Dual fibration of a projectiveLagrangian fibration

Yasunari Nagai

(the University of Tokyo)

November 2005

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Irreducible symplecticmanifolds

Definition 1. n: a positive integerA compact Kahler manifold X of dimension 2n

is irreducible symplectic if(i) X has a holomorphic symplectic form σ,i.e. σ is a holomorphic 2-formsuch that σ∧n is nowhere vanishing.(ii) H0(X,Ω2

X) = C · σ(iii) X is simply connected.

Example 2. 2 dim. irreducible symplectic = K3

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Very few examples of irreducible symplecticmanifolds.

Example 3. (i) (Fujiki, Beauville) S : K3.X = Hilbn(S) : the moduli space of 0-dim.subschemes of length n.=⇒ X is an irreducible symplectic manifold ofdimension 2n. (Hilbert type)More generally, a compact connected modulispace of stable coherent sheaves on a K3surface is an irred. sympl. mfd, which isdeformation equivalent to Hilbn(K3)(Mukai, Huybrechts, Yoshioka...).

(ii) (Beauville) A : Abelian surface.Y = Hilbn+1(A)→ A : Albanese morphismThis family is isotrivial and the fiberX = Kumn(A) is an irred. sympl. mfd of dim. 2n.(Generalized Kummver variety)

(iii) O’Grady’s Examples (also related to modulispace of semi-stable sheaves on K3 or Abeliansurfaces).

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Fiber space

Important to understand possible fiber spacestructures on an irreducible symplecticmanifold.

Definition 4. X : a 2n dim. complex manifoldwith a hol. symplectic form σX

(i.e., non-degenerate hol. 2-form σX withdσX = 0).A morphism f : X→ S is a Lagrangian fibrationif f is of relative dimension n and σX|F = 0 forany irreducible component F of any fiber of f .

Theorem 5 (Matushita). X : a projectiveirreducible symplecticf : X→ S : a morphism with a projective baseS such that f∗OX = OS and 0 < dimS < dimX.=⇒ f is a Lagrangian fibration and its generalfiber is an Abelian variety.

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Examples of Lagrangian fibration

Example 6. S→ P1 : an elliptic K3.=⇒ the compositionHilbn(S)→ Symn(S)→ Symn(P1) Pn

is a Lagrangian fibration.

Example 7. S : K3 containinga smooth curve C of genus g.Zd : moduli space of stable sheaves F

with suppF ∈ |C| and χ(F) = 1− g + d.=⇒ the natural morphismZd → |C| Pg is a Lagrangian fibration.(compactified relative Picard space)

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∃ examples of Lagrangian fibration onirreducible sympelctic manifolds of generalizedKummer type and O’Grady’s examples.

In any of these examples, the base of thefibration is a projective space.

Problem . X irred. sympl. of dim 2n.f : X→ S : a Lagrangian fibration.=⇒ S Pn?

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Theorem 8(Cho–Miyaoka–Shepehered-Barron). X : 2ndim projective irred. sympl.f : X→ S : a Lagrangian fibrationwith a global sectionand the projective base S.=⇒ S Pn.

Remark . In the Example 7,Z0 has a global section,while Zd (d < (2g− 2)Z) does not have one.

=⇒We want to consider Lagrangian fibrationswhich have no global sections.

My approach : consider the dual fibration.

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Dual fibration

Basing on Raynaud’s construction.

f : X→ S : projective morphism satisfying

Assumption (R). (i) f is flat(ii) f∗OX = OS

(iii) f has analytic local sections.

Under this assumption (R),∃ the analytic/algebraic space PicX/S → S

which represents the relative Picard functor(Grothendieck, Artin).

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F : general fiber of f .P ⊂ PicX/S connected componentwhich contains a point [OF ].P→ S is a group space (group object in thecategory of analytic/algebraic spaces).

Remark . If f has a reducible fibre,P is not separated in general.

E = the closure of a section induced by OX.E ⊂ P is a group subspace.Q = P/E→ S : a separated group space.We call it the maximal separated quotient ofP→ S.

Definition 9. If the general fiber of f : X→ S

an Abelian variety (an Abelian fibration)satisfying Assumption (R),we define the dual fibraion of f

as Q→ S defined above.

Note that the dual fibration has a canonicalglobal section.

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One of the important feature of the dualfibration is the Neron mapping property.

Proposition 10 (Neron mapping property).f : X→ S : projective morphism satisfying thecondition (P).∆ ⊂ S proper closed subset s.t.f is smooth over S0 = S\∆.g : Y→ S : a smooth morphismπ : Q→ S : the biggest separeted quotient ofthe Picard space.

=⇒ S0-morphism YS0 → QS0 extends uniquelyto an S morphism Y→ Q.

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Main Theorem . X a complex manifold,σX a holomorphic symplectic form on X,f : X→ S a projective Lagrangian fibration.Assume (i) f has analytic local sections(ii) f∗OX = OS

(iii) R1 f∗OX is locally free.

=⇒ (1) the dual fibration π : Q→ S

is a smooth group space.(2) ∃ a natural symplectic form σQ on Q.(3) π : Q→ S is Lagrangian w.r.t. σQ in (2).

Remark . a Lagrangian fibration isequidimensional (Matsushita).By the condition (i), S is smooth so that f isflat. Therefore we can consider the dualfibration Q→ S in the theorem.

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As an application, we have a slightgeneralization ofCho–Miyaoka–Shephered-Barron’s result.

Collorary 11. X : a 2n dim. projectiveirreducible symplectic.f : X→ S : a Lagrangian fibrationadmitting analytic local sections with theprojective base S.=⇒ S Pn.

Proof. ApplyCho–Miyaoka–Shephered-Barron’s argumentto our dual fibration π : Q→ S.

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Related results

(1) (Mukai, ’84) S : K3 or Abelian.M a moduli space of simple sheaves on S.=⇒ M is smooth and admits a naturalsymplectic form.(Proof: cohomological, Serre duality)

(2) (Kobayashi, ’86)X : compact symplectic Kahler.M = E | simple vector bundle withH2(End0(E)) = 0=⇒ ∃ natural symplectic form on M.(Proof: holomorphic symplectic reduction)

(3) (Verbitski, ’92)X : compact hyper-Kahler manifoldM : moduli space of hyper-complex bundles=⇒ M is a hyper-complex space.(Proof: Hodge theory on hyper-complexbundles)

(4) Donagi–Markman’s result

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Comments on the Main Theorem

(1) In Raynaud’s construction of the dualfibration π : Q→ S, π is not proper.We get a smooth space as compensation.

(2) “Compactified dual fibration” will havesingularities on the boundary.Analysis of the singularities seems to becomplicated for the time being.

(3) No idea to handle Lagrangian fibrationswith multiple fibres.

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(4) Our proof of the existence of a symplecticform is geometric.There should be a more conceptual proof.(←→ (3)).

F : a fiber of a projective Lagrangian f : X→ S,and L be a line bundle on F.We can consider the alternating biliner form

Ext1OX(L,L) × Ext1OX

(L,L)∪−→ Ext2OX

(L,L)

tr−→ H2(X,OX) C

as in Mukai’s argument.Is this non-degenerate?...

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... seems complicated.

Example 12 (Donagi–Markman). A : Abelianvariety of even dimension > 4

Y ⊂ A : a smooth ample divisors.t. dim |Y| is odd.M : connected component of the moduli ofstable sheaves containing OY.=⇒ dim M = 2 dimA + dim |Z| is odd.a fortiori, M does not admit a symplectic form.

The existence of a symplectic form on the dualshould be strongly related to the structure ofLagrangian fibrations.

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F : a smooth fiber of Lagrangian f : X→ S.L : a line bundle on F.Consider the spectral sequence

Epq2 = Hp(X,Extq

OX(L,L)) =⇒ Extp+q

OX(L,L).

Koszul resolution implies ExtqOX

(L,L) ∧qNF/X

and the symplectic form induces ∧qNF/X ΩqF

since F is Lagrangian.

E.g. we knowExt1OX

(L,L) = H1(F,OF) ⊕ H0(F,Ω1F)

Ext2OX(L,L) → H2(F,OF)⊕H1(F,Ω1

F)⊕H0(F,Ω2F)

etc.

=⇒ the data of deformation is encoded in thisspectral sequence... ?

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Idea of the proof of Main Theorem

f : X→ S as in Main Theorem,and let ∆ ⊂ S the discriminant locus of f .∆0 = the smooth points of ∆.

STEP1 Consider the case f : X→ S issmooth and admitting a global section(take a regular value s < ∆ and replace S by asmall enough neighbourhood of s).

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H = f∗HomOX(Ω1X/S,OX)

Λ =⋃

s∈S H1(Xs,Z) : family of lattices in H.=⇒ X H/Λ (by the global section).

f -ample line bundle L

⇒ non-degenerate alternatingbilinear E : Λ ×S Λ→ ZS.

Matrix representation E =

0 δ

−δ 0

where δ = diag(d1, . . . , dn), di ∈ Z,and e(δ) = l.c.m.(d1, . . . , dn).Let Λ = l ∈ Λ ⊗ R | E(l, λ) ∈ e(δ)Z.Then H ⊃ Λ ⊃ Λ.

Obviousely the dual Q H/Λ and we have anatural finite etale morphism

ϕL : Q→ X

over S.Just take σQ = ϕ∗LσX, then σQ is a symplecticform on Q. Cleary π : Q→ S is Lagrangianw.r.t. σQ.

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STEP2 Patching.Taking S1,S2 ⊂ S\∆ small enough,∃ si : Si → X a Lagrangian section of f over Si .(STEP1)⇒ we have a finite etale morphismϕi : QSi → XSi using si .

QS1∩S2ϕ1

zzuuuu ϕ2

$$IIII

XS1∩S2

ψ12 // XS1∩S2

.

ψ12 is a translation by s2 regarding s1 as0-section.

Since f is Lagrangian fibration,

ψ∗12σX + f ∗s∗1σX − f ∗s∗2σX = σX,

where the second equality follows from the factthat si is Lagrangian. This implies ϕ∗1σX andϕ∗2σX can be patched.

=⇒ ∃ symplectic form σQ on QS\∆.

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STEP3 Take s ∈ ∆0.Assume ∆ = ∆0 and f admits a global sectionby shrinking S.Furthermore, Assume f is a toroidaldegeneration along ∆0.

The condition that f is Lagrangian⇒ the degree of unipotency of the monodromymust be 6 2 (Matsushita).

Let D a unit disc and D0 a punctured disc.Assume S Dn and S0 = S\∆ = D0 × Dn−1.M0 := Cn−1 × C∗ × D0 × Dn−1.

Periods induces an action of Γ = Zn−1 on M0

and we have X0 = f −1(S0) M0/Γ.

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As in the smooth case, we can construct thedualized action of Γ = Zn−1 on M and weconstruct X = M/Γ.

Since KX/S is trivial by the construction,

X] = x ∈ X | f is smooth at x → S

has the Neron mapping property (⇐ MMP).=⇒ Q X].

f -ample line bundle induces a finite etalemorphism

ϕL : Q = X] → X],

extending ϕL on the smooth part. Therefore σQ

on the smooth part extends to the singular fiberover ∆0 and π : Q→ S is Lagrangian w.r.t. σQ.

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STEP4 “Additive reduction”Take s ∈ ∆0.Assume ∆ = ∆0 and f admits a global sectionby shrinking S.Now consider the general degeneration over ∆.

Matsushita showed that for some cycliccovering T → S = T/G (G : finite cyclic group),the following explicit reduction process:

Wαzz

zz β

ÁÁ>>>

>

Xν1 //

f ²²

X/G

²²

Xf

²²T

ν2 // T/G S

where f : X→ T is smooth or toroidaldegeneration, α is a family of minimalresolution of surface quotinent singularities onX/G and β is birational.

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We can dualize whole the reduction processand chasing the following diagram,

Wβ

¢¢¤¤¤¤ α

ÃÃBBBB

Wα||

|| β

ÀÀ;;;

;

Xf ²²

ˆX/G

²²

ˆXν1oo

ˆf²²

ϕL // Xν1//

f ²²

X/G

²²

Xf²²

S T/G Tν2oo T

ν2// T/G S

we can extend the symplectic form σQ on thesmooth part to the singular fiber over ∆0.π : Q→ S is Lagrangian w.r.t. σQ.

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STEP5 Final Step.∆ is the discriminant locus of f .Let S′ = S\∆sing.By the arguments in STEPs 1-4,We have a natural symplectic form σQ onQS′ = π−1(S′).

Since Q is smooth and codimQ\QS′ > 2,σQ on QS′ extends to the whole Q. π : Q→ S isautomatically Lagrangian since so is πQ′ and πis smooth.

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Further interests

• Compactification of the dual fibration.

• Derived equivalence between theLagrangian fibration and its dual ...involving the compactification and themoduli of twisted sheaves.(cf. Sawon’s works)

• Multiple fibres, approach by cohomologicalmethods, e.g. Hodge theory.

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