INTRODUCTION TO TWISTOR D-MODULES
KYOTO, JANUARY 2008
Claude Sabbah
Contents
1. Around the Hard Lefschetz Theorem (first talk). . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1. The classical Hard Lefschetz theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2. The Hard Lefschetz theorem for local systems. . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3. The notion of a harmonic metric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4. Twistor structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5. Polarization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.6. Variations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2. Polarizable twistor D-modules (second talk). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1. Extension of local systems of flat holomorphic bundles. . . . . . . . . . . . . . . . . 7
2.2. z-connections and RX -modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3. Extension of z-connections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4. The category R- Triples(X). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5. Direct images in R- Triples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6. Specialization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
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1. Around the Hard Lefschetz Theorem (first talk)
1.1. The classical Hard Lefschetz theorem. Let X be a smooth complex
projective variety of dimension n, or more generally a compact Kähler manifold,
and let ω be the Kähler class. The Lefschetz operator L is the operator ω∪
acting on the space of differential forms or on the de Rham cohomology of X.
It can also be defined in order to act on the rational cohomology of X.
Theorem (Hard Lefschetz Theorem). For any k > 0, Lk induces an isomorphism H n−k (X, Q) → H n+k (X, Q).
Let me recall the main ingredient of the analytic proof. Considering the
Laplacians with respect to d, d0 , d00 , we have 12 ∆ = ∆0 = ∆00 , and L commutes
with them, therefore acts on the space of harmonic forms.
1.2. The Hard Lefschetz theorem for local systems. Let now V be a
local system on X, corresponding to a linear representation of π1 (X). One can
define the Lefschetz operator acting on H ∗ (X, V ). For instance, one consider
a holomorphic vector bundle V with flat connection ∇ such that ker ∇ = V ,
and realize the cohomology as the cohomology of the C ∞ de Rham complex
(Γ(X, A • ⊗OX V ), ∇ + d00 ) and ω∧ acts on this complex.
The Hard Lefschetz theorem for H ∗ (X, V ) holds when V is a unitary local
systems (with the same proof), but may not be satisfied otherwise.
Example. Let V1 be rank-one non constant local system on a Riemann surface of
genus g > 2 and let σ be a nonzero element of H 1 (X, V1 ) (which has dimension
2g − 2). Then σ defines an extension
0 −→ V1 −→ V −→ CX −→ 0.
One can show that dim H 2 (X, V ) 6= dim H 0 (X, V ), hence HL cannot be satisfied.
Theorem (Simpson [9]). If X is smooth projective or compact Kähler, and if V
is irreducible (or semisimple), then H ∗ (X, V ) satisfies Hard Lefschetz.
1.3. The notion of a harmonic metric. Let V be a holomorphic vector
bundle on a complex manifold, and let ∇ be a flat holomorphic connection
on V (if X is a Riemann surface, any holomorphic connection is flat). Let us
denote by H the associated C ∞ bundle, and by D the associated C ∞ connection,
with D00 = d00 and D0 is induced by ∇. Hence V = ker D00 .
On the other hand, let h be a Hermitian metric on H.
INTRODUCTION TO TWISTOR D-MODULES
3
(1,0)
Lemma. There exists a unique θ0 : H → AX ⊗ H such that, if θ00 denotes the
h-adjoint of θ0 and if we set θ = θ0 + θ00 , then the connection
DE = D − θ = D0 − θ0 + D00 − θ00
is compatible with h.
Note that D is compatible with h iff θ = 0.
Definition. The hermitian flat bundle (V, ∇, h) is harmonic if (DE00 + θ0 )2 = 0.
Note that we know that D = (DE00 + θ0 ) + (DE0 + θ00 ) has square 0. The harmonicity condition is equivalent to the following relations, according to types:
DE002 = 0,
D00 (θ0 ) = 0,
θ0 ∧ θ0 = 0.
Hence DE00 defines a new holomorphic structure on H, and if we set E = ker DE00 ,
then θ0 is a holomorphic Higgs field on E.
Theorem (Corlette [2]). If X is compact Kähler or smooth projective, then a
holomorphic flat bundle (V, ∇) admits a harmonic metric iff it is semisimple.
If (V, ∇) is irreducible, this metric is unique up to a positive constant.
Sketch of proof of the theorem of Simpson. By the theorem of Corlette, the flat
bundle (V, ∇) corresponding to V admits a harmonic metric. For any z ∈ C,
déf
let ∆z be the Laplacian of the connection Dz = (DE00 + θ0 ) + z(DE0 + θ00 ) with
respect to h and to the Kähler form on X. The theorem is then a consequence
of
Lemma (Simpson [9]). If h is harmonic, the following Kähler identities hold
on X:
∆1 = 2∆0 = 2∆∞ .
1.4. Twistor structures. The harmonic metric provides each H k (X, V ) with
a positive definite Hermitian form, which plays the role of a polarization. However, there is no a priori analogue of a Hodge structure. In particular, we are
not able to distinguish the various cohomology groups by their weight, as the
weight is not defined yet.
Definition. A twistor structure is a vector bundle on P1 . The twistor structure
is pure of weight w if the bundle is isomorphic to OP1 (w)d .
1
Let σ : P1 → P be the antilinear involution z 7→ −1/z. We have σ(0) = ∞.
If f ∈ O(Ω), where Ω is an open set of P1 , then f ∈ O(σ(Ω)) is defined by
f (z) = f (−1/z). If H is a holomorphic vector bundle on Ω, we denote by
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C. SABBAH
H the conjugate antiholomorphic bundle, and by H the holomorphic bundle
σ ∗ H on σ(Ω).
It will be convenient to represent a vector bundle on P1 as coming from the
gluing of two holomorphic vector bundles, that we denote H 0∨ and H 00 , where
H 0 , H 00 are holomorphic vector bundles on a neighbourhood Ω0 of {|z| 6 1} in
déf
P1 , along the circle S = {z = 1}, where the gluing is given by a nondegenerate
-sesquilinear pairing
C : H|S0 ⊗O|S H|S00 −→ O|S .
We say that we have a triple T = (H 0 , H 00 , C).
Example (Hodge structure). Let H be a vector space with a decomposition
L p,w−p
H =
and a sesquilinear pairing k such that the decomposition is
pH
L
k-orthogonal. We set F p = r>p H r,w−r , F 0p = F [w]p = F w+p and F 00p = F p .
We associate to these data the Rees C[z]-modules
M
M
déf
déf
H 0 = RF 0 H =
F 0k z −k , H 00 = RF 00 H =
F 00k z −k .
k
k
We finally set
C = RF k : H 0 [z −1 ] ⊗C[z,z −1 ] H 00 [z −1 ] −→ C[z, z −1 ].
We have H 0 [z −1 ] = H ⊗C C[z, z −1 ] and
P 0 i P 00 j P
C
= i,j k(m0i , m00j )(−1)j z i−j .
i mi z ,
j mj z
Then (H 0 , H 00 , C) is a pure twistor of weight w.
Remark. We have the notion of
(1) morphism of triples:
ϕ = (ϕ0 , ϕ00 ) : T1 −→ T2 ,
ϕ0 : H20 −→ H10 ,
ϕ00 : H100 −→ H200 ,
with C2 (m02 , ϕ00 (m001 )) = C1 (ϕ0 (m02 ), m001 );
(2) adjunction: T ∗ = (H 00 , H 0 , C ∗ ) with C ∗ (m00 , m0 ) = C(m0 , m00 );
(3) Tate twist: for ` ∈ 12 Z, T (`) = (H 0 , H 00 , (iz)−2` C).
If T is pure of weight w, then T ∗ is pure of weight −w and T (`) is pure of
weight w − 2`. There is no nonzero morphism between pure twistors T1 and
T2 if w1 > w2 .
INTRODUCTION TO TWISTOR D-MODULES
5
1.5. Polarization. Let T be a pure twistor structure of weight w. A Hermitian duality is an isomorphism S : T → T ∗ (−w) which is Hermitian,
i.e., S ∗ = (−1)w S . If S = (S 0 , S 00 ), we set hS = (iz)−w C ◦ (S 00 ⊗ Id) :
H|S00 ⊗O|S H|S00 → O|S .
We say that S is a polarization if there exists a global frame of H 00 on Ω0
such that the matrix of hS in this frame is constant and equal to Id.
Lemma. Giving a polarized pure twistor structure of weight 0 is equivalent to
giving a finite dimensional complex vector space with a positive definite Hermitian form.
1.6. Variations. Let X be a complex manifold. A variation of twistor structures parameterized by X consists of a triple (H 0 , H 00 , C), where H 0 , H 00 are
holomorphic bundles on X × Ω0 with a flat relative connection (z∇ is usually
called “a z-connection”)
∇ : H 0 (00 ) −→
1 1
ΩX×Ω0 /Ω0 ⊗ H 0 (00 )
z
∞,an
is a nondegenerate sesquilinear pairing comand C : H|S0 ⊗OX×S H|S00 → CX×S
patible with the relative connection ∇.
The previous definitions apply to the case of a variation by restricting to each
point of X.
Remark. In tt∗ geometry, such variations appear with a supplementary property
and a supplementary structure. The supplementary property is integrability,
saying that ∇ comes from an absolute flat connection on H 0 and H 00 , having
Poincaré rank one, and C is compatible with the absolute connection. The
supplementary structure is a real structure, satisfying some compatibilities with
the previous structures (cf. [3]).
Lemma (Simpson [10]). Giving a polarized variation of pure twistor structures
of weight 0 is equivalent to giving a flat complex vector bundle with a harmonic
metric.
Sketch of proof. We will give one way of the correspondence. Let us start with
a flat complex vector bundle with a harmonic metric. We denote by H the
associated C ∞ bundle and by D = D0 + D00 the flat connection. Let θ0 be
the Higgs filed and let θ00 be its h-adjoint. Lastly, we set DE0 = D0 − θ0 and
DE00 = D00 − θ00 . Let π : X × Ω0 → X denote the projection. On the C ∞ bundle
π ∗ H = H , we consider the holomorphic structure H 0 defined by d00z +DE00 +zθE00 ,
and we set H 00 = H 0 . We will also fix the polarization to be S = (Id, Id).
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C. SABBAH
∞,an
The pairing C : H|S0 ⊗O|S H|S00 → CX×S
is the restriction of the pull-back of the
metric h : H ⊗CX∞ H → CX∞ .
Theorem (Hodge-Simpson). Let X be smooth projective or compact Kähler and
let (T , S ) be a polarized variation of twistor structure of weight w. Then, if aX
denotes the constant map to a point, H k aX,+ (T , S ) is a polarized pure twistor
structure of weight w + k and the Hard Lefschetz theorem holds.
INTRODUCTION TO TWISTOR D-MODULES
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2. Polarizable twistor D-modules (second talk)
We now consider variations of twistor structures with singularities.
2.1. Extension of local systems of flat holomorphic bundles. Let U
be the complement of a divisor D in a complex manifold X. Assume that
we are given a local system V on U , with associated flat holomorphic bundle
(V, ∇). It is well known (Deligne) that there exists a unique meromorphic
bundle (Ve , ∇) on X (i.e., a locally free OX (∗D)-module) with meromorphic
connection, extending (V, ∇) and such that, near any point of D, the entries
of the matrix of the base change from a meromorphic frame of Ve to a flat
frame have moderate growth along D, in other words, such that ∇ has regular
singularities along D. We have therefore an equivalence of categories
Deligne ext.
(V, ∇) m
(Ve , ∇)
restriction to U
Remark.
(1) There usually exist other meromorphic extensions, on which ∇ has irregular singularities.
(2) On the other hand, other extensions as a regular holonomic DX -module
possibly exist. They correspond to the various extensions of V as a perverse
sheaf on X. These various extension occur when the local monodromies around
D have 1 as an eigenvalue. The Deligne meromorphic extension is the maximal
possible extension. There also exists the intermediate (or minimal) extension
Vemin , whose de Rham complex is the intersection cohomology complex IC(V ).
We also have an equivalence
minimal ext. .
(V, ∇) m
(Vemin , ∇)
restriction to U
(3) If we are given a Hermitian metric on V , various conflicts with the Deligne
meromorphic extension can occur:
(a) the norm of flat sections may or may not have moderate growth near
the divisor; in the latter case, we say that the metric is wild, otherwise it
is tame;
(b) in the tame case, and assuming that D has normal crossings, let us
choose a local frame for which the matrix of the connection has logarithmic
poles; then the norm of each section of the frame has moderate growth;
but there may be a discrepancy between the exponents occurring in the
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C. SABBAH
asymptotic expansion of the norm and the eigenvalues of the residue of the
connection: this is the parabolic structure.
2.2. z-connections and RX -modules. Let H be a holomorphic bundle on
déf
X = X × Ω0 . An operator ∇ as the one defined for variations is called a
z-connection (more precisely, z∇ is the z-connection). Restricting (H , ∇) to
z = 1 gives a flat bundle (V, ∇). Restricting H to z = 0 and taking the residue
of the z-connection gives a Higgs bundle (E, θ).
Example. Let (V, ∇) be a flat holomorphic bundle with a decreasing filtration
F • V by subbundles satisfying the Griffiths transversality property ∇F p V ⊂
déf L
p
−p
F p−1 V . The Rees module RF V =
is a OX [z]-module equipped
pF Vz
with a z-connection. The restriction to z = 1 is (V, ∇) and the restriction to
z = 0 is (gr•F V, θ = gr−1 ∇).
Let RX be the sheaf on X of z-differential operators: in local coordinates,
RX = OX hz∂x1 , . . . , z∂xn i. Then, a OX -module with a flat z-connection is
nothing but a left RX -module. Restriction to z = 1 gives DX . Restriction to
z = 0 gives grF DX (forgetting the gradation), where F• is the filtration of DX
by the order of differential operators. We have grF DX = OX [T X].
Examples.
(1) Let (M, F• ) be a coherent left DX -module with a good filtration. Then
RF M is a coherent RF DX -module, hence, after tensoring with OX , one gets a
coherent left RX -module. Restricting to z = 1 gives M back (without filtration) and restricting to z = 0 gives grF M . The support of the restriction to
z = 0 is the characteristic variety.
(2) On the disc with coordinate t, consider the z-connection with matrix
1
dt
(eα,a (z) Id +N )
z
t
where N is a constant nilpotent matrix and eα,a (z) is a degree two polynomial in z of the form 12 (αz 2 − α) + az, with α ∈ C and a ∈ R. Restricting to z = 1 gives a meromorphic bundle with connection ∇ having matrix
[(a + i Im α) Id +N ]dt/t, while restricting to z = 0 gives a meromorphic Higgs
bundle with Higgs field θ having matrix (−α Id +N )dt/t. If τ is the coordinate on T ∗ X dual to t, the support of the corresponding OX [T X]-module is
{tτ + α = 0}. This is the spectral curve of the Higgs bundle on X r {0}. Notice
that it is not homogeneous with respect to the C∗ action on T ∗ X, but it is
Lagrangian.
INTRODUCTION TO TWISTOR D-MODULES
9
2.3. Extension of z-connections. Given a holomorphic bundle with
z-connection on U ⊂ X , we wish to extend it to X , either as a meromorphic bundle with meromorphic z-connection, or as a left RX -module, in
a minimal way. One is aware that the problem is not trivial: in the case of
variations of Hodge structures for instance, the existence relies on the theorems
of Schmid (in dimension one) or Cattani-Kaplan-Schmid and Kashiwara-Kawai
in dimension > 2. On the other hand, it is not easy to characterize the RX modules one obtains by extending a variation of polarized Hodge structure.
These are polarized Hodge modules (defined by M. Saito, [6]) if we consider
the minimal extension, or mixed Hodge modules (cf. [7]) if one considers
the maximal extension as meromorphic bundles. In this sense, the minimal
extension behaves in a better way, so we will only consider it below.
Theorem (Simpson [8], Biquard [1], T. Mochizuki [4]). Let X be a complex
manifold, D a normally crossing divisor, and T be a polarized variation of
twistor structures of weight w on X r D (i.e., a flat holomorphic bundle with
harmonic metric if w = 0). Then, if the corresponding harmonic metric is tame
along D, this variation extends as a polarized twistor D-module of weight w.
Using the direct image theorem for twistor D-modules (cf. below), we more
precisely get an equivalence statement:
Theorem. Let Z be a closed analytic subset of X, and let T ⊂ Z be a closed
analytic subset such that Z rT is smooth. Giving a polarized variation of twistor
structures of weight w on Z r T is equivalent to giving a polarized twistor
D-module of weight w on X supported on Z with singular support contained
in T .
2.4. The category R- Triples(X). It is now time to give the definition of
a twistor D-module. The definition is parallel to that of a polarizable Hodge
D-module. One first considers the category R- Triples(X) of triples on X: an
object in this category is a triple (M 0 , M 00 , C), where M 0 , M 00 are left RX modules, and C is a sesquilinear pairing
C : M|S0 ⊗O|S M|S00 −→ DbXR ×S/S ,
where DbXR ×S/S is the sheaf of distributions on X × S which are continuous
with respect to z ∈ S (so that we can restrict such a distribution to any given
value z o of z in S).
The notions that we have introduced in the case of R- Triples(pt) easily extend to the previous situation, in particular that of a sesquilinear duality of
weight w, denoted S : T → T ∗ (−w).
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C. SABBAH
Remark. This category is much too big for our purpose, but is a good framework
to work in. One would want to consider only strict objects, i.e., for which M 0
and M 00 have no OΩ0 -torsion, but this property, which is important, is not
preserved a priori by various operations. It is a theorem that it is preserved by
natural operations in the subcategory of polarizable twistor D-modules.
2.5. Direct images in R- Triples. Let f : X → Y be a projective morphism
and let T = (M 0 , M 00 , C) be an object of R- Triples(X) such that M 0 , M 00 are
coherent (and good). The direct images H j f+ T are then defined as objects
of R- Triples(Y ):
−j
• H
f+ M 0 and H j f+ M 00 are defined as in the theory of D-modules.
j
• H f+ C is defined by using integration of currents.
The behaviour with respect to adjunction is: H j f+ (T ∗ ) = (H −j f+ T )∗ . If
we are given a sesquilinear duality S : T → T ∗ (−w), we obtain various
morphisms H j f+ S : H j f+ T → (H −j f+ T )∗ (−w).
If we are given a relatively ample line bundle on X, there is associated to it
a Lefschetz morphism L : H j f+ T → H j+2 f+ T .
Theorem (C.S. [5]). Let (T , S ) be a polarized twistor D-module of weight w on
X. Then, for f as above, the ‘Hodge-Lefschetz properties’ are fulfilled:
j
• Each H f+ T is a pure twistor D-module of weight w + j,
L
j
• the Hard Lefschetz Theorem holds for (
j H f+ T , L ),
j
• S induces a polarization on the L -primitive part of H f+ T in the usual
way.
2.6. Specialization. Defining the operation of specialization and the subcategory of R- Triples(X) consisting of objects which are strictly specializable is
one of the main point in the definition of polarizable twistor D-modules. A polarizable twistor D-module of weight w will be an object of R- Triples(X) for
which the ‘restriction’ to any point of X is a polarized twistor structure of a
suitable weight. However, the operation of restriction has to be replaced with
that of specialization when there are singularities.
Let M be a DX -module and let f be a holomorphic function on X. Then
one defines, for any α ∈ C, functors ψf,α M with the help of the KashiwaraMalgrange filtration of if,+ M along {t = 0}:
i
X
 f /
X ×C
HH
HH
HH
HH
t
f HH# C
INTRODUCTION TO TWISTOR D-MODULES
11
These functors are defined for holonomic D-modules, and, more generally, to
D-modules which are strictly specializable along f = 0.
In a similar way, one can define functors ψf,a,α . The subcategory on which
these functors are defined is the category of strictly specializable triples. The
object ψf,a,α T is supported in {f = 0}, so, by iterating, we finally reach an
object supported on a point. In order to keep control on weights, it is moreover necessary to grading by the monodromy filtration M• at each step. We
can therefore give a definition analogous to that of polarizable Hodge Modules
(M. Saito [6]) by induction on the dimension of the support:
Definition. A pure twistor D-module of weight w is a S-decomposable object of
R- Triples(X) such that, for any local holomorphic function f and any (a, α),
the object grM
` ψf,a,α T is a pure twistor D-module of weight w + ` (supported
on {f = 0}).
There is also, similarly, the notion of polarization.
Theorem (C.S. [5]).
(1) The category of pure twistor D-modules of weight w is abelian.
(2) The category of pure polarized twistor D-modules of weight w is semisimple.
We say that the polarizable twistor D-module is of Deligne type (or has a
purely imaginary parabolic structure) if for any f , ψf,α,a T vanishes if α is
not purely imaginary, and the same property holds for any iterated specialized
objects. Then we have the following analogue of Corlette’s theorem:
Theorem (Mochizuki [4]). If X is projective, the restriction to z = 1 is an equivalence between polarizable twistor D-modules of weight 0 on X which are of
Deligne type and semisimple regular holonomic DX -modules (i.e., according to
the Riemann-Hilbert correspondence via the de Rham functor, to semisimple
perverse sheaves on X).
Remark. According to the direct image theorem above, this result gives a proof
of the decomposition theorem for the direct image of a semisimple perverse
sheaf on X by a projective morphism, a result conjectured (in the more general
situation of possibly irregular holonomic DX -modules) by M. Kashiwara, and
which has been given an arithmetic proof by V. Drinfeld.
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C. SABBAH
References
[1] O. Biquard – Fibrés de Higgs et connexions intégrables: le cas logarithmique (diviseur lisse),
Ann. Sci. École Norm. Sup. (4) 30 (1997), p. 41–96.
[2] K. Corlette – Flat G-bundles with canonical metrics, J. Differential Geom. 28 (1988), p. 361–
382.
[3] C. Hertling – tt∗ geometry, Frobenius manifolds, their connections, and the construction for
singularities, J. reine angew. Math. 555 (2003), p. 77–161.
[4] T. Mochizuki – Asymptotic behaviour of tame harmonic bundles and an application to pure
twistor D-modules, vol. 185, Mem. Amer. Math. Soc., no. 869-870, American Mathematical Society, Providence, RI, 2007.
[5] C. Sabbah – Polarizable twistor D-modules, Astérisque, vol. 300, Société Mathématique de
France, Paris, 2005.
[6] M. Saito – Modules de Hodge polarisables, Publ. RIMS, Kyoto Univ. 24 (1988), p. 849–995.
[7]
, Mixed Hodge Modules, Publ. RIMS, Kyoto Univ. 26 (1990), p. 221–333.
[8] C. Simpson – Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3 (1990), p. 713–
770.
[9]
, Higgs bundles and local systems, Publ. Math. Inst. Hautes Études Sci. 75 (1992), p. 5–95.
[10]
, Mixed twistor structures, Prépublication Université de Toulouse & arXiv: math.AG/
9705006, 1997.
C. Sabbah, UMR 7640 du CNRS, Centre de Mathématiques Laurent Schwartz, École polytechnique,
F–91128 Palaiseau cedex, France • E-mail : sabbah@math.polytechnique.fr
Url : http://www.math.polytechnique.fr/~sabbah