On periods of Gushel–Mukai Varieties

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On periods of Gushel–Mukai Varieties

Olivier DEBARRE (joint with Alexander KUZNETSOV)

Ecole normale sup´´ erieure

Carry-le-Rouet, mai 2016

Olivier DEBARRE On periods of Gushel–Mukai Varieties 1 of 17

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Definition of GM varieties

A Gushel–Mukai variety of dimensionn is (ordinary type) a transverse intersection

X :=G(2,V5)∩P(Wn+5)∩Q ⊂P(V₂ V5)

(special type) a double cover

X −→M :=G(2,V₅)∩P(W_n+4)⊂P(V₂ V₅) branched along M∩Q.

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Characterization of GM varieties

One hasKX ≡

lin(2−n)H andHⁿ= 10. GM manifolds are:

(n=1) genus-6 curves with Clifford index 2 (not hyperelliptic, not trigonal, not a plane quintic), special ⇔bielliptic;

(n=2) degree-10 K3 surfaces whose polarization contains a genus-6 curve with Clifford index 2, special ⇔hyperelliptic;

(n=3,4,5,6) prime Fano n-folds of degree 10, coindex 3;

(n=6) only special case occurs.

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Hodge structures of GM manifolds

The integral cohomology is torsion-free and the upper Hodge diamonds are

(n = 1) (n= 2) (n= 3) (n= 4)

616

0 1 0 1 20 1

0 1 0

0 10 10 0

0 1 0

0 0 0 0

0 1 22 1 0

(n= 5) (n = 6)

0 1 0

0 0 0 0

0 0 2 0 0

0 0 10 10 0 0

0 1 0

0 0 0 0

0 0 2 0 0

0 0 0 0 0 0

0 0 1 22 1 0 0

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Period maps for GM manifolds

Whenn is odd, the Hodge structure has level 1 and there are (p.p.) intermediate Jacobians and period maps

℘₁:M1−→A6 , ℘₃:M3−→A10 , ℘₅:M5−→A10. Whenn is even, the Hodge structure is of K3 type and the vanishing cohomology defines period maps

℘₂:M2 −→D19 , ℘₄:M4 −→D20 , ℘₆:M6−→D20, whereDm is the quasi-projective quotient of a bounded symmetric domain of dimensionm by a discrete group of automorphisms.

We know that℘1 and℘2 are closed embeddings. What about the other maps?

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Period maps for GM manifolds

The dimensions are

n 1 2 3 4 5 6

dim(Mn) 15 19 22 24 25 25 dim (period space) 21 19 55 20 55 20 We will show that℘_n is not injective forn ≥3 and describe its fibers.

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The magic trick (O’Grady, Iliev–Manivel)

Given a smooth (ordinary) GMn-fold

X :=G(2,V5)∩P(Wn+5)∩Q ⊂P(V₂ V5), one can consider and construct

the space V₆ of quadrics in P(W_n+5) containingX; the hyperplane V₅ ⊂V₆ of “Pl¨ucker quadrics”;

a Lagrangian subspace A⊂V₃

V6 such that dim(V₃

V5∩A) = 5−n.

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Parametrization of GM manifolds

This construction can be reversed and gives, forn≥3, a bijection (V₆ is a fixed 6-dimensional vector space)

smooth ordinary GMn-folds

. isom.

LagrangianA⊂V₃ V₆ with no decomposable

vectors and hyperplane

V₅⊂V₆ such that dim(V₃

V5∩A) = 5−n .

PGL(V₆)

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EPW stratification

Given a LagrangianA⊂V₃

V6, we define the (dual) EPW stratification

Y_A^`⊥ :={[V₅]∈P(V₆^∨)|dim(V₃

V5∩A) =`}.

We deduce from the correspondence above a morphism

moduli space Mn^ord

of smooth ordinary GM n-folds

pn //

moduli spaceE of LagrangiansA⊂V₃

V6

with no decomposable vectors

where the fiber of [A] is Y_A⁵⁻ⁿ⊥ (modulo automorphisms).

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EPW sextics

Theorem (O’Grady)

If the Lagrangian subspace A⊂V₃

V₆ contains no decomposable vectors,

Y_A^⊥:=Y_A^≥1⊥ ⊂P(V₆^∨) is an integral sextic hypersurface;

Y_A^≥2⊥ = Sing(Y_A^⊥)is an integral normal surface;

Y_A^≥3⊥ is finite and smooth, empty for A general;

Y_A^≥4⊥ is empty.

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Putting everything together, we have

p5:M5^ord→E, fibers are complements of hypersurfaces inP⁵ (both M₅^ord andE are affine);

p4:M4^ord→E, fibers are complements of surfaces in hypersurfaces inP⁵;

p₃:M₃^ord→E, fibers are surfaces (minus a finite set).

Ignoring stacky issues, these maps fit together and yield morphisms pn:Mn=M_nôrdtM_n^spe =M_nôrdtMn−1ôrd −→E.

Corollary

For each n∈ {3,4,5}, there exist non-isotrivial families of smooth GM varieties of dimension n parametrized by a proper curve.

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Shafarevich conjecture and hyperbolicity

The Shafarevich conjecture does not hold for Fano threefolds:

the set of ¯Q-isomorphism classes of Fano threefolds with Picard number 2, defined overQ and with good reduction outside {2,29}, is infinite (the moduli space of blow ups of any line in a smooth intersection of two quadrics inP⁵_Q contains abelian surfaces; Javanpeykar–Loughran). What about GM manifolds?

This is related (via the Lang–Vojta conjecture) to

hyperbolicity of moduli spaces: do they contain entire curves?

In our case, M4 andM5 do contain many (proper) rational or elliptic curves (M6 is affine). This is in contrast with a result of Viehweg–Zuo, which says that this cannot happen with moduli space of smooth projective varieties withample

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Main theorem

Theorem (D.–Iliev–Manivel, D.–Kuznetsov)

The period map℘n for GM manifolds of dimension n∈ {3,4,6}

factors through p_n:Mn→E. In particular, it has positive dimensional fibers.

Whenn= 3 ([DIM]): explicit birational isomorphisms between GM threefolds with sameA.

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Periods of double EPW sextics

There is a canonical double coverYe_A^⊥ →Y_A^⊥. Theorem (O’Grady)

If A has no decomposable vectors and Y_A^≥3⊥ =∅, the fourfoldYe_A^⊥ is a HK manifold of K3^[2]-type.

Theorem (D.–Kuznetsov)

If X is a GM manifold of dimension n∈ {4,6}, with Lagrangian A, there is an isomorphism of polarized Hodge structures

Hⁿ(X;Z)₀₀∼=H²(Ye_A^⊥;Z)₀((−1)^n/2−1).

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Periods of double EPW sextics

Corollary (D.–Kuznetsov)

When n∈ {4,6}, there is a factorization

℘_n:Mn pn

−−→E −−→^℘ D20, where the period map℘ is an open embedding.

The last statement is a theorem of Verbitsky.

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Method of proof

We consider the dual situationYe_A→Y_A ⊂P(V6) and the inverse imageYe_A,V₅⊂Ye_A of the hyperplane V₅ ⊂V₆.

As for cubic fourfolds, the isomorphism in the theorem is given by a correspondence, using

when n= 4, the (smooth) variety of lines inX, a small resolution ofYe_A,V₅;

when n= 6, the (smooth) variety ofσ-planes in X, a P¹-bundle over Ye_A,V₅.

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Birationalities

Theorem (D.–Iliev–Manivel, D.–Kuznetsov)

Any GM manifolds of the same dimension with isomorphic associated Lagrangians are birationally isomorphic.

In particular, the rationality of a GM manifold only depends on its associated Lagrangian, hence on its period point.

Whenn= 3, a general GM manifold is not rational (use intermediate Jacobian).

Whenn∈ {5,6}, all GM manifolds are rational.

Whenn= 4, the situation is analogous to that of cubic fourfolds:

some rational examples are known, but one expects very general GM fourfolds to be irrational.