SCHEMATIC HOMOTOPY TYPES
ANTHONY BLANC, TING CHEN, TOBIAS BARTHEL
Remark This document is based on notes taken by Hiro Tanaka.
0. Introduction
The goal in the first part of our talk is to construct a certain universal stack pX b Cqsch
for a pointed and connected homotopy type X, the schematization of X over C. In the
second part, by specializing to homotopy types of smooth projective varieties, we will
equip pX bCqsch with a C -action, which then induces the Hodge structures on completed
fundamental groups, higher homotopy groups and cohomology encountered earlier.
Our main references are Champs affine by Toën and Schematic homotopy types and
non-abelian Hodge theory by Katzarkov, Pantev and Toën.
1. The schematization problem
Let us start by introducing Grothendieck’s schematization problem described in his
manuscript Pursuing stacks, as interpreted by Toën.
To this end, denote by Hopk q the homotopy category of simplicial presheaves on the
site of affine schemes over a field k with the flat topology. Note that, if A P Hopk q is
an abelian group stack, we can define its classifying stack BA by applying the classifying space construction level-wise, yielding another abelian group stack. Iterating this
procedure, we obtain Eilenberg-MacLane stacks:
Definition 1. Let A P Hopk q and define the Eilenberg-MacLane stacks K pA, iq for
i ¥ 0 by setting K pA, 0q A and K pA, i 1q BK pA, iq.
From Grothendieck’s perspective, the stacks K pGa , iq should be viewed as fundamental examples of schematic homotopy types, even though he did not give a rigorous
definition. However, the following one can be extracted from his ideas:
Definition 2. Given a subcategory C of the homotopy category of stacks, Hopk q, we
call it a schematic homotopy type category if it contains K pGa , iq for all i, and is stable
under homotopy limits.
Recall that we have the right derived functor of global section
RΓ : Hopk q Ñ HopsSetq
so that we can consider its restriction to a subcategory C as above. We are now ready
to state the schematization problem:
Schematization Problem. Find a schematic homotopy type category C such that
(1) the functor RΓ restricted to C has a left adjoint, called b k : HopsSetq Ñ C
Date: 2011 Talbot Workshop.
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ANTHONY BLANC, TING CHEN, TOBIAS BARTHEL
(2) the functor is fully faithful when restricted to connected rational homotopy types
of finite type.
2. Affine Stacks
In short, the theory of affine stacks provides a solution to the schematization problem.
In this section, we will work over a field k of characteristic 0. Denote by Algk the
category of commutative k-algebras, and by Schk the category of schemes over k. To
every k-algebra A we associate the affine scheme Spec A P Schk .
To get a homotopical version of affine schemes, we replace this category by a model
category: Let Alg∆
k be the category of cosimplicial algebras equipped with the following
model structure:
A weak equivalence is a map A Ñ B such that H ipN pAqq Ñ H ipN pB qq is an
isomorphism for all i, where N is the normalized chain complex functor.
Fibrations are defined to be level-wise surjections.
Cofibrations are the induced ones, i.e., maps with the left-lifting property with
respect to all trivial fibrations.
We now replace Schk by the category sPrk of simplicial presheaves over the site of
affine schemes over k with the flat topology. This extends the theory of ordinary schemes
which is included via the Yoneda embedding.
Proposition 1. There exists a local model structure on sPrk such that
op
Spec : pAlg∆
kq
Ñ sPrk ,
defined by the simplicial Hom, Spec ApSpec B qn HompAn , B q, is a right Quillen functor. Note that B is just an algebra, cosimplicial in a trivial way.
Therefore, we can consider its right derived functor eluded to above,
op
RSpec : pAlg∆
kq
Ñ sPrk ,
which turns out to be fully faithful. So, as in ordinary algebraic geometry, we have a
good notion of affines.
Definition 3. An affine stack is an object in the essential image of RSpec.
Example Let S piq, to be thought of as an analogue of the i-sphere, be the cosimplicial algebra generated by Ski , the cosimplicial k-module corresponding to the complex
with just k in degree i. Then the Eilenberg-MacLane stack K pGa , iq is equivalent to
RSpec S piq and hence is an example of an affine stack.
More generally, one can give the following characterization of affine stacks in terms of
cohomology with coefficients in Ga .
Proposition 2. Every affine stack is a homotopy limit of K pGa , iqs.
Another equivalent definition uses the notion of O-locality: A morphism between
stacks is called an O-local equivalence if it induces an isomorphism in cohomology with
coefficients in Ga .
Proposition 3. A stack A is affine if and only if it is sub-affine, i.e., representable by
some simplicial affine scheme, and O-local.
SCHEMATIC HOMOTOPY TYPES
3
3. Schematic homotopy types
Given a stack F P Hopk q, we can talk about the affinization pF b k quni of F , which is
the universal affine stack equipped with a map from F . An explicit model for affinization
of F is given by the following formula
pF b kquni RSpec OpF q,
where O is the right adjoint to RSpec. In fact, Toën showed that affinization provides
a solution to the schematization problem, but it has some deficiencies. Essentially, the
problem is that the fundamental group of the affinization of a topological space turns out
to be represented by a unipotent group scheme, implying that pF bk quni for F P HopsSetq
contains no information about irreducible local systems of higher rank on F .
To get rid of this restriction, Toën introduces the theory of schematic homotopy
types. The idea is to replace O-equivalence by another notion of equivalence called P equivalence. These are morphisms which induce isomorphisms in cohomology, but now
with coefficients in all K pA, V, nq, where A is an affine group scheme and V can be
any linear representation of A of finite dimension. Here, K pA, V, nq is the semi-direct
product of K pV, nq with A via the action of A on V .
Definition 4. A schematic homotopy type is a pointed connected stack F
that
(1) the stack RΩ F of loops on F is affine, and
(2) F is P -local
P Hopkq such
Schematic homotopy types are characterized among pointed and connected stacks by
the property that their fundamental group and higher homotopy sheaves are represented
by affine group schemes and unipotent group schemes, respectively, thereby avoiding the
aforementioned restrictive nature of affine stacks. The main result in Toën’s paper can
be summarized as follows.
Theorem 1. For every pointed and connected homotopy type X P HopsSetq, there exists
a schematization
pX b Cqsch
of X over C, the universal schematic homotopy type equipped with a map from X.
Moreover, schematization gives a solution to the schematization problem as stated above.
4. Properties of the schematization
As shown above, for X a smooth, projective variety over C, associated to its underlying
topological space is a schematic homotopy pX b Cqsch P HopCq, called the schematization
of X. By universality, one can deduce the following properties.
Proposition 4. The schematization pX b Cqsch of any connected and pointed homotopy
type X satisfies the following three properties:
(1) We can recover the cohomology of X with coefficients in C, H ppX b Cqsch , Ga q H pX, Cq
(2) For X simply connected and i ¡ 1, πi ppX b Cqsch , q πi pX, q b C
(3) Moreover, we have π1 ppX b Cqsch , q π1 pX, qalg .
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ANTHONY BLANC, TING CHEN, TOBIAS BARTHEL
The key idea in the paper by Katzarkov, Pantev and Toën is to define a C -action
on pX b Cqsch such that the induced action on the homotopy and cohomology groups
will recover the Hodge filtration. To explain this, recall that a Hodge filtration on the
cohomology space H n pX, Cq is the same thing as a C -action on H n pX, Cq. To be more
precise, when y P H p,q , then for λ P C , λ acts on y by λpy q λp y.
Similarly, Morgan in 1978 described a mixed Hodge structure on the complexified
homotopy groups, πi pX, xq b C, at least in the simply connected case. Later in 1998,
Simpson defined a C -action on the π1 pX, xqred , the pro-reductive completion of π1 , i.e.,
the maximal reductive quotient of the pro-algebraic completion of π1 .
Remark In fact, Morgan’s mixed Hodge structure on higher homotopy groups coincides
with the one exhibited by Hain, which was introduced in one of the previous talks. Note,
however, that their constructions differ on fundamental groups.
5. The C -action on pX
b Cqsch
We are now ready to state the main theorem.
Theorem 2. There exists a C -action on pX b Cqsch which recovers the ordinary action
on
(1) the cohomology groups H pX, Cq,
(2) the higher homotopy groups, assuming X to be simply connected, and on
(3) the completed fundamental group π1 pX, xqred .
This gives a pure Hodge structure on pX b Cqsch , but there exists also mixed Hodge
structure. Recall that, on the level of cohomology, there is a weight filtration such that
the associated graded will be a pure Hodge structure. For stacks, instead of a weight
filtration, we have a weight tower on pX b Cqsch .
For F pX b Cqsch , there exists a tower in Hopk q
F
Ñ . . . Ñ Fi Ñ . . . Ñ F1 Ñ F0 of pointed stacks with homotopy fiber W i1 over Fi . The associated long exact sequence
of homotopy groups
π F
/ ...
/ π Fi
O
/ ...
u
u
u
uu
uu
u
uz u
π pWi1 q
/ π F1
O
/ π F0
u
u
uu
uu
u
uz u
π pW0 q
gives rise to a spectral sequence, the weight spectral sequence associated with pX b Cqsch .
This has two applications: First of all, the spectral sequence can be used to calculate
the homotopy groups of the schematization, which appear to be quite mysterious in
general. Furthermore, it can be used to exhibit constraints on the homotopy types
of projective varieties. We refer to the paper mentioned before for the details of a
particular counterexample - a homotopy type which is not the homotopy type of any
smooth projective variety.
SCHEMATIC HOMOTOPY TYPES
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6. The construction of the Hodge filtration
Let us now talk about the actual construction of the C -action. Note that we have
not done any non-abelian Hodge theory yet, but it will come up during the construction
of this action.
To this end, fix a smooth projective variety X over C and let LDol be the category
of Higgs bundles on X, polystable and with vanishing ci as always. The objects of
this category are pairs pV, D2 q with V a smooth vector bundle and D2 B θ. Then
D2 : V Ñ A1 b V such that D2 pa, sq B a s aD2 psq, where A is the sheaf of smooth
differential forms on X. Furthermore, TDol is the category of ind-objects in LDol ; you
can think of this as a completion of LDol with respect to the inductive limits. Similarly
we have LDR the category of semi-simple local systems on X where objects are pairs
pV, ∇q and its ind-category TDR .
The non-abelian Hodge theorem tells us that these categories are equivalent:
LDR
L ,
ÝÑ
Dol
TDR
T ,
ÝÑ
Dol
in the version proved by Simpson.
Consider the fiber functor ωX : LDol Ñ Vect which first extends to TDol and then
admits a right adjoint called p. We would like to give a more explicit description of the
value of p on the 1-dimensional vector space 1: by passing through the above equivalence,
pp1q can be identified with the regular representation of the group GX π1 pX, xqred ,
i.e., the smallest reductive group containing π1 . Here we use the fact that an object
pV, ∇q on the deRham side is the same thing as a representation of GX .
Now there is a C -action on LDol by rescaling Higgs field
pV, B
θq ÞÑ pV, B
λθq,
which extends naturally to the ind-category. Since p respects the symmetric monoidal
structure on both categories and is C -invariant, pp1q is in fact a C -fixed point in the
category of commutative monoids in TDol .
To pV, D2 q an object in the category LDol , you can associate its Dolbeault complex,
which is
A0 pV q ÝÝÑ A1 pV q ÝÝÑ A2 pV q ÝÝÑ ...
D2
D2
D2
with Ai pV q the global sections of V b An . By passing to the inductive limit of these
Higgs bundles, one can similarly construct the Dolbeault complex of pp1q, which will be
denoted by ADol ppp1q, D2 q.
Now let us define the C -action on this Dolbeault complex of pp1q. Firstly, the
µλ
action of λ P C induces an isomorphism pp1q ÝÑ
λ pp1q, because pp1q is fixed by the
C -action. So this gives rise to a map
λ
ADol ppp1q, D2 q ÝÑ
ADol pλ pp1q, D2 q.
µ
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ANTHONY BLANC, TING CHEN, TOBIAS BARTHEL
Secondly, there is a map called rλs, which is multiplication by λp on pp, q q-forms, and
fits into the following diagram
/ A λ p 1 , D 2
Dol
O
iSSS SSS
SSS
r
λs
SSS
SS
λaction
ADol ppp1q, D2 q
µλ
p pq
q
ADol ppp1q, D2 q.
The composition of rλs with the inverse of µλ is the desired action of λ on ADol ppp1q,
2
D q. Moreover, it can be checked that this C -action is compatible with the one by
GX . Therefore, by the Dold-Kan correspondence, we can take the RSpec of A and pass
to the quotient with respect to the natural GX -action. On the one hand, the resulting
differential stack, denoted pX b Cqdiff , still carries a C -action. On the other hand, by
using the non-abelian Hodge correspondence and mimicking the above construction, one
can show:
Theorem 3. For X a smooth projective variety, the differential stack of X gives an
explicit model for the schematization of X,
pX b Cqdiff pX b Cqsch;
Thus, it comes equipped with a C -action inducing the Hodge structures discussed before.