Complex Lagrangians in a hyperKaehler manifold and the relative Albanese

Let $M$ be the moduli space of complex Lagrangian submanifolds of a hyperK\"ahler manifold $X$, and let $\varpi : \widehat{\mathcal{A}} \rightarrow M$ be the relative Albanese over $M$. We prove that $\widehat{\mathcal{A}}$ has a natural holomorphic symplectic structure. The projection $\varpi$ defines a completely integrable structure on the symplectic manifold $\widehat{\mathcal{A}}$. In particular, the fibers of $\varpi$ are complex Lagrangians with respect to the symplectic form on $\widehat{\mathcal{A}}$. We also prove analogous results for the relative Picard over $M$.


Introduction
A compact Kähler manifold admits a holomorphic symplectic form if and only if it admits a hyperKähler structure [Be], [Ya]. To explain this, let X be a compact manifold equipped with almost complex structures J 1 , J 2 , J 3 , and let g be a Riemannian metric on X, such that (X, J 1 , J 2 , J 3 , g) is a hyperKähler manifold. Then g defines a C ∞ isomorphism, T 0,1 X g 1 symplectic form Φ on X. It is known that Φ is parallel (meaning, covariant constant) with respect to the Levi-Civita connection on X associated to ω [Be,p. 760,"Principe de Bochner"], [YB,p. 142].
Let M denote the moduli space of compact complex submanifolds of X that are Lagrangian with respect to the symplectic form Φ. Consider the corresponding universal family of Lagrangians Z −→ M .
(1.1) Let ̟ : A −→ M be the relative Albanese over M. So for any Lagrangian L ∈ M, the fiber of A over L is Alb(L) = H 0 (L, Ω 1 L ) * /H 1 (L, Z). This ϕ is a holomorphic family of compact complex tori over M.
We prove the following (see Theorem 3.3): The complex manifold A has a natural holomorphic symplectic form.
The symplectic form on A is constructed using the canonical Liouville symplectic form on the holomorphic cotangent bundle T * M of M. The symplectic form Φ on X is implicitly used in the construction of the symplectic form on A. Recall that the Lagrangian submanifolds, and hence M, are defined using Φ.
We prove the following (see Lemma 3.4): The projection ̟ : A −→ M defines a completely integrable structure on the symplectic manifold A. In particular, the fibers of ̟ are Lagrangians with respect to the symplectic form on A.
In Section 4 we consider the relative Picard bundle over M for the family Z in (1.1). Let We prove that A is equipped with a natural holomorphic symplectic structure; see Proposition 4.3.
Let Θ A denote the above mentioned holomorphic symplectic structure on A. The following lemma is proved (see Lemma 4.4): The projection ̟ 0 : A −→ M defines a completely integrable structure on A for the symplectic form Θ A .
These results are natural generalizations of some known cases of integrable systems, such as the Hitchin system [Hi1] or the Mukai system [Mu] (see also [DEL]). One of our motivations has been mirror symmetry; we hope to come back to the study of A and A from the point of view of hyperKähler geometry.

Cotangent bundle of family of Lagrangians
Let X be a compact Kähler Ricci-flat manifold of complex dimension 2d equipped with a Kähler form ω. Let Φ be a holomorphic symplectic form on X which is parallel with respect to the Levi-Civita connection on X given by the Kähler metric on X associated to ω.
is the inclusion map.
It is known that the infinitesimal deformations of a complex Lagrangian submanifold of X are unobstructed [Mc], [Vo], [Hi2]. Furthermore, the moduli space of complex Lagrangian submanifolds of X is a special Kähler manifold [Hi2,p. 84,Theorem 3].
Let M be the moduli space of complex Lagrangian submanifolds of X. Let L ⊂ X be a complex Lagrangian submanifold. The point of M representing L will also be denoted by L. Let N L −→ L be the normal bundle of L ⊂ X; it is a quotient bundle of ι * T X of rank d, where ι is the map in (2.1). The infinitesimal deformations of the complex submanifold L are parametrized by H 0 (L, N L ). Since L is complex Lagrangian, the holomorphic symplectic form Φ on X produces a holomorphic isomorphism is the holomorphic tangent (respectively, cotangent) bundle of L. Using this isomorphism we have H 0 (L, N L ) = H 0 (L, Ω 1 L ) . Among the infinitesimal deformations of the complex submanifold L, there are those which arise from deformations within the category of complex Lagrangian submanifolds, meaning those arise from deformations of complex Lagrangian submanifolds as Lagrangian submanifolds. An infinitesimal deformation α ∈ H 0 (L, N L ) = H 0 (L, Ω 1 L ) of L lies in this subclass if and only if the holomorphic 1-form α on L is closed [Hi2,. But any holomorphic 1-form on L is closed because L is Kähler. Therefore, M is in fact an open subset of the corresponding Douady space for X.
Consider the Ricci-flat Kähler form ω on X. The form on L is also Kähler, where ι is the map in (2.1). Therefore, the pairing is nondegenerate. We shall identify H 1 (L, O L ) with H 0 (L, Ω 1 L ) * using this nondegenerate pairing.
It was noted above that Using the pairing in (2.3) and (2.4), we have be the universal family of complex Lagrangians over M. So Z is the locus of all (x, L ′ ) ∈ X × M such that x ∈ L ′ ⊂ X. Consider the natural projections p X : X × M −→ X and p M : be the restrictions of p X and p M respectively to the submanifold Z ⊂ X × M. So p(q −1 (L ′ )) ⊂ X for every L ′ ∈ M is the Lagrangian L ′ itself.
Let Ω 1 Z/M −→ Z be the relative cotangent bundle for the projection q to M in (2.7). It fits in the short exact sequence of holomorphic vector bundles where C is the constant sheaf on Z with stalk C. The direct image R 1 q * C is a flat complex vector bundle, equipped with the Gauss-Manin connection. We briefly recall the construction of the Gauss-Manin connection. For any point L ′ = y ∈ M, let U y ⊂ M be a contractible open neighborhood of y. Since U y is contractible, the inverse image q −1 (U y ) is diffeomorphic to U y × L ′ such that the diffeomorphism between q −1 (U y ) and U y × L ′ takes q to the natural projection from U y × L ′ to U y . Using this diffeomorphism, the restriction of R 1 q * C to U y coincides with the trivial vector bundle with fiber H 1 (L ′ , C). Using this isomorphism between (R 1 q * C)| Uy and the trivial vector bundle U y × H 1 (L ′ , C) in (2.9), the trivial connection on the trivial vector bundle in (2.9) produces a flat connection on (R 1 q * C)| Uy . This connection on (R 1 q * C)| Uy does not depend on the choice of the diffeomorphism between q −1 (U y ) and U y × L ′ . Consequently, these locally defined flat connections on R 1 q * C patch together compatibly to define a flat connection on R 1 q * C. This flat connection is the Gauss-Manin connection mentioned above.
Since the Gauss-Manin connection on R 1 q * C is flat, and M is a complex manifold, the Gauss-Manin connection produces a natural holomorphic structure on the C ∞ vector bundle R 1 q * C. The direct image R 0 q * Ω 1 Z/M is a holomorphic subbundle of R 1 q * C, but it is not preserved by the flat connection in general. The holomorphic structure on the quotient R 1 q * O Z induced by that of R 1 q * C coincides with its own holomorphic structure; the fiber of Since ω L in (2.2) is the restriction of a global Kähler form on X, the section of is covariant constant with respect to the Gauss-Manin connection on R 2 q * C. Consequently, the homomorphism is also covariant constant with respect to the connection on (R 1 q * C) ⊗ (R 1 q * C) induced by the Gauss-Manin connection on R 1 q * C. Hence the pairing in (2.10) produces a holomorphic isomorphism of vector bundles On the other hand, the pointwise isomorphisms in (2.4) combine together to produce a holomorphic isomorphism of vector bundles on M. Composing the isomorphisms in (2.11) and (2.12), we obtain a holomorphic isomorphism of vector bundles For notational convenience, the total space of R 1 q * O Z will be denoted by Y. Let be the natural projection. Consider the canonical Liouville 1-form on Ω 1 M . Using the isomorphism χ in (2.13) this Liouville 1-form on Ω 1 M gives a holomorphic 1-form on Y. Let be the holomorphic 1-form given by the Liouville 1-form on Ω 1 M . Note that for any L ∈ M, and any v ∈ γ −1 (L), the form coincides with the composition of homomorphisms where φ L is the bilinear pairing constructed in (2.3), and dγ is the differential of the projection γ in (2.14); the above identification is the one constructed in (2.4).
It is straight-forward to check that the 2-form is a holomorphic symplectic form on the manifold Y in (2.14). Indeed, dθ evidently coincides with the 2-form on Y given by the Liouville symplectic form on Ω 1 M via the isomorphism χ in (2.13).

The family of Albanese tori
Take a compact complex Lagrangian submanifold L ⊂ X represented by a point of M. We know that T * L M ∼ = H 0 (L, Ω 1 L ) * (see (2.4)). Note that the non-degenerate pairing which is also the Serre duality pairing, yields an isomorphism For a fixed L, we have the Hodge decomposition but if we move L in the family M, meaning if we consider the universal family q : Z −→ M in (2.6), then only R d−1 q * Ω d Z/M is a holomorphic subbundle of R 2d−1 q * C, and we have the short exact sequence of holomorphic vector bundles The holomorphic vector bundle R 2d−1 q * C is equipped with the Gauss-Manin connection, which is an integrable connection. The quotient is covariant constant with respect to the connection on (R 1 q * C) ⊗ (R 2d−1 q * C) induced by the Gauss-Manin connections on R 1 q * C and R 2d−1 q * C. Consequently, the pairing in (3.2) yields a holomorphic isomorphism of vector bundles Combining this isomorphism with the isomorphism in (2.12) we get a holomorphic isomorphism of vector bundles We shall denote the total space of the holomorphic vector bundle R d q * Ω d−1 Z/M by W, so is a holomorphic fiber bundle.
As in Section 2, consider the canonical Liouville holomorphic 1-form on the total space of Ω 1 M . Using the isomorphism in (3.3), this Liouville 1-form on the total space of Ω 1 M produces a holomorphic 1-form on W in (3.4). Let θ ′ ∈ H 0 (W, Ω 1 W ) be this holomorphic 1-form on W. We note that is a holomorphic symplectic form on W. Indeed, the isomorphism in (3.3) takes dθ ′ to the Liouville symplectic form on the total space of Ω 1 M .
Remark 3.1. Note that while the Kähler form ω on X was used in the construction of the symplectic form dθ on Y in (2.16) (see the pairing in (2.10)), the construction of dθ ′ in (3.5) does not use the Kähler form ω on X. We recall that the isomorphism in (2.12) is constructed from the the pointwise isomorphisms in (2.4). Note that the isomorphism in (2.4) does not depend on the Kähler form ω.
The Albanese Alb(Y ) of a compact Kähler manifold Y is defined to be [GH,p. 331]). It is a compact complex torus.
For each point L ∈ M, consider the composition of homomorphisms where W is defined in (3.4).
From this it follows immediately that the C ∞ submanifold R 2d−1 q * Z ⊂ W in (3.6) is in fact a complex submanifold.

The quotient
for the homomorphism in (3.6) is in fact a holomorphic family of compact complex tori over M. Note that the fiber of A over each L ∈ M is the Albanese torus For any complex Lagrangian L ∈ M, using Serre duality, Using Poincaré duality for L, we have H 2d−1 (L, Z)/Torsion = H 1 (L, Z)/Torsion. Consequently, A in (3.7) admits the following isomorphism: where H 1 (Z) is the local system on M whose stalk over any L ∈ M is H 1 (L, Z)/Torsion. Also we have the isomorphism of real tori where H 1 (R) is the local system on M whose stalk over any L ∈ M is H 1 (L, R).
such that the fiber of the S 1 -bundle B over L 0 represents the homology class c in (3.10); recall that the fiber of Z over the point L 0 ∈ M is the Lagrangian L 0 itself.
Consider the symplectic form Φ on X. Integrating ι * U p * Re(Φ) along the fibers of B, where ι U and p are the maps in (3.11) and (2.7) respectively, we get a closed 1-form ξ c on U. Let f c be the unique function on U such that f c (L 0 ) = 0 and df c = ξ c . Now, let be the function uniquely determined by the condition that φ Lx (c ⊗ µ(x)) = f c (x) for all x ∈ U and c ∈ H 1 (L 0 , Z)/Torsion, where φ Lx is the pairing constructed as in (2.3) for the complex Lagrangian L x = q −1 (x) ⊂ X, where q is the projection in (2.7). This µ is a local diffeomorphism [Hi2,p. 79,Theorem 2].
Using the Kähler form ω L 0 := ω| L 0 on L 0 (see (2.2)), we identify H 1 (L 0 , R) with H 1 (L 0 , R) as follows. Since the pairing (3.12) On the other hand, there is the natural homomorphism be the subgroup that corresponds to H 1 (L 0 , Z) by the isomorphism in (3.12).
Using the above coordinate function µ on U, we have where T * denotes the real cotangent bundle.
The Liouville symplectic form on T * µ(U) is clearly the constant 2-form on given by the natural isomorphism of H 1 (L 0 , R) with H 1 (L 0 , R) * . From this it follows immediately that is a Lagrangian submanifold with respect to the Liouville symplectic form on T * µ(U), where Γ defined in (3.13).
It was noted in Remark 3.2 that R 2d−1 q * Z is a complex submanifold of W. Consequently, the 2-form on R 2d−1 q * Z obtained by restricting the holomorphic 2-form dθ ′ on W is also holomorphic. Since the real part of the holomorphic 2-from on R 2d−1 q * Z given by dθ ′ vanishes identically, we conclude that the holomorphic 2-from on R 2d−1 q * Z, given by dθ ′ , itself vanishes identically. Therefore, R 2d−1 q * Z is a Lagrangian submanifold of the holomorphic symplectic manifold W equipped with the holomorphic symplectic form dθ ′ .
To complete the proof we recall a general property of the Liouville symplectic form.
Let N be a manifold and α a 1-form on N. Let t : T * N −→ T * N be the diffeomorphism that sends any v ∈ T * n N to v + α(n). If ψ is the Liouville symplectic form on T * N, the t * ψ = ψ + dα .
In particular, the map t preserves ψ if and only if the form α is closed. Also, the image of t is Lagrangian submanifold of T * N for ψ if and only α is closed.
Since R 2d−1 q * Z is a Lagrangian submanifold of the symplectic manifold (W, dθ ′ ), from the above property of the Liouville symplectic form it follows immediately that the 2-form dθ ′ on W descends to the quotient space A in (3.7). Let be the quotient map (see (3.7)). From Theorem 3.3 we know that there is a unique 2-form where q A is the map in (3.14). Since dθ ′ is a holomorphic symplectic form, it follows immediately that Θ A is a holomorphic symplectic form on A.
The projection γ : W −→ M in (3.4) clearly descends to a map from A to M. Let be the map given by γ; so we have where q A is constructed in (3.14).
Lemma 3.4. The projection ̟ in (3.17) defines a completely integrable structure on A for the symplectic form Θ A constructed in (3.15). In particular, the fibers of ̟ are Lagrangians with respect to the symplectic form Θ A .
Proof. Recall that W is holomorphically identified with Ω 1 M by the map χ in (3.3) (see (3.4)). This map χ takes the Liouville symplectic form on Ω 1 M to the symplectic form dθ ′ on W. Therefore, from (3.14) and (3.16) we conclude that A is locally isomorphic to Ω 1 M such that the projection ̟ is taken to the natural projection Ω 1 M −→ M, and the symplectic form Θ A on A is taken to the Liouville symplectic form on Ω 1 M . The lemma follows immediately from these, because the natural projection Ω 1 M −→ M defines a completely integrable structure on Ω 1 M for the Liouville symplectic form.

The relative Picard group
For any L ∈ M, consider the homomorphisms where H 1 (L, Z) −→ H 1 (L, C) is the natural homomorphism given by the inclusion of Z in C, and the projection H 1 (L, C) −→ H 1 (L, O L ) corresponds to the isomorphism H 1 (L, C)/H 0 (L, Ω 1 L ) = H 1 (L, O L ) (see (2.8)). The image of the composition of homomorphisms in (4.1) is actually a cocompact lattice in H 1 (L, O L ); so H 1 (L, O L )/H 1 (L, Z) is a compact complex torus. We note that the composition of homomorphisms in (4.1) is in fact injective. When the compact complex manifold L is a complex projective variety, then H 1 (L, O L )/H 1 (L, Z) is in fact an abelian variety.
As L moves over the family M, these cocompact lattices fit together to produce a C ∞ submanifold of the complex manifold Y in (2.14).
The Gauss-Manin connection on R 1 q * C −→ M evidently preserves the above bundle of cocompact lattices R 1 q * Z ⊂ R 1 q * C. From this it follows immediately that the above C ∞ submanifold R 1 q * Z ⊂ Y is in fact a complex submanifold.
Taking fiber-wise quotients, we conclude that is a holomorphic family of compact complex tori over M.
Remark 4.1. Let Y be a compact Kähler manifold. Consider the short exact sequence of sheaves on Y given by the exponential map Recall that we have the holomorphic symplectic form dθ on Y, where θ is constructed in (2.15).
Remark 4.2. Recall from Remark 3.1 that dθ does depend on the Kähler form ω on X.
Proposition 4.3. The 2-form dθ on Y descends to the quotient space A in (4.2), or in other words, dθ is the pullback of a 2-form on A.
Proof. The proof of the proposition is very similar to the proof of Theorem 3.3. As before, the local coordinate functions on M constructed in [Hi2] play a crucial role. We omit the details of the proof.
Let q A : Y −→ A := Y/(R 1 q * Z) (4.3) be the quotient map (see (4.2)). Let Θ A ∈ H 0 (A, Ω 2 A ) be the holomorphic symplectic form given by Proposition 4.3, so where q A is the map in (4.3).