By means of a Fourier-Mukai transform we embed moduli spaces of stable
bundles on an algebraic curve C as isotropic subvarieties of moduli spaces of
mu-stable bundles on the Jacobian variety J(C). When g(C)=2 this provides new
examples of special Lagrangian submanifolds.Comment: 9 pages, no figures, to appear in Diff. Geom. App

F. Pioli

Harvey

Hilton

Hitchin

Itoh

Kobayashi

McLean

Mukai

Newstead

Simpson

U. Bruzzo

English

arXiv

AbstractBy means of a Fourier–Mukai transform we embed moduli spaces MC(r,d) of stable bundles on an algebraic curve C of genus g(C)≥2 as isotropic subvarieties of moduli spaces of μ-stable bundles on the Jacobian variety J(C). When g(C)=2 this provides new examples of special Lagrangian submanifold

Bruzzo, U.

Pioli, F.

Elsevier - Publisher Connector 

Complex Lagrangian embeddings of moduli spaces of vector bundles 

Differential Geometry and its Applications 14 (2001) 151–156 151
North-Holland
www.elsevier.com/locate/difgeo
Complex Lagrangian embeddings of moduli
spaces of vector bundles
U. Bruzzo1 and F. Pioli2
Dipartimento di Matematica, Universita degli Studi di Genova, Via Dodecaneso 35, 16146 Genova, Italy
Communicated by B.A. Dubrovin
Received 12 December 1999
Abstract: By means of a Fourier–Mukai transform we embed moduli spaces MC (r, d) of stable bundles on
an algebraic curve C of genus g(C)  2 as isotropic subvarieties of moduli spaces of µ-stable bundles on the
Jacobian variety J (C). When g(C) = 2 this provides new examples of special Lagrangian submanifolds.
Keywords: Moduli spaces of stable bundles, Fourier–Mukai transform, complex Lagrangian submanifolds.
MS classification: 14D20, 14J60, 53C42.
1. Introduction
Throughout this paper we shall fixC as the ground field. Let C be a smooth algebraic curve of
genus g > 1, and denote by J (C) its Jacobian variety and by  ∈ H 2(J (C),Z) the cohomology
class corresponding to the theta divisor. Fix coprime positive integers r , d such that d > 2rg,
and letMC(r, d) be the moduli space of stable vector bundles on C of Chern character (r, d). We
show thatMC(r, d) can be embedded as an isotropic holomorphic submanifold of the complex
symplectic variety MµJ (C)(r, d) = MµJ (C)(d + r(1 − g), −r, 0, . . . , 0)—the moduli space of
µ-stable vector bundles on J (C) with Chern character (d + r(1− g), −r, 0, . . . , 0) (cf. The-
orem 2.1 for a precise statement). When g(C) = 2 one has dimMµJ (C)(r, d) = 2 dimMC(r, d),
and by using the hyper-Ka¨hler structure ofMµJ (C)(r, d), one can choose on this space a complex
structure such thatMC(r, d) embeds as a special Lagrangian submanifold, thus providing new
examples of such objects.
We recall a few facts about the Fourier–Mukai transform in the context of Abelian vari-
eties [8]. Let X be an Abelian variety and X̂ = Pic0(X) its dual variety. LetP be the normalized
Poincare´ bundle on X × X̂ . The Mukai functor is defined as
RS: D(X ) → D( X̂ ) , RS(·) = RπX̂ ,∗(π∗X(·) ⊗ P)
where D(X) and D(X̂) are the bounded derived categories of coherent sheaves on X and X̂ ,
respectively. Mukai has shown that the functor RS is invertibile and preserves families of
1 E-mail:bruzzo@sissa.it.
2 E-mail:pioli@sissa.it.
0926-2245/01/$ – see frontmatter c©2001 Elsevier Science B.V. All rights reserved
PI I S0926-2245(00)00040-1
152 U. Bruzzo, F. Pioli
sheaves (cf. [8, 10]). If E is a WITi sheaf on X , that is, a sheaf whose transform is concentrated
in degree i , then the functor RS preserves the Ext groups
Ext jX (E, E ) ∼= Ext jX̂
(
Eˆ, Eˆ
)
for every j ,
where Eˆ indicates the transform of E .
Let C be a smooth projective curve of genus g > 1 and J (C) the Jacobian of C . If we
fix a base point x0 on C , and let αx0 : C → J (C) be the Abel–Jacobi embedding given by
αx0(x) = OC(x − x0), the normalized Poincare´ bundle PC on C × J (C) is the pullback of the
Poincare´ bundle on J (C) × J (C), where we identify J (C) with Ĵ (C) via the isomorphism
−φ: J (C) → Ĵ (C). The Poincare´ bundle on C × J (C) gives rise to a derived functor (which
is not invertible)
RC : D(C ) → D( J(C )) , RC (−) = RπJ (C),∗(π∗C(−) ⊗ PC ).
Since αx0 is a closed immersion we have a natural isomorphism of functors
RC ∼= RS ◦ αx0,∗. (1)
Thus the study of the transforms of bundles F on C with respect to RC is equivalent to studying
the transforms of sheaves of pure dimension 1 of the form αx0,∗(F) with respect to RS. We
recall the following fact which is proven in [6].
Proposition 1.1. If E is a stable bundle on C of rank r and degree d such that d > 2rg, then
E is WIT0, and the transformed sheaf Eˆ = R0C(E) is locally free and µ-stable with respect
to the theta divisor on J (C).
2. Complex Lagrangian embeddings
If we consider the moduli space MC(r, d) of stable bundles of rank r and degree d on a
projective smooth curve of genus g > 1 such that d > 2rg and r, d are coprime, the functor
RC gives rise to an injective morphism
˜ :MC (r, d ) → MµJ (C)(r, d ) = MµJ (C)(d + r(1 − g), −r, 0, . . . , 0)
where the sheaves in MµJ (C)(r, d) are stable with respect to the polarization .
Before studying the morphism ˜ we need to recall some elementary facts about the Yoneda
product of Ext groups. Let A be an abelian category with enough injectives. The elements of
Ext 1A(E, E) are identified with equivalence classes of exact sequences 0 → E → F → E → 0
with respect to the usual relation. This can be generalized to the groups Ext 2A(E, E) as follows.
We refer to [2] for proofs and details.
Consider the following commutative diagram with exact rows:
E : 0 −→ B −→ G1 −→ G2 −→ A −→ 0
↓IdB ↓ ↓ ↓IdA
E ′ : 0 −→ B −→ G ′1 −→ G ′2 −→ A −→ 0.
(2)
Complex Lagrangian embeddings of moduli spaces of vector bundles 153
We write E  E ′ when such a diagram holds. The relation is not symmetric, but it generates
the following equivalence relation: E ∼ E ′ if and only if there exists a chain of sequences E0,
E1, . . . , Ek such that
E = E0  E1  E2  · · · Ek = E ′.
Let Yext2A(−, −) the set of such equivalence classes.
There is a canonical isomorphism
Yext 2A(−, −) ∼= Ext 2A(−, −).
From now on we shall identify the above groups. Observe that the identity of Ext 2A(A, B) is
given by the class of the sequence
0 −→ B IdB−→ B 0−→ A IdA−→ A −→ 0.
Moreover the Yoneda product
Ext 1A(B, A) × Ext 1A( A, C ) → Ext 2A(B, C )
is obtained in the following way: let E and E ′ be two elements of Ext 1A(B, A) and Ext 1A(A, C)
represented respectively by the sequences
E : 0 −→ A ν−→ F p−→ B −→ 0 , E ′ : 0 −→ C i−→ G λ−→ A −→ 0.
Then the class of the exact sequence
0 −→ C i−→ G ν◦λ−→ F p−→ B −→ 0
in Ext 2A(B, C) is the image of (E, E ′) with respect to the Yoneda product.
We shall also need to introduce a moduli space of stable sheaves in Simpson’s sense [13].
For simplicity we denote the Abel–Jacobi map as j : C → J (C). Observe that if E is a stable
bundle on C then j∗(E) is a stable sheaf of pure dimension 1 on J (C) with respect to the
polarization . Let MpureJ (C)(r, d) be the moduli space of all stable pure sheaves on J (C) with
Chern character (0, . . . , 0, r, d + r(1 − g)). If E is a flat family of vector bundles on C
parametrized by a Noetherian scheme S, then jS,∗(E) is a flat family of sheaves on J (C) × S
over S, where jS: C × S → J (C)× S is the embedding j × IdS . Therefore one has a morphism
of moduli spaces
j∗ : M(r, d ) → MpureJ (C)(r, d ). (3)
Lemma 2.1. The morphism ˜ :MC(r, d) → MµJ (C)(r, d) is an immersion (i.e., its tangent map
is injective).
Proof. From the isomorphism given by eq. (1) and recalling that the transform RS preserves the
Ext groups of WIT sheaves, it is enough to show that the same claim holds for the morphism (3).
By the very definition of the Kodaira–Spencer map, the tangent map to j∗ may be identified
with the map
Ext 1C (E, E )
φ
↪→ Ext 1J (C)( j∗(E ), j∗(E ))
154 U. Bruzzo, F. Pioli
obtained in the following way. Let
A : 0 −→ E −→ F −→ E −→ 0 (4)
be a sequence representing an element of Ext 1C(E, E). If we apply the functor j∗ to the above
sequence we obtain the exact sequence
B : 0 −→ j∗(E ) −→ j∗(F ) −→ j∗(E ) −→ 0. (5)
One checks immediately that the map φ([A]) = [B] is well defined. If φ([A]) = 0 then φ([A])
is represented by the extension
0 −→ j∗(E ) −→ j∗(E ) ⊕ j∗(E ) −→ j∗(E ) −→ 0. (6)
Now applying the functor j∗ to the above sequence and noting that j∗( j∗(H)) ∼= H for every
vector bundle H on C we obtain the split exact sequence
0 −→ E −→ E ⊕ E −→ E −→ 0. (7)
Therefore φ([A]) = 0 implies [A] = 0 and φ is injective. 
Mukai proved that the moduli space of simple sheaves on an abelian surface X is symplectic;
more precisely, the Yoneda pairing
υ: Ext 1X (E, E ) × Ext 1X (E, E ) → Ext 2X (E, E ) ∼= C
defines a holomorphic symplectic form on the moduli of simple sheaves on X (cf. [9, 11]).
When dim X = 2n > 2 to define a symplectic form on the smooth locus of the moduli space
one needs to choose a symplectic form ω on X . The symplectic form is then defined by the
compositions (cf. [5])
Ext 1X (E, E ) × Ext 1X (E, E ) → Ext 2X (E, E )
tr−−→ H 2(X, OX )
∼→ H 0,2(X, C) λ−−→ H n,n(X, C) ∼= C
(8)
where tr is the trace morphisms and the map λ is obtained by wedging by ωn ∧ ω¯n−1.
Theorem 2.1. If g(C) is even, and the map ˜ embedsM(r, d) into the smooth locusM0J (C)(r, d)
of MµJ (C)(r, d), then the subvarietiesMC(r, d) are isotropic with respect to any of the symplectic
forms defined by equation (8). In particular, when g(C) = 2 the subvarieties MC(r, d) are
Lagrangian with respect to the Mukai form of MµJ (C)(r, d).
Proof. Since M0J (C)(r, d) is smooth, and ˜ :M(r, d) → M0J (C)(r, d) is injective and is an
immersion, it is also an embedding. Now, let E ∈ MC(r, d). It is enough to show that the
Yoneda product
Ext 1J (C)( j∗(E ), j∗(E )) × Ext 1J (C)( j∗(E ), j∗(E )) −→ Ext 2J (C)( j∗(E ), j∗(E ))
vanishes when applied to pairs ([A], [B]) of elements in Ext 1J (C)( j∗(E), j∗(E)) where [A] and
Complex Lagrangian embeddings of moduli spaces of vector bundles 155
[B] are represented, respectively, by the sequences
A : 0 −→ j∗(E )
ν−→ j∗(F )
p−→ j∗(E ) −→ 0 ,
B : 0 −→ j∗(E )
i−→ j∗(G )
λ−→ j∗(E ) −→ 0
with F, G ∈ MC(r, d). It is enough to remark that the product of the classes of the sequences
of sheaves on C
0 → E → F → E → 0 , 0 → E → G → E → 0
is zero for dimensional reasons, and apply the functor j∗.
In the case g(C) = 2 the moduli space is smooth by the results in [9]; moreover,
dimMµJ (C)(r, d ) = 2(r2 + 1) = 2 dimMC (r, d ). 
Remark 3. If we consider the moduli space MC(r, ξ) of stable bundles on C of rank r and
fixed determinant isomorphic to ξ , then the result is trivial: the variety MC(r, ξ) is Fano, so
that it carries no holomorphic forms.
3. The case g(C) = 2
In this section we elaborate on the case g(C) = 2. One can characterize situations where the
moduli space MµJ (C)(r, d) is compact. This happens for instance in the following case.
Proposition 3.1. Assume g(C) = 2, d > 4r and that ρ = d − r is a prime number. Then
every Gieseker-semistable sheaf on J (C) with Chern character (d − r, −r, 0) is µ-stable.
Moreover, if d > r2 +r , every such sheaf is locally free (this always happens when r = 1, 2, 3).
Proof. Since d−r is prime, every sheaf inMJ (C)(r, d) is properly stable. Let [F] ∈ MJ (C)(r, d)
and assume that the subsheaf G destabilizes F . Let ch(G) = (σ, ξ, s). Standard computations
show that if F is not µ-stable then
ξ · 
σ
= − 2r
ρ
and s < 0.
Setting n = ξ ·  we have |n| = 2rσ/ρ, with σ < ρ and ρ > 3r . This is impossible whenever
ρ is prime.
The statement about local freeness follows from the Bogomolov inequality. 
In the case g(C) = 2 the complex Lagrangian embedding ˜ :MC(r, d) → MµJ (C)(r, d)
provides new examples of special Lagrangian submanifolds. We refer to [1, 7] for the definition
and the main properties of these objects. Now, if X is a hyper-Ka¨hler manifold of complex
dimension 2n, let I , J , K be three complex structures compatible with the hyper-Ka¨hler metric,
and such that I J = K . Let ωI , ωJ , ωK be the corresponding Ka¨hler forms. Then the 2-form
 = ωI + iωJ is a holomorphic symplectic form in the complex structure K . It is easy to check
that a K -complex n-dimensional submanifold which is Lagrangian with respect to  is special
Lagrangian in the structure J [3].
156 U. Bruzzo, F. Pioli
One should notice that via the Hitchin–Kobayashi correspondence (which identifies µ-stable
bundles on a Ka¨hler manifold with irreducible Einstein–Hermite bundles, cf. [5]), the space
M
µ
J (C)(r, d) acquires a hyper-Ka¨hler structure, compatible with a Ka¨hler form provided by the
Weil–Petersson metric, and with a holomorphic symplectic form which may be identified with
the Mukai form [4].
Therefore we obtain the following result.
Proposition 3.2. The space MµJ (C)(r, d) has a complex structure such that ˜ :MC(r, d) →
M
µ
J (C)(r, d) is a special Lagrangian submanifold.
The elements of the Jacobian variety J (C) act on the embedding j : C → J (C) by transla-
tion, so that for every x ∈ J (C) we have a special Lagrangian submanifold ˜x :MC(r, d) →
M
µ
J (C)(r, d). This provides a family of deformations of ˜ (MC(r, d)) through special Lagrangian
submanifolds. One easily shows that this embeds J (C) into the moduli space MSL of special
Lagrangian deformations of ˜ (MC(r, d)) (notice that dimRMSL = b1(MC(r, d)) = 4 =
dimR(J (C))) [12]. The case r = 1 is somehow trivial because MµJ (C)(1, d)  J (C) × J (C)
by a result of Mukai [8].
Acknowledgements
We thank A. Maciocia and M.S. Narasimhan for useful suggestions or remarks. The second
author acknowledges financial support from SISSA.
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By means of a Fourier-Mukai transform we embed moduli spaces M-C(r, d) of stable bundles on an algebraic curve C of genus g(C) greater than or equal to 2 as isotropic subvarieties of moduli spaces of mu -stable bundles on the Jacobian variety I(C). When g(C) = 2 this provides new examples of special Lagrangian submanifolds

Complex Lagrangian embeddings of moduli spaces of vector bundles

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