A Fourier-Mukai transform for real torus bundles

0.4 MB·

by  James F. Glazebrook

A Fourier–Mukai transform for real torus bundles
James F. Glazebrook
Eastern Illinois University
Department of Mathematics and Computer Science
4
Charleston, IL 61920 USA
0
0 [email protected]
2

n Marcos Jardim
a
University of Massachusetts at Amherst
J
6 Department of Mathematics and Statistics
1 Amherst, MA 01003–9305 USA

[email protected]
]
G
D Franz W. Kamber
. University of Illinois at Urbana–Champaign
h
t Department of Mathematics
a
m Urbana, IL 61801 USA
[ [email protected]

2
February 1, 2008
v
9
9
1
7 Abstract
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We constructa Fourier–Mukaitransformfor smooth complex vector bundles E overa torus
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0 bundle π : M → B , the vector bundles being endowed with various structures of increasing
/ complexity. Ataminimum,weconsidervectorbundlesE withaflatpartialunitaryconnection,
h
thatisfamiliesordeformationsofflatvectorbundles(orunitarylocalsystems)onthe torusT .
t
a
This leads to a correspondence between such objects on M and relative skyscraper sheaves S
m
supported on a spectral covering Σ֒→M , where πˆ :M →B is the flat dual fiber bundle. Ad-
v: ditionalstructures on (E,∇) (flatness, acnti-self-dualityc)will be reflectedby correspondingdata
i on the transform (S,Σ) . Several variations of this construction will be presented, emphasizing
X
the aspects of foliation theory which enter into this picture.
r Keywords: Fourier-Mukai transforms; foliation theory; unitary local systems; instantons;
a
monopoles.
2000 MSC: 65R10; 53C12;14D21.
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Contents
1 Introduction 3
2 Local systems on tori 4
3 The Fourier–Mukai transform 7
3.1 Transforming foliated bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 The inverse transform for relative skyscrapers . . . . . . . . . . . . . . . . . . . . . . 14
3.3 The main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 The Fourier–Mukai transform for vector bundles withPoincar´e basic connections 17
4.1 Transforming bundles with Poincar´e basic connections . . . . . . . . . . . . . . . . . 17
4.2 The inverse transform for connections . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.3 The main theorem for bundles with Poincar´e basic connections . . . . . . . . . . . . 25
5 Applications and examples 27
5.1 Local systems and the representation variety . . . . . . . . . . . . . . . . . . . . . . 27
5.2 Instantons on T1–fibered 4–manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.3 Monopoles on T1–fibered 3–manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2
1 Introduction
The construction nowadays known as the Fourier–Mukai transform first appeared in a seminal
work by Mukai [11], whereit was shownthat the derived categories of sheaves on an abelian variety
(e.g. a complex torus) is equivalent to the derived category of coherent sheaves on the dual abelian
variety.
Since then, the Fourier–Mukai transform has been generalized in different ways, and has led to
a number of interesting results concerning not only the derived categories of coherent sheaves, but
also the moduli spaces of stable sheaves on abelian varieties, K3 surfaces and elliptic surfaces.
ThispaperdrawsontwotypesofgeneralizationofMukai’soriginalideas. First,onecanconsider
families of abelian varieties, anddefineatransformthat takes (complexes of)sheaves on afamily of
abelian varieties to (complexes of) sheaves on the correspondingdual family. This has been applied
with great success to the study of stable sheaves on elliptic surfaces (i.e. holomorphic families of
elliptic curves parametrized by an algebraic curve), see for instance [2] and the references there.
In particular, given an elliptic surface X, it can be shown that there exists a 1–1 correspondence
between vector bundles on X which are stable with respect to some suitable polarization, and
certain torsion sheaves (spectral data) on the relative Jacobian surface (see [6, 7] for details).
Ontheother hand,Mukai’s construction can begeneralized from complex toritoreal tori. Such
a generalization, first considered by Arinkin & Polishchuk in [1] is briefly described in Section 2
below.
Building on previous work by Arinkin & Polishchuk [1] and by Bruzzo, Marelli & Pioli [3, 4],
we consider in this paper a Fourier–Mukai transform for vector bundles with (partial) connections
on families of real tori. Rather than restricting ourselves to flat connections on Lagrangian families
of real tori as in [1, 4], we study a broader class of connections on vector bundles over a (not
necessarily symplectic) manifold M with the structure of a flat torus bundle.
After briefly reviewing the Fourier–Mukai transform for real tori, following [3], we start in
Section 3 by defining a Fourier–Mukai transform for foliated bundles, which in our context can be
viewed as families of flat bundles on the fibers of the torus bundle M → B. This takes foliated
Hermitian vector bundles over M into certain torsion sheaves on the dual fibration M → B. We
c
then introduce the concept of Poincar´e basic connections in Section 4, and extend our construction
to include vector bundles provided with such connections. We then conclude in Section 5 by
3
applying our techniques to three different examples: flat connections, that is unitary local systems,
instantons on 4–dimensional circle fibrations, and monopoles on 3-dimensional circle fibrations.
It is a somewhat surprising fact that certain concepts and techniques from foliation theory
occur quite naturally in thecontext of the Fourier–Mukai transform. Besides thenotions of foliated
bundle and Poincar´e basic connection which refer to the torus fibration, that is to a foliation which
is rather trivial from the point of view of foliation theory, there is also a canonical foliation on
the dual fibration M → B, transverse to the fibers which has a more complicated structure. For
c
locally trivial families of flat bundles on the fibers of the torus bundle M → B, it turns out that
the supportsΣ ֒→ M of the Fourier–Mukai transform are (finite unions of) leaves of this transverse
c
foliation. This allows us to give in section 5.1 an explicit parametrization of the representation
variety R (n) of M in terms of leaves with transversal holonomy of order ℓ such that ℓ|n .
M
The main motivation behind [1] and [3, 4] comes from string theory and the Strominger-Yau-
Zaslowapproachtomirrorsymmetry,withthemaingoalofunderstandingKontsevich’shomological
mirror symmetry conjecture. In a sense, the two main results here presented may also be relevant
to the understanding of Kontsevich’s conjecture. Although it seems reasonable to expect that the
ideas explored in this paper might provide some interesting connections with String Theory and
mirror symmetry, we do not elaborate on them, leaving such a task to mathematical physicists.
Notation. Weworkonthecategory ofrealC∞-manifolds. Byavectorbundleover amanifoldX,
we mean a C∞ vector bundle over X. We will also identify a vector bundle with the corresponding
sheaf of C∞ sections. By the same token, a sheaf on X should be understood as a sheaf of modules
over the algebra of C∞ functions on X.
2 Local systems on tori
Let us begin by briefly recalling the Fourier–Mukai transform for real tori, as defined by Arinkin
and Polishchuk [1] and Bruzzo, Marelli and Pioli [3]; the interested reader should refer to these
papers for the details of this construction.
Let T be the d–dimensional real torus, that is T = Rd/Λ for the rank d integral lattice Λ ⊂ Rd .
The associated dual torus is defined as T = (Rd)∗/(Λ)∗, where
b
(Λ)∗ := {z ∈ (Rd)∗ : z(y) ∈ Z , ∀y ∈ Λ} , (2.1)
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is the dual lattice. From the exact sequence
0 → HomZ(Λ,U(1)) −→ H1(T,O∗)−c→1 H2(T,Z)→ 0 , (2.2)
T
we see that, upto gauge equivalence, points in T parametrize flatunitary connections on thetrivial
b
line bundle C = T ×C −→ T, since we have :
exp
T = H1(T,R)/H1(T,Z) ∼= HomZ(Λ,U(1)) ∼= U(1)d . (2.3)
b
For ξ ∈ T , x ∈Rd , a∈ Λ and λ ∈ C , consider the equivalence relation
b
Rd×T ×C −→ Rd×T ×C/∼ ,
b b (2.4)
(x+a,ξ,λ) ∼ (x,ξ,exp(ξ(a))λ) .
The quotient space under ‘∼’ defines the Poincar´e line bundle P −→ T ×T . Let p and pˆ denote
b
the natural projections of T ×T onto its first and second factors, respectively. In accordance with
b
(2.4), the bundle P has the property that for ξ ∈ T, the restriction P | pˆ−1(ξ) ∼= L , where the
ξ
b
latter denotes the flat line bundle parametrized by ξ . It is straightforward to see that
Ω1 = p∗Ω1 ⊕pˆ∗Ω1 . (2.5)
T×T T T
b b
Corresponding to the definition of P and its above property, it is shown in [3] that there exists a
canonical connection ∇ :P −→ P ⊗Ω1 , whose connection form is given by
P
T×T
b
d
A =2πι ξ dzj , (2.6)
X j
j=1
where {zj} are (flat) coordinates on T and {ξ } are dual (flat) coordinates on T . The connection
j
b
∇ splits as the sum ∇r ⊕∇t , where
P P P
∇r =(1 ⊗r)◦∇ , ∇t = (1 ⊗t)◦∇ , (2.7)
P P P P P P
with natural maps r :Ω1 −→ p∗Ω1, and t :Ω1 −→ pˆ∗Ω1 .
T×T T T×T T
b b b
For later purposes we shall denote the dual of any complex vector bundle E by E∨ and in
particular the dual line bundle of P by P∨ .
Now consider the categories Sky(T) and Loc(T) defined as follows (see [3]) :
b
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• Loc(T) is the category of unitary local systems on T . Its objects are pairs (E,∇) consisting
of a smooth complex vector bundle E −→ T and a flat unitary connection ∇ . Morphisms
are simply bundle maps compatible with the connections.
• Sky(T)isthecategory ofskyscrapersheaves onT offinitelength, thatis,dimH0(T,S) < ∞,
b b b
for all S ∈ Ob(Sky(T)) .
b
The Fourier–Mukai transform is the invertible functor
F :Loc(T)−→ Sky(T) , (2.8)
b
which we now describe. Given (E,∇) ∈Ob(Loc(T)), we have the relative connection
∇r :p∗E ⊗P∨ −→ p∗E ⊗P∨⊗p∗Ω1 ,
E T
(2.9)
∇rE = (1(E⊗P∨)⊗r)◦(∇⊗1P∨ +1E ⊗∇P∨) ,
and the transversal connection
∇t : p∗E⊗P∨ −→ p∗E⊗P∨⊗p∗Ω1 ,
E T
(2.10)
b
∇tE = (1(E⊗P∨)⊗t)◦(∇⊗1P∨ +1E ⊗∇P∨) .
As a section of End (p∗E⊗P∨)⊗Ω2 , the commutator satisfies (see e.g. [10]) :
s
T×T
b
∇rE∇tE +∇tE∇rE = 1E ⊗∇2P∨ . (2.11)
Lemma 2.1. [3] If (E,∇) ∈Ob(Loc(T)), then :
(1) Rjpˆ (ker∇r )= 0 , for 0 ≤ j ≤ d−1 .
∗ E
(2) S = Rdpˆ (ker∇r )∈ Ob(Sky(T)) .
∗ E
b
Moreover, dimH0(T,S) = rank E .
b
We say that S = F(E,∇) is the Fourier–Mukai transform of the local system (E,∇) .
Conversely, take S ∈ Ob(Sky(T)), and let σ be the support of S . Clearly, pˆ∗S ⊗P as a sheaf
b
on T ×T, is supported on T ×σ . Thus
b
Rjp (pˆ∗S ⊗P)= 0 , for 0 < j ≤ d , (2.12)
∗
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while E = R0p (pˆ∗S⊗P) is a locally–free sheaf of rank dimH0(T,S) . In order to get a connection
∗
b
on E , consider again the relative connection :
1 ⊗∇r : pˆ∗S ⊗P −→ pˆ∗S ⊗P ⊗p∗Ω1 . (2.13)
S P T
Pushing it down to T, we get a connection
∇ = R0p (1 ⊗∇r ): E −→ E ⊗Ω1 , (2.14)
∗ S P T
since R0p (pˆ∗S⊗P ⊗p∗Ω1) = E⊗Ω1 , by the projection formula. Since (∇r )2 = 0 , we conclude
∗ T T P
that ∇ is indeed flat, hence (E,∇) ∈ Ob(Loc(T)), as desired. We usethe notation (E,∇) = F(S) .
b
In summary, referring once more to [3], we have :
Proposition 2.2. The functorsF and F are inverse toeach other, and yieldan equivalence between
b
the categories Loc(T) and Sky(T) .
b
3 The Fourier–Mukai transform
Let M be a smooth manifold of dimension m, which is the total space of a d–torus bundle over a
(m−d)–dimensional connected manifold B , that is
Td ֒→ M −π→ B . (3.1)
Given a point b ∈ B , we define T = π−1(b) to be the fiber over b , where the point o(b) marks
b
the origin of T . Regarded as a bundle of groups, π : M −→ B admits a discrete structure group
b
Aut(T) ∼= GL(d,Z) , and so the former has the structure of a flat fiber bundle and admits a 0–
section o :B −→ M . Since the fiber T is compact, this flat structure is determined by a holonomy
homomorphism ρ:π (B)→ Aut(T) ∼= GL(d,Z) as a twisted product
1
M ∼= B × T . (3.2)
ρ
e
Remark 3.1. For the purpose of this paper, we may weaken the structure of the fiber bundle
π : M → B as follows. Let Diff(T,o) be the group of diffeomorphisms of T which fix the origin.
Then Diff(T) is given as a crossed productDiff(T) ∼= T× Diff(T,o) , where T acts by translations
ϕ
and Diff(T,o) acts on T in the obvious way. Moreover the canonical homomorphism Diff(T,o) →
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π (Diff(T,o)) to the mapping class group is realized as a deformation retraction
0
Diff(T,o) −−−−→ Aut(Λ)
b
⊂x ∼=x (3.3)
 
Aut(T) −−−∼=−→ GL(d,Z)
via ϕ 7→ ϕ 7→ ϕ = ϕ|Λ ∈ Aut(Λ) , where ϕ is the unique equivariant lift of ϕ ∈ Diff(T,o) to
Diff(Rd,o)eand bϕ coiencides with the automoerphism ϕ induced by ϕ on the fundamental group
∗
π (T,o) ∼= Λ . Tbhe statement about the deformation retraction follows from the fact that any
1
diffeomorphism (actually any homeomorphism) which fixes the lattice Λ is isotopic to the identity,
andin facttheconnected componentDiff (T,o)is contractible totheidentity; anelementary result
e
which is stated in the 1960’s thesis of John Franks (as pointed out to us by Keith Burns). This
said, we may start with a fiber bundle π : M → B with structure group Diff(T,o) . This still
guarantees the existence of the section o : B → M and the previous holonomy homomorphism
ρ : π (B) → Aut(T) is now recovered as the canonical homomorphism π (B) → π (Diff(T,o))
1 1 0
associated tothefiberbundleπ :M → B .Formula(3.4)forπ (M)remainsvalid,aswellastheflat
1
structure (3.7) of the dual fiber bundle πˆ : M → B , the latter property being a consequence of the
c
homotopy invariance of singular cohomology. Infact, theabove deformation retraction implies that
the structure group of a Diff(T,o)–torus bundle admits a unique reduction to Aut(T) ⊂ Diff(T,o)
and so π : M → B is still diffeomorphic to a flat fiber bundle of the form (3.2).
The fundamental group of M is determined as a crossed product
0 → Λ= π (T)−→ π (M) ∼= π (T)× π (B)−←→ π (B)→ 1 , (3.4)
1 1 1 ρ∗ 1 1
where ρ is given by the induced action of π (B) on Λ via the isomorphism GL(d,Z) ∼= Aut(Λ) .
∗ 1
We have the exact sequence
0 → T(π) −→ TM −→ π∗TB → 0 , (3.5)
and the dual sequence of 1–forms
0→ π∗Ω1 −→ Ω1 −→ Ω1 → 0 . (3.6)
B M M/B
Observe that a flat structure of π : M → B defines a splitting of the exact sequences (3.5) and
(3.6).
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Thedual fiberbundleM → B isgivenbythetotalspaceofR1π R/R1π Zasa(locallyconstant)
∗ ∗
c
sheaf on B . If πˆ :M −→ B is the natural projection, it is easy to see that πˆ−1(b) = T . Note that
b
c b
this projection also has a 0–section σ : B → M . It follows that πˆ : M → B is given by the flat
0
c c
bundle of fiber cohomologies
M ∼= B ×ρ∗ T , (3.7)
c e b
where ρ∗ is the induced action of π (B) on T = H1(T,R)/H1(T,Z) . Furthermore, R1πˆ R/R1πˆ Z
1 ∗ ∗
b
coincides with M as sheaves on B, and we have M ∼= M .
c
c
LetZ = M× M bethefiberproduct,withitsnaturalprojectionsp :Z −→ M andpˆ: Z −→ M
B
c c
onto the first and second factors. Clearly, π◦p = π◦pˆand (π◦p)−1(b) = T ×T .
b b
b b
Z (3.8)
@
~ @
(cid:127)(cid:127)~~~~~p~~ @@@pˆ@@(cid:31)(cid:31)
M M
@
@@@@π@ ~~~~c
@@(cid:31)(cid:31) (cid:127)(cid:127)~~~ πˆ
B
It is also easy to see that p−1(x) = T , for all x ∈M and pˆ−1(y)= T , for all y ∈ M .
π(x) πˆ(y)
b c
Defining Ω1 = Ω1/pˆ∗Ω1 , recall that the Gauss–Manin connection yields a splitting of the
Z/M Z M
short exact sequecnce c
0 → pˆ∗Ω1 −→ Ω1 −→r Ω1 → 0 , (3.9)
M Z Z/M
c c
such that we have the decomposition
Ω1 = pˆ∗Ω1 ⊕Ω1 . (3.10)
Z M Z/M
c c
From (3.6) it follows that
Ω1 = p∗Ω1 , (3.11)
Z/M M/B
c
since pˆ: Z → M is the pull–back fibration of π :M → B along πˆ .
c
There exists a line bundle P over Z = M × M, with the property that P|(π◦p)−1(b) is just
B
c
the Poincar´e line bundle P over T ×T , for all b ∈ B (see [4]). We call P the relative Poincar´e
b b b
b
line bundle. Just as in the absolute case, it has the property that for ξ ∈ M, the restriction
c
P | pˆ−1(ξ) ∼= L , where the latter denotes the flat line bundle parametrized by ξ ∈ T ⊂ M .
ξ b
b c
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There is a canonical connection on P which we denote by ∇ . Following [4], we can write its
P
connection matrix A in a suitable gauge on an open subset U ×T ×T ⊂ Z as follows
b
d
A =2πι ξ dzj , (3.12)
X j
j=1
where{zj}are(flat)coordinatesonT and{ξ }aredual(flat)coordinatesonT . Insuchcoordinates
j
b
the curvature F = ∇2 is then given by :
P
d
F= 2πι dξ ∧dzj . (3.13)
X j
j=1
In the same coordinate system, we have ∇2 = −F .
P∨
3.1 Transforming foliated bundles
Let E → M be a Hermitian vector bundle of rank n . With reference to (3.5), we assume a foliated
bundle structure on E given by a flat partial unitary connection [8] :
o
∇ : E −→ E ⊗Ω1 = E ⊗Ω1 /π∗Ω1 , (3.14)
E M/B M B
o
satisfying (∇ )2 = 0 .
E
o
The local structure of a foliated bundle (E,∇ ) on M is described next.
E
π
Example 3.2. Localstructureof foliated bundlesonM : Intuitively, afoliated bundleonM → B
isafamilyofflatbundles(unitarylocalsystems)(E ,∇ )onthefibersT ,parametrizedbyb ∈ B .
b Eb b
Of course, the topology of E has to be taken into account. The local description is quite similar to
that of the Poincar´e line bundle in (2.4), which is of course an example of a foliated bundle. Thus
for sufficiently small open sets U ⊂ B , there are isomorphisms
U ×(Rd× Cn) −−−∼=−→ E | π−1(U)
Λ
id×τ′ τ|π−1U (3.15)
 
U ×y T −−−∼=−→ π−1y(U) ,
where the identification on the LHS is given by
(b,x+a,λ) ∼ (b,x,exp(ξ (a))λ) , (3.16)
b
10