP⁵

is birational onto its image, a smooth quadric: X is rational.

2) IfX contains aτ-quadric surface Σ (linear section ofG(2,V4)), the projection

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Examples (Roth, Prokhorov)

2) IfX contains aτ-quadric surface Σ (linear section ofG(2,V4)),

the projection

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Examples (Roth, Prokhorov)

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Examples (Roth, Prokhorov)

3) IfX contains a plane P :=G(2,V₃),

the projection X 99KP⁵

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Examples (Roth, Prokhorov)

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Period domain

X ⊂P⁸ as above.

“Vanishing” Hodge structure is of K3 type

H⁴(X,C)van = H^1,3(X) ⊕ H^2,2(X)0 ⊕ H^3,1(X)

dimensions 1 20 1

Lattices

H⁴(X,Z) ' I22,2

H⁴(X,Z)van ' 2E8⊕2U⊕2A1 =: Λ1

Period domain (20-dimensional, bounded symmetric domain of type IV)

Q1 :={ω∈P(Λ₁⊗C)|ω·ω = 0, ω·ω <¯ 0}

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Period domain

X ⊂P⁸ as above.

H⁴(X,C)van = H^1,3(X) ⊕ H^2,2(X)0 ⊕ H^3,1(X)

dimensions 1 20 1

Lattices

H⁴(X,Z) ' I22,2

H⁴(X,Z)van ' 2E8⊕2U⊕2A1 =: Λ1

Q1 :={ω∈P(Λ₁⊗C)|ω·ω = 0, ω·ω <¯ 0}

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Period domain

X ⊂P⁸ as above.

H⁴(X,C)van = H^1,3(X) ⊕ H^2,2(X)0 ⊕ H^3,1(X)

dimensions 1 20 1

Lattices

H⁴(X,Z) ' I22,2

H⁴(X,Z)van ' 2E8⊕2U⊕2A1 =: Λ1

Q1:={ω ∈P(Λ₁⊗C)|ω·ω = 0, ω·ω <¯ 0}

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Period map

“Moduli space” is a 24-dimensional irreducible stackN

with period map

p:N →D1 := Γ₁\Q1 where Γ₁ ⊂O(Λ₁) Period domainD1 is a quasi-projective variety with projective Baily-Borel compactificationD^BB1

Theorem

The map p is dominant with fibers smooth of dimension 4. Questions

1) What is the image? 2) What are the fibers?

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Period map

“Moduli space” is a 24-dimensional irreducible stackN with period map

p:N →D1 := Γ₁\Q1 where Γ₁ ⊂O(Λ₁)

Period domainD1 is a quasi-projective variety with projective Baily-Borel compactificationD^BB1

Theorem

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Period map

Theorem

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Period map

Theorem

The map p is dominant with fibers smooth of dimension 4.

Questions

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Period map

Theorem

Questions

1) What is the image?

2) What are the fibers?

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Period map

Theorem

Questions

1) What is the image?

2) What are the fibers?

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Noether-Lefschetz locus

Again,

H⁴(X,Z)∩H^2,2(X,Z) =H⁴(G(2,V₅),Z) for X very general.

H⁴(G(2,V5),Z)⊂K ⊂I22,2

define

Q^K₁ :={ω∈Q1 |ω·K = 0}irreducible hypersurface in Q1, image D₁^d ⊂D1: it depends only ond := disc(K), except when d ≡2 (mod 8), where it has two components D₁^0d and D₁^00d.

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Noether-Lefschetz locus

Again,

H⁴(G(2,V5),Z)⊂K ⊂I22,2

define

Q^K₁ :={ω∈Q1 |ω·K = 0}irreducible hypersurface in Q1, image D₁^d ⊂D1: it depends only ond := disc(K), except when d ≡2 (mod 8), where it has two components D₁^0d and D₁^00d.

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Noether-Lefschetz locus

Again,

H⁴(G(2,V5),Z)⊂K ⊂I22,2

define

Q^K₁ :={ω∈Q1 |ω·K = 0}irreducible hypersurface in Q1,

image D₁^d ⊂D1: it depends only ond := disc(K), except when d ≡2 (mod 8), where it has two components D₁^0d and D₁^00d.

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Noether-Lefschetz locus

Again,

H⁴(G(2,V5),Z)⊂K ⊂I22,2

define

Q^K₁ :={ω∈Q1 |ω·K = 0}irreducible hypersurface in Q1, image D₁^d ⊂D1: it depends only ond := disc(K),

except when d ≡2 (mod 8), where it has two components D₁^0d and D₁^00d.

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Noether-Lefschetz locus

Again,

H⁴(G(2,V5),Z)⊂K ⊂I22,2

define

Q^K₁ :={ω∈Q1 |ω·K = 0}irreducible hypersurface in Q1, image D₁^d ⊂D1: it depends only ond := disc(K), except when d ≡2 (mod 8), where it has two components D₁^0d and D1^00d.

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Examples

One has:

D₁^d 6=∅iff d >0 andd ≡0,2,4 (mod 8);

image of p meetsD₁^d for all such d ≥10. Examples

1) IfX contains a plane P :=P(V1∧V4), we are inQ^K₁ with K =





2 0 0 0 2 1 0 1 3



. The period is a general point of D₁⁰⁰¹⁰. 2) IfX contains aτ-quadric surface, we are in Q₁^K with K =





2 0 1 0 2 0 1 0 3



. The period is a general point of D1⁰¹⁰. 3) IfX contains a plane P :=G(2,V₃), the period is a general point ofD₁¹².

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Examples

One has:

D₁^d 6=∅iff d >0 andd ≡0,2,4 (mod 8);

image of p meetsD₁^d for all such d ≥10.

Examples





2 0 0 0 2 1 0 1 3







2 0 1 0 2 0 1 0 3



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Examples

One has:

D₁^d 6=∅iff d >0 andd ≡0,2,4 (mod 8);

Examples





2 0 0 0 2 1 0 1 3



. The period is a general point of D₁⁰⁰¹⁰.

2) IfX contains aτ-quadric surface, we are in Q₁^K with K =





2 0 1 0 2 0 1 0 3



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Examples

One has:

D₁^d 6=∅iff d >0 andd ≡0,2,4 (mod 8);

Examples





2 0 0 0 2 1 0 1 3







2 0 1 0 2 0 1 0 3



. The period is a general point of D1⁰¹⁰.

3) IfX contains a plane P :=G(2,V₃), the period is a general point ofD₁¹².

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Examples

One has:

D₁^d 6=∅iff d >0 andd ≡0,2,4 (mod 8);

Examples





2 0 0 0 2 1 0 1 3







2 0 1 0 2 0 1 0 3



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Image of the period map

Conjecture

The image ofp should be

D^BB₁ D²₁ D⁴₁ D⁸₁.

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Associated irreducible symplectic variety

ForX general,varietyF(X) of conics inX is smooth of dim. 5.

Canonical morphismα:F(X)→P(H⁰(P⁸,IX(2))^∨)'P⁵, with Stein factorization

α:F(X)→^β YeX

γ YX ⊂P⁵ Theorem

Ye_X is an irreducible symplectic fourfold (double EPW sextic). The incidence correspondance induces a factorization

H⁴(X,Z)van

→a H²(YeX,Z)0(−1) ^β

∗

,→H²(F(X),Z)(−1) where a is an isomorphism of polarized Hodge structures. The groupH²(Ye_X,Z)₀ is endowed with the Beauville quad. form.

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Associated irreducible symplectic variety

Canonical morphismα:F(X)→P(H⁰(P⁸,IX(2))^∨)'P⁵,

with Stein factorization

α:F(X)→^β YeX

γ YX ⊂P⁵ Theorem

H⁴(X,Z)van

→a H²(YeX,Z)0(−1) ^β

∗

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Associated irreducible symplectic variety

α:F(X)→^β YeX

γ YX ⊂P⁵

Theorem

H⁴(X,Z)van

→a H²(YeX,Z)0(−1) ^β

∗

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Associated irreducible symplectic variety

α:F(X)→^β YeX

γ YX ⊂P⁵

Theorem

H⁴(X,Z)van

→a H²(YeX,Z)0(−1) ^β

∗