Formality and splitting of real non-abelian mixed Hodge structures

56 Pages·2009·0.48 MB·English

Formality and splitting of real non-abelian mixed Hodge
structures
9
0
0
J.P.Pridham∗
2

b
February 4, 2009
e
F

4

]
G
Abstract
A
. We define and construct mixed Hodge structures on real schematic homotopy
h
typesofcomplexprojectivevarieties,givingmixedHodgestructuresontheirhomo-
t
a topygroups. WealsoshowthatthesesplitontensoringwiththeringR[x]equipped
m
withtheHodgefiltrationgivenbypowersof(x i),givingnewresultsevenforsim-
−
[ ply connected varieties. These split mixed Hodge structures can be recoveredfrom

structures on cohomology groups of local systems. Real Deligne cohomology and
1
v ArchimedeancohomologycanalsoberecoveredfromthesemixedHodgestructures.
0
7
7 Contents
0
.
2
Introduction 2
0
9
0 1 Non-abelian structures 4
:
v 1.1 Hodge filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
i
X 1.1.1 SL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2
r 1.2 Twistor filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
a
1.3 Mixed Hodge structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Mixed twistor structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Relative Malcev homotopy types 12
2.1 Review of pro-algebraic homotopy types . . . . . . . . . . . . . . . . . . 13
2.1.1 Pro-algebraic groupoids . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.2 The pro-algebraic homotopy type of a topological space . . . . . 15
2.1.3 Relative Malcev homotopy types . . . . . . . . . . . . . . . . . . 16
2.2 Equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Relative homotopy types . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.1 General base schemes . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.2 Derived pullbacks and base change . . . . . . . . . . . . . . . . . 22
∗ThisworkwassupportedbyTrinityCollege,Cambridge;andtheEngineeringandPhysicalSciences
Research Council [grant numberEP/F043570/1].
1
3 Structures on relative Malcev homotopy types 24
3.1 Hodge filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.1 Extensions and SL . . . . . . . . . . . . . . . . . . . . . . . . . 26
2
3.1.2 Analogies with limit Hodge structures . . . . . . . . . . . . . . . 27
3.2 Twistor filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Mixed Hodge structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Mixed twistor structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Mixed Hodge structures on relative Malcev homotopy types of com-
pact K¨ahler manifolds 30
4.1 Real homotopy types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.1.1 The Hodge filtration . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.1.2 Mixed Hodge structures . . . . . . . . . . . . . . . . . . . . . . . 31
4.1.3 The family of formality quasi-isomorphisms . . . . . . . . . . . . 32
4.1.4 Comparison with Morgan . . . . . . . . . . . . . . . . . . . . . . 35
4.1.5 Real Deligne cohomology . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Relative Malcev homotopy types . . . . . . . . . . . . . . . . . . . . . . 37
4.2.1 The reductive fundamental groupoid is pure of weight 0 . . . . . 37
4.2.2 Variations of Hodge structure . . . . . . . . . . . . . . . . . . . . 38
4.2.3 Mixed Hodge structures . . . . . . . . . . . . . . . . . . . . . . . 40
4.2.4 Archimedean cohomology . . . . . . . . . . . . . . . . . . . . . . 42
5 Mixed twistor structures on relative Malcev homotopy types of com-
pact K¨ahler manifolds 43
5.1 Unitary actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.1.1 Evaluation maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.1.2 Analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6 Simplicial and singular varieties 48
6.1 Semisimple local systems. . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.2 The Malcev homotopy type . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.3 Mixed Hodge structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.4 Mixed twistor structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.5 Enriching twistor structures . . . . . . . . . . . . . . . . . . . . . . . . . 53
Bibliography 54
Introduction
The main aims of this paper are to construct mixed Hodge structures on the real
schematic homotopy types of complex varieties, and to investigate how far these can be
recovered from the structures on cohomology groups of local systems.
In [Mor], Morgan established the existence of natural mixed Hodge structures on
the minimal model of the rational homotopy type of a smooth variety X, and used this
to define natural mixed Hodge structures on the rational homotopy groups π (X Q)
∗
⊗
of X. This construction was extended to singular varieties by Hain in [Hai1].
2
WhenX isalsoprojective,[DGMS]showedthatitsrationalhomotopytypeisformal;
in particular, this means that the rational homotopy groups can be recovered from the
cohomology ring H∗(X,Q). However, in [CCM], examples were given to show that
the mixed Hodge structure on homotopy groups could not be recovered from that on
cohomology. We will describe how formality interacts with the mixed Hodge structure,
showing the extent to which the mixed Hodge structure on π (X Q) can be recovered
∗
⊗
from the pure Hodge structure on H∗(X,Q).
This problem was suggested to the author by Carlos Simpson, who asked what
happens when we vary the formality quasi-isomorphism. The answer (Corollary 4.14)
is that, if we define the ring := R[x] to be pure of weight 0, with the Hodge filtration
S
on C given by powers of (x i), then there is an -linear isomorphism
R
S ⊗ − S
π∗(X ⊗Q)⊗QS ∼= π∗(H∗(X,Q))⊗QS,
preserving the Hodge and weight filtrations, where the homotopy groups π (H∗(X,Q))
∗
are given the Hodge structurecoming from the Hodgestructureon thecohomology ring
H∗(X,Q), regarded as a rational homotopy type.
We also extend these results to schematic (and relative Malcev) homotopy types,
giving more information when X is not simply connected (Corollaries 4.35 and 5.16).
ThestartingpointistheHodgestructuredefinedonthereductivecomplexpro-algebraic
fundamental group ̟(X,x)red in [Sim3], in the form of a discrete C∗-action. We only
C
makeuseoftheinducedaction ofU C∗, sincethispreservestherealform̟(X,x)red,
1 ⊂ R
respects the harmonic metric, and has has the important property that the map
π (X,x) U ̟(X,x)red
1 1 R
× →
isrealanalytic. Weregardthisasakindofpureweight0Hodgestructureon̟(X,x)red,
R
since a pure weight 0 Hodge structure is the same as an algebraic U -action. We
1
extend this to a mixed Hodge structureon theschematic (or relative Malcev) homotopy
type (Theorem 4.32 and Proposition 5.3). In some contexts, the unitary action is
incompatible with the homotopy type. In these cases, we instead only have mixed
twistor structures (as defined in [Sim2]) on the homotopy groups (Corollary 5.2). We
also extend these results to singular varieties.
The structure of the paper is as follows.
In Section 1, we introduce our non-abelian notions of algebraic mixed Hodge and
twistor structures. If we define C∗ = ( A1) 0 = A2 0 and S = G by
C/R −{ } ∼ −{ } C/R m
Weil restriction of scalars, then our firQst major observation (Corollary 1.8)Qis that real
vector spaces V equipped with filtrations F on V C correspond to flat quasi-coherent
⊗
modulesonthestack[C∗/S],viaaReesmoduleconstruction,withV beingthepullback
along 1 C∗. This motivates us to define an algebraic Hodge filtration on a real object
∈
Z as an extension of Z over thebasestack [C∗/S]. This is similar to theapproach taken
by Kapranov to define mixed Hodge structures in [Kap]; see Remark 1.9 for details.
Similarly, filtered vector spaces correspond to flat quasi-coherent modules on the
stack [A1/G ], so we define an algebraic mixed Hodge structure on Z to consist of
m
an extension Z over [A1/G ] [C∗/S], with additional data corresponding to an
MHS m
×
opposedness condition (Definition 1.29). This gives rise to non-abelian mixed Hodge
structures in the sense of [KPS], as explained in Remark 1.32. In some cases, a mixed
3
Hodge structure is too much to expect, and we then give an extension over [A1/G ]
m
×
[C∗/G ]: an algebraic mixed twistor structure. For vector bundles, algebraic mixed
m
Hodge and twistor structures coincide with the classical definitions (Propositions 1.34
and 1.42).
Section 2 is mostly a review of the relative Malcev homotopy types introduced in
[Pri2], generalising both schematic and real homotopy types, with some new results in
2.3 on homotopy types over general bases (rather than just fields). In Section 3, the
§
constructions of Section 1 are then extended to homotopy types. The main results are
Propositions3.14and3.17, showinghownon-abelianalgebraicmixedHodgeandtwistor
structures on relative Malcev homotopy types give rise to such structures on homotopy
groups.
In the next two sections, Theorems 4.32 and 5.1 establish the existence of algebraic
mixedHodgeandmixedtwistorstructuresonvariousrelativeMalcevhomotopytypesof
compact Ka¨hler manifolds, while Corollary 5.2 establishes a mixed twistor structure on
homotopy groups, with Corollaries 4.35 and 5.16 establishing mixed Hodge structures
on homotopy groups. Proposition 4.18 shows that these are compatible with Morgan’s
mixed Hodge structures on rational homotopy types and groups. However, Remarks 4.7
and 5.4 show that these differ substantially from the complex Hodge structures defined
in [KPT2], while indicating how some of their structure can be recovered from ours.
Moreover, there is an S-equivariant morphism row : SL C∗ corresponding to
1 2
→
projection of the first row; all of the structures split on pulling back along row , and
1
these pullbacks can be recovered from cohomology of local systems. This is because
the principle of two types (or the ddc-lemma) holds for any pair ud + vdc,xd + ydc
of operators, provided (u v) GL . The pullback row corresponds to tensoring with
x y 2 1
∈
the algebra described above. Proposition 3.7 shows how this pullback to SL can be
2
S
regarded as an analogue of the limit mixed Hodge structure, while Proposition 4.19,
Corollary 4.40 and Proposition 4.21 show how it is closely related to real Deligne co-
homology, Consani’s Archimedean cohomology and Deninger’s Γ-factor of X at the
Archimedean place. Remark 4.17 explains how the mixed Hodge structure corresponds
to a locally nilpotent derivation on the split Hodge structure over SL .
2
In Section 6, we extend the results of Sections 4 and 5 to simplicial compact Ka¨hler
manifolds, and hence to singular proper complex varieties.
I would like to thank Carlos Simpson for drawing my attention to the questions
addressed in this paper, and for much useful discussion. I would also like to thank Jack
Morava for suggesting that non-abelian mixed Hodge structures should be related to
Archimedean Γ-factors.
1 Non-abelian structures
1.1 Hodge filtrations
In this section, we will define algebraic Hodge filtrations on real affine schemes. This
construction is essentially thatof [Sim1] 5, with thedifferencethat weareworkingover
§
the reals.
Definition 1.1. Define C to be the real affine scheme A1 obtained from A1 by
C/R C
Q
4
restrictionofscalars, soforanyrealalgebraA, C(A)= A1C(A⊗RC)∼= A⊗RC. Choosing
i C gives an isomorphism C = A2, and we let C∗ be the quasi-affine scheme C 0 .
∈ ∼ R −{ }
Define S to be the real algebraic group G obtained as in [Del1] 2.1.2 from
C/R m
Gm,C by restriction of scalars. Note that thQere is a canonical inclusion Gm ֒ S, and
→
that S acts on C and C∗ by inverse multiplication, i.e.
S C C
× →
(λ,w) (λ−1w).
7→
Remark 1.2. AmorestandardS-action isgivenbytheinclusionS ֒ A2 = C. However,
∼
→
we wish to regard C as being S , so C is C( 1) for the C∗-action.
∪{∞} −
Fix an isomorphism C = A2, with co-ordinates u,v on C so that the isomorphism
∼
C(R) = C is given by (u,v) u+iv. Thus the algebra O(C) associated to C is the
∼
7→
polynomial ring C = R[u,v]. S is isomorphic to the scheme A2 (u,v) : u2+v2 = 0 .
R−{ }
Definition 1.3. Given an affine scheme X over R, we define an algebraic Hodge filtra-
tion X on X to consist of the following data:
F
1. an S-equivariant affine morphism X C∗,
F
→
2. an isomorphism X ∼= XF,1 := XF×C∗,1SpecR.
Definition 1.4. A real splitting of the Hodge filtration X consists of an S-action on
F
X, and an S-equivariant isomorphism
X ×C∗ ∼= XF
over C∗.
Remark 1.5. Note that giving X as above is equivalent to giving the affine morphism
F
[X /S] [C∗/S] of stacks.This fits inwith theideain [KPS]thatifOBJis an -stack
F
→ ∞
parametrising some -groupoid of objects, then the groupoid of non-abelian filtrations
∞
of this object is om([A1/G ],OBJ).
m
H
Now, we may regard a quasi-coherent sheaf F on a stack X as equivalent to the
affine cogroup Spec(O F) over X. This gives us a notion of an algebraic Hodge
X
⊕
filtration on a real vector space. We now show how this is equivalent to the standard
definition.
Lemma 1.6. There is an equivalence of categories between flat quasi-coherent G -
m
equivariant sheaves on A1, and exhaustive filtered vector spaces, where G acts on A1
m
via the standard embedding G ֒ A1.
m
→
Proof. Lettbetheco-ordinateonA1,andM globalsectionsofthesheaf. SinceM isflat,
t
0 M M M k 0 is exact, so t is an injective endomorphism. The G -
k[t],0 m
→ −→ → ⊗ →
actionisequivalenttogivingadecompositionM = M ,andwehavet :M ֒ M .
n n n+1
→
Thus the images of Mn n∈Z give a filtration on ML k[t],1k.
{ } ⊗
Conversely, set M to be the Rees module Rees(V,F) := F V. If I is a k[t]-
n
ideal, then I = (f), since k[t] is a principal ideal domain. ThLe map M I M is
⊗ →
thus isomorphic to f : M M. Writing f = a tn, we see that it is injective on
n
→
M = Mn. Thus M I M is injective. ThuPs M is flat by [Mat] Theorem 7.7.
⊗ →
L
5
Remark 1.7. We might also ask what happens if we relax or strengthen the condition
that the filtration be flat, since non-flat structures might sometimes arise as quotients.
An arbitrary algebraic filtration on a real vector space V is a system W of complex
r
vector spaces with (not necessarily injective) linear maps s : W W , such that
r r+1
→
lim W = V.
r→∞ r ∼
−→
Corollary 1.8. The category of flat algebraic Hodge filtrations on real vector spaces
is equivalent to the category of pairs (V,F), where V is a real vector space and F an
exhaustive decreasing filtration on V C. A real splitting of the Hodge filtration is
R
⊗
equivalent to giving a real Hodge structure on V (i.e. an S-action).
Proof. The flat algebraic Hodge filtration on V gives an S-module M on C∗, with
M|1 = V. Observe that C∗ ⊗R C ∼= A2C −{0}, and S ⊗R C ∼= Gm ×Gm, compatible
with the usual actions, the isomorphism given by (u,v) (u + iv,u iv). Writing
7→ −
A2 0 = (A1 G ) (G A1), we see that giving M C amounts to giving two
C −{ } × m ∪ m× ⊗
filtrations (F,F′) on V C, which is the fibre over (1,1) in the new co-ordinates. The
R
⊗
real structure determines behaviour under complex conjugation, with F′ = F¯.
Remark 1.9. Althoughflatmoduleson[C∗/S]also correspondtoflatmoduleson[C/S],
we do not follow [Kap] in working over the latter, since many natural non-flat objects
arise on [C/S] whose behaviour over 0 C is pathological. However, our approach has
∈
the disadvantage that we cannot simply describe the bigraded vector space gr gr V,
F F¯
which would otherwise be given by pulling back along [0/S] [C∗/S].
→
The motivating example comes from the embedding ∗ A• of real harmonic
H →
forms into the real de Rham algebra of a compact Ka¨hler manifold. This gives a quasi-
isomorphism of the associated complexes on [C∗/S], since the maps Fp( ∗ C)
H ⊗ →
Fp(A• C) are quasi-isomorphisms. However, the associated map on [C/S] is not a
⊗
quasi-isomorphism, as this would force the derived pullbacks to 0 C to be quasi-
∈
isomorphic, implying that the maps pq Apq are isomorphisms.
H →
Remark 1.10. We might also ask what happens if we relax or strengthen the condition
that the Hodge filtration be flat.
An arbitrary algebraic Hodge filtration on a real vector space V is a system Fp of
complex vector spaces with (not necessarily injective) linear maps s :Fp Fp−1, such
→
that lim Fp = V C.
p→−∞ ∼ ⊗
Fo−→r a flat algebraic Hodge filtration M on real vector spaces to be a locally free
moduleover C∗ is equivalent to saying that thefiltered complex vector space (V C,F)
⊗
admits a splitting.
Definition 1.11. For C∗ as in 1.1, fix an isomorphism C = A2, with co-ordinates
§ ∼
(u,v), so that the isomorphism C(R)= C is given by (u,v) u+iv. Let C∗ C∗ be
∼
7→ →
the ´etale covering of C∗ given by cutting out the divisor u iv = 0 from C∗ C.
R
{ − } f ⊗
Lemma 1.12. There is an equivalence of categories between flat S-equivariant quasi-
coherent sheaves on C∗, and exhaustively filtered complex vector spaces.
Proof. First, observefthat there is an isomorphism C∗ ∼= A1C×Gm,C, given by (u,v) 7→
(u+iv,u−iv). As in Corollary 1.8, SC ∼= Gm,C ×Gfm,C under the same isomorphism.
Thus S-equivariant quasi-coherent sheaves on C∗ are equivalent to G 1-equivariant
m,C
×
f
6
quasi-coherent sheaves onthe schemeA1 C∗ given byu iv = 1. Now apply Lemma
C ⊂ 0 −
1.6.
f
1.1.1 SL
2
Definition 1.13. Define maps row ,row : GL A2 by projecting onto the first and
1 2 2
→
second rows, respectively. If we make the identification C = A2 of Definition 1.1, then
these are equivariant with respect to the right S-action GL S GL , given by
2 2
(A,λ) A ℜλ ℑλ −1. × →
7→ −ℑλℜλ
(cid:0) (cid:1)
Definition 1.14. Define an S-action on SL to be given on the top row as right mul-
2
tiplication by λ ℜλ ℑλ −1, and on the bottom row by λ ℜλ −ℑλ .
7→ −ℑλℜλ 7→ ℑλ ℜλ
Let row : SL (cid:0) C∗ b(cid:1)e the S-equivariant map given by pro(cid:0)jection (cid:1)onto the first
1 2
→
row.
Remark 1.15. Observe that, as an S-equivariant scheme over C∗, we may decompose
row : GL C∗ as GL = SL 1 0 , where the S-action on G has λ acting as
1 2 → 2 2 × 0 Gm m
multiplication by λ 2. (cid:0) (cid:1)
| |
We may also write C∗ = [SL /G ], where G acts on SL as left multiplication by
2 a a 2
1 0 , where G is given the S-action for which the standard co-ordinate is of type
Ga 1 a
(cid:0)(1,1).(cid:1)
Lemma 1.16. The morphism row : SL C∗ is weakly final in the category of S-
1 2
→
equivariant affine schemes over C∗.
Proof. We need to show that for any affine scheme U equipped with an S-equivariant
morphism f : U C∗, there exists a (not necessarily unique) S-equivariant morphism
→
g :U SL such that f = row g.
2 1
→ ◦
If U = SpecA, then A is a O(C) = R[u,v]-algebra, with the ideal (u,v) = A, so
A
there exist a,b A with ua vb = 1. Thus the map factors through row : SL C∗.
1 2
∈ − →
Complexifying, and writing w = u+iv,w¯ = u iv gives an expression αw+βw¯ = 1.
−
Now splitting α,β into types, we have α10w +β01w¯ = 1. Similarly, 1(α10 +β01)w +
2
1(β01+α10)w¯ = 1, on conjugating and averaging. Write this as α′w+β′w¯ = 1. Finally,
2
note that y := α′+β′, x:= iα′ iβ′ are both real, giving uy vx= 1, with x,y having
− − −
the appropriate S-action to regard A as an O(SL )-algebra when SL has co-ordinates
2 2
(u v).
x y
Lemma 1.17. The affine scheme SL row1 C∗ is a flat algebraic Hodge filtration,
2
−−−→
corresponding to the algebra
:= R[x],
S
with filtration Fp( C) = (x i)pC[x].
S ⊗ −
Proof. Since row is flat and equivariant for the inverse right S-action, we know by
1
Lemma 1.8 that we have a filtration on S ⊗ C, for SpecS = SL2 ×row1,C∗,1 SpecR.
Spec consists of invertible matrices (1 0), giving the ring structure claimed.
S x 1 S
Todescribethefiltration,weuseLemma1.12, consideringthepullbackofrow along
1
C∗ C∗. The scheme SL := SL C∗ is isomorphic to C∗ A1, with projection
→ 2 2×row1,C ×
f g f f
7
onto A1 given by (u v) x iy. This isomorphism is moreover S -equivariant over
C x y 7→ − C
C∗, when we set the co-ordinates of A1 to be of type (1,0).
The filtration F on C then just comes from the decomposition on C[x iy]
f S ⊗ −
associated to the action of G 1 S , giving
m,C C
×{ } ⊂
FpC[x iy]= (x iy)−p′C.
− −
pM′≥p
The filtration on C is given by evaluating this at y = 1, giving Fp( C) =
R ⊗ R ⊗
(x i)pC[x], as required.
−
Remark 1.18. We may now reinterpret Lemma 1.16 in terms of Hodge filtrations. An
S-equivariant affine scheme, flat over C∗ is equivalent to a real algebra A, equipped
with an exhaustive decreasing filtration F on A C, such that gr gr (A C).
⊗R F F¯ ⊗R
This last condition is equivalent to saying that 1 F1 +F¯1, or even that there exists
∈
α F1(A C) with α = 1. We then define a homomorphism f : A by setting
R
∈ ⊗ ℜ S →
f(x)= α, noting that f(1+ix) = α F1(A C), as required.
R
ℑ ∈ ⊗
We may make use of the covering row :SL C∗ to give an explicit description of
1 2
→
the derived direct image Rj∗OC∗ as a DG algebra on C, for j :C∗ C, as follows.
→
The G -action on SL of Remark 1.15 gives rise to an action of the associated Lie
a 2
algebra g = R on O(SL ). Explicitly, the standard generator N g acts as the
a ∼ 2 a
derivation with Nx = u,Ny = v,Nu = Nv = 0, for co-ordinates (u∈v) on SL . The
x y 2
Nǫ
DG algebra O(SL ) O(SL )ǫ, for ǫ of degree 1, is an algebra over O(C) = R[u,v],
2 2
−−→
so we may consider the DG algebra j−1O(SL ) Nǫ j−1O(SL )ǫ on C∗, for j : C∗ C.
2 2
−−→ →
This is an acyclic resolution of the structure sheaf OC∗, so
Rj∗OC∗ j∗(j−1O(SL2) Nǫ j−1O(SL2)ǫ)= (O(SL2) Nǫ O(SL2)ǫ),
≃ −−→ −−→
regarded as an O(C)-algebra. This construction can moreover be made S-equivariant
by defining λ S(R) to act on ǫ as multiplication by λ 2.
∈ | |
Nǫ
Definition 1.19. From now on, we willdenote the DG algebra O(SL ) O(SL )ǫ by
2 2
−−→
Rj∗OC∗, thereby making a canonical choice of representative in this equivalence class.
1.2 Twistor filtrations
Definition 1.20. GivenanaffineschemeX over R,wedefineanalgebraic(real)twistor
filtration X on X to consist of the following data:
T
1. a G -equivariant affine morphism T :X C∗,
m T
→
2. an isomorphism X ∼= XT,1 := XT C∗,1SpecR.
×
Definition 1.21. A real splitting of the twistor filtration X consists of a G -action
T m
on X, and an G -equivariant isomorphism
m
X ×C∗ ∼= XT
over C∗.
8
Definition1.22. Adapting[Sim2] 1fromcomplextorealstructures,saythatatwistor
§
structure on a real vector space V consists of a vector bundle E on P1, with an isomor-
R
phism V = E , the fibre of E over 1 P1.
∼ 1
∈
Proposition 1.23. The category of finite flat algebraic twistor filtrations on real vector
spaces is equivalent to the category of twistor structures.
Proof. Theflatalgebraic twistor filtration isaflatG -moduleM onC∗, withM = V.
m 1
|
Taking the quotient by the right G -action, M corresponds to a flat module M on
m Gm
[C∗/Gm]. Now, [C∗/Gm] ∼= [(A2 −{0})/Gm] = P1, so Lemma 1.6 implies that MGm
corresponds to a flat module E on P1. Note that E1 = (M|Gm)Gm ∼= M1 ∼= V, as
required.
Definition 1.24. Define the real algebraic group U to be the unitary group, whose
1
A-valued points are given by (a,b) A2 : a2+b2 = 1 . Note that U ֒ S, and that
1
{ ∈ } →
S/G = U . This latter S-action gives U a split Hodge filtration.
m ∼ 1 1
Lemma 1.25. There is an equivalence of categories between algebraic twistor filtrations
XT on X, and extensions X˜U1 of X over U1 (with X = (X˜U1)1) equipped with algebraic
Hodge filtrations (X˜U1)F, compatible with the standard Hodge filtration on U1.
Proof. Given an algebraic Hodge filtration (X˜U1)F over U1 C∗, take
×
X := (X˜U1) SpecR,
T F×U1,1
and observe that this satisfies the axioms of an algebraic twistor filtration. Conversely,
given an algebraic twistor filtration X (over C∗), set
T
(X˜U1) = (X U )/( 1, 1),
F T 1
× − −
with projection π(x,t) = (pr(x)t−1,t2) C∗ U .
1
∈ ×
Corollary 1.26. A flat algebraic twistor filtration on a real vector space V is equivalent
to the data of a flat O(U1)-module V˜U1 with V˜U1 ⊗O(U1) R = V, together with an
exhaustive decreasing filtration F on(V˜U1) C, with the morphism O(U1) RV˜U1 V˜U1
⊗ ⊗ →
respecting the filtrations (for the standard Hodge filtration on O(U ) C). In particular,
1
⊗
the filtration is given by Fp(V˜U1 C)= (a+ib)pF0(V˜U1 C).
⊗ ⊗
Definition 1.27. Given a flat algebraic twistor filtration on a real vector space V as
aOb(oUv1e),-mdeofidnuele,graFnV˜dUd1efitonebgerTthVe:=rea(lgrpFaV˜rtU1o)f⊗gOrF(Ug1r)F¯R(.V˜U1 ⊗ C). Note that this is an
These results have the following trivial converse.
Lemma 1.28. Analgebraic Hodge filtration X C∗ onX isequivalentto analgebraic
F
→
twistor filtration T : X C∗ on X, together with a U -action on X with respect to
T 1 T
→
which T is equivariant, and for which 1 U acts as 1 G .
1 m
− ∈ − ∈
Proof. The subgroups U and G of S satisfy (G U )/( 1, 1) = S.
1 m m× 1 − − ∼
9
1.3 Mixed Hodge structures
We now define algebraic mixed Hodge structures on real affine schemes.
Definition 1.29. GivenanaffineschemeX over R,wedefineanalgebraicmixedHodge
structure X on X to consist of the following data:
MHS
1. an G S-equivariant affine morphism X A1 C∗,
m MHS
× → ×
2. a real affine scheme grX equipped with an S-action,
MHS
3. an isomorphism X ∼= XMHS (A1×C∗),(1,1)SpecR,
×
4. a G S-equivariant isomorphism grX C∗ = X SpecR, where
m MHS ∼ MHS A1,0
× × ×
G acts on grX via the inclusion G ֒ S. This is called the opposedness
m MHS m
→
isomorphism.
Definition 1.30. Given an algebraic mixed Hodge structure X on X, define
MHS
grWX := X SpecR, noting that this is isomorphic to grX C∗. We
MHS MHS A1,0 MHS
× ×
also define X := X SpecR, noting that this is a Hodge filtration on X.
F MHS A1,1
×
Definition 1.31. A real splitting of the mixed Hodge structure X is a G S-
MHS m
×
equivariant isomorphism
A1 grX C∗ = X ,
× MHS× ∼ MHS
giving the opposedness isomorphism on pulling back along 0 A1.
{ } →
Remarks 1.32. 1. Note that giving X as above is equivalent to giving the affine
MHS
morphisms [X /G S] [A1/G ] [C∗/S] and grX BS of stacks,
MHS m m MHS
× → × →
satisfying an opposedness condition.
2. To compare this with the non-abelian mixed Hodge structures postulated in
[KPS], note that pulling back along the morphism C∗ C∗ gives an object
→
over [A1/Gm] × [C∗/SC] ∼= [A1/Gm] × [A1/Gm]C; thfis is essentially the stack
X of [KPS]. The stack X of [KPS] corresponds to pulling back along
dR f B,R
1 : SpecR C∗. Thus our algebraic mixed Hodge structures give rise to pre-
→
non-abelian mixed Hodge structures in the sense of [KPS]. Our treatment of
the opposedness condition is also similar to the linearization condition for a pre-
NAMHS, by introducing additional data corresponding to the associated graded
object.
AsforHodgefiltrations, thisgives usanotionofanalgebraic mixedHodgestructure
on a real vector space. We now show how this is equivalent to the standard definition.
Definition 1.33. Recall from [Del1] Definition 2.3.1 that a real mixed Hodge structure
is a triple (V,W,F), where V is a real vector space, W a finite increasing filtration on
V, and F a finite decreasing filtration on V C, satisfying the opposedness condition
R
⊗
grWgri grj (V C)= 0
n F F¯ ⊗
for i+j = n.
6
10