C OMPOSITIO M ATHEMATICA
J AMES A. C ARLSON
The one-motif of an algebraic surface
Compositio Mathematica, tome 56, n
o3 (1985), p. 271-314
<http://www.numdam.org/item?id=CM_1985__56_3_271_0>
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271
THE ONE-MOTIF OF AN ALGEBRAIC SURFACE
James A. Carlson
© 1985 Martinus Nijhoff Publishers, Dordrecht. Printed in The Netherlands.
1. Introduction
One of the
striking aspects
of thetheory
ofalgebraic
curves is that theHodge
structure on thecohomology
hasgeometric meaning:
Abel’stheorem
gives
a naturalisomorphism
between thecohomologically
de-fined Albanese
variety
and thegeometrically
defined Picardvariety.
Thepurpose of this paper is to discuss a
partial analogue
of Abel’s theorem forsufficiently singular algebraic
surfaces.To
place
theproblem
in its proper context, recall that acomplex
one-motif
[D,
section10]
consists of:(1)
acompact complex
torus A which can be made into an abelianvariety
(2)
acomplex multiplicative
group G(a
groupisomorphic
to(C *)’").
(3)
an extension J of Aby
G:(4)
a lattice L(a
groupisomorphic
to?l m ) (5)
ahomomorphism
A
morphism
of one motifs us u’ isgiven by
a commutativediagram
where
fj
preserves the canonical extension. If bothf,
andfj
areisomorphisms,
then uisomorphic
to u’. Atypical example
of such anobject
is furnishedby
a stable curve X with adistinguished
finitepoint
set
M c Xreg.
If one makes the definitionsZM = f divisors
ofdegree
zerosupported
inM},
P =
generalized
Picard ofX,
* Partially supported by NSF Grant MCS 8102745.
then the map
which sends a divisor to its linear
equivalence
class is a one-motif.According
toDeligne [D,
section10]
thecategory
of one-motifs isisomorphic
to thecategory
of torsion-free mixedHodge
structures oftype { ( p, p ), ( p, p - 1 ), ( p - 1, p ), ( p - 1, p - 1)}.
For aprojective variety X
there is aunique largest
such substructure ofH2( X )
withp =
1,
and hence aunique "largest" one-motif,
which we denote qx(the
one-motif of
Hodge).
Our main result is that for analgebraic surface,
ex isisomorphic
to ageometrically
definedobject,
the tracehomomorphism
Tx.
The
general
definition of the trace, which we shallgive
in section4, requires
thetheory
ofsemisimplicial
resolutions. Insimple
cases, how-ever, it is
possible
togive simple
answers. Consider therefore a surface Xwhich consists of two smooth
components
A and Bmeeting transversely
in a curve C. Assume further that A and B are
simply
connected. Letbe the
normalization,
and letbe a divisor on X which meets each copy of C
properly.
Define the traceof Z on C to be the
zero-cycle
obtainedby
difference of intersections:By analogy
with the smooth case, define the Neron-Severi group of X to beThen the trace induces a
homomorphism
THEOREM A
[C2] :
There is a naturalisomorphism
between the trace and theone-motif of H 2 ( X ),
modulo the torsion
of N S( X ).
The main result of the
present
paper is that Theorem A holds for anysurface,
once the trace has beenproperly
defined.We remark that when the second
cohomology
of X isentirely
one-mo-tivic,
as it is for K3degenerations,
that thegeometric
consequences areparticularly
strong.Typical
is thefollowing:
THEOREM
G[C2]:
Let X be the unionof
a smooth cubicsurface
and aplane,
tangent in at most one
point.
Then X is determined up toisomorphism by
the
polarized
mixedHodge
structure on theprimitive cohomology 01 H2( X).
Although
theproof
of the theoremgiven
in[C2] depends
onanalytic
methods
(integrals
andcurrents),
theproof given
here reliesentirely
onthe
theory
of sheaves andcomplexes
of sheaves on asemisimplicial
space[D].
Thistheory
is reviewedbriefly
in sections3, 6,
and8,
as progres-sively larger segments
of it are needed.Applications
of the main theoremare
given
in sections12, 13,
and 14.They
may be readindependently
ofthe
proof
of the maintheorem,
which isgiven
in sections 6through
11.We remark that as a dividend of our sheaf-theoretic method of
proof,
weobtain a second
interpretation
of the motif qx: it measures, for normalcrossing surfaces,
the obstruction that a Weil divisor faces inbecoming
aCartier divisor.
The author wishes to thank the
University
of Leiden for itshospitality during
the Fall of1981,
when much of the work of thepresent
paper was done.2. One-motif s and mixed
Hodge
structuresIn this section we recall a construction of the one-motif associated to a
torsion-free mixed
Hodge
structure of level one[C1 J.
Since the level is thelenth of the
Hodge filtration,
a level one structure is oftype
{( p, p), ( p, p - 1), ( p - 1, p), (p - 1, p - I)j
for aunique
p. Tobegin
the construction we define the Griffiths Jacobian
[GP]
of a mixedHodge
structure
by
These Jacobians admit a canonical filtration
by weight:
One verifies that
The
weight
filtration then induces a canonical filtrationwhere the
right-hand
term iscompact.
IfW2p-1 H
has level one, thenJP H is a semi-abelian
variety:
theright-hand
term ispolarizable
and theleft-hand term is of
multiplicative type:
Define next a
finitely generated
abelian groupWhen this group is torsion-free there is a canonical
homomorphism
To construct the canonical
homomorphism,
consider the extension of mixedHodge
structuresand choose sections sg
and sF
of gr overLez,
where the first preserves theinteger
lattice and where the second preserves theHodge
filtration.Let 4,
= S Z - S F be the differencehomomorphism,
and setfor all x in Lu. The
resulting
map from L pH to JPH is a homomor-phism independent
of the choice of sections whichgives
the obstruction torepresenting x by
anintegral
element ofW2 p
which also lies in FP.When
W2p -}
1 has leveloyez
carries the additional structure of aone-motif.
3.
Semisimplicial
spaces(geometry)
A
semisimplicial
space X. is a sequence of spaces(simplices) Xo,..., Xn
connected
by morphisms (face operators)
which
satisfy
the commutation relationsREMARK: We recall from
[D],
section5,
that asimplicial
space has both face anddegeneracy operators. Thus,
asemi simplicial
space is " half" ofa
simplicial
space.An
augmentation
of X. toward Y isgiven by
a mapsuch that E o
80
= E: 081.
If allsimplices
and face maps of X. are smooth(algebraic, etc.),
we say that X. is smooth(algebraic, etc.).
To every
simplicial
space isfunctorially
associated ageometric
realiza-tion,
which is anordinary topological
space. To defineit,
let0p
be thestandard
geometric p-simplex,
and define the realizationby
where R is the relation
generated by
where
is the linear inclusion of
AP-1
1 in the i-th face ofLlp.
Because the construction isfunctorial,
amorphism
induces a continuous map
We say that
f
is ahomotopy equivalence
ofsemisimplicial
spacesif 1 f. 1
is an
ordinary homotopy equivalence
oftopological
spaces.If X. is
augmented
towardsY,
then there is an induced mapThis map is a resolution of Y
Then 1 E: 1
is ahomotopy equivalence.
Notethat X. resolves Y
if 1 E: 1
has contractiblefibers,
and note thatEXAMPLE 1: Let
be a normal
crossing variety,
and letso that X. is the nerve of the
covering {du}.
LetSince
for
all y
inY p),
thegiven semisimplicial
space resolves Y. The corre-sponding simplicial
space isgiven by
thedisjoint
union withi0
...i p .
It therefore contains
degenerate simplices,
e.g.Dk
r1Dk
n...n Dk .
EXAMPLE 2: Let Y be a
variety
withsingular
locus E. Let be adesingularization,
and letE
be thepullback
of E to Y. Then theMayer-Vietoris diagram
defines a
semisimplicial
resolution of Y.If y
is a smoothpoint,
then thefiber of E over y is a
point. If y
is asingular point,
then the fiberof 1 E: 1
over y is a cone over
03C0-1(y).
Unlike the firstexample,
this resolution is notnecessarily
smooth.In the discussion
below,
it is useful to define a restricted class ofsemisimplicial
spaces.First,
we say that X. has natural dimension n ifhere dimension is taken as the maximum of the dimensions of the
components. Second,
we say that X. isnondegenerate
if for all pFinally,
we say that X. isdistinguished
if it is bothnondegenerate
and ofnatural dimension n. Both of the
examples
considered above are dis-tinguished,
and one can in fact show that everyvariety
hasdistinguished
resolutions
[C3].
The
technique is, roughly said,
to resolve thesingularities
of theMayer-Vietoris diagram,
then to take themapping
cone of the resolutionin the
category
ofpolyhedral
spaces. Each time this basicstep
isapplied,
the dimension of the
singular
locus of the resolutiondrops by
one.Repetition
of the basicstep
at most ntimes,
where n is the dimension ofY,
results in a smooth resolution in thecategory
ofpolyhedral objects.
4. The trace
In this section we shall define a functorial trace
homomorphism
for an
arbitrary semisimplicial
manifold. When this manifold is theMayer-Vietoris diagram
the construction agrees with that of the introduction. The proper gener- alization of Theorem A is therefore an assertion about
semisimplicial
spaces:
THEOREM A: Let
X.
be a smoothprojective semisimplicial
space, letT( X.)
be the trace, and let
11( X.)
be theone-motif
associated toH 2 ( X.).
Thenthere is a natural
isomorphism
betweenT( X.)
and1I( X.),
modulo torsion inNS(X.).
If
f:
X.- X.’ is amorphism,
one has a commutativediagram relating
T and 11 on the two spaces:
If in addition
f
is ahomotopy equivalence,
then11 (X:) and q(X.)
areisomorphic,
so thatT( XJ)
andT ( X.)
areisomorphic
modulo the torsionpart
ofNS( X.’).
Since the trace motif modulo torsion is ahomotopy invariant,
we are allowed to define the trace motif of a surface Y to be the trace motif of anysemisimplicial
resolution.Furthermore,
we areallowed,
if we sodesire,
to assume that the resolution used for Y isdistinguished.
Let us
begin
with the construction of the torusP( X.),
which mustspecialize
toPic ° ( C )
in theMayer-Vietoris
case. Define the group of divisors ingood position on Xp by
where 8’ is the map determined
by pullback
of localdefining equations.
Thus we have
homomorphisms
Define a
decreasing
chain of groups of divisorson Xp by
Define also groups of
meromorphic
functions:In the last
definition,
When
X.
has zero first Bettinumber,
thegeneralized
torus which weseek is then
where c is the function-to-divisor map. There is a natural
subgroup
with
compact quotient
groupPROPOSITION B :
If
X. is adistinguished semisimplicial surface,
then thereis a natural
isomorphism
PROOF: Because X. is a
distinguished semisimplicial surface, dim X2
=0,
and so
M*(X2)=C*(X2).
Because of thenondegeneracy assumption,
themap
is
surjective.
But then 8* induces therequired isomorphism,
since theimage
ofM;(X1) is, by definition, 8*C*(X1).
COROLLARY C:
If
X. is adistinguished semisimplicial surface,
thenm(X.)
is a
multiplicative
torus.For the
general
case in whichXl
haspossibly
nonzero first Bettinumber,
we definewhere
Pic0(X0)
is the group of divisors onXo
which arehomologous
tozero modulo those which are
linearly equivalent
to zero. Since8*Pic°( Xo )
is
compact
andconnected, Pm(X.)
and03B4*Pic0(X0)
intersectonly
in theidentity
element. Thus we may defineto obtain a canonical extension
where
Pm
is ofmultiplicative type,
whereP,
is apolarizable
abelianvariety.
Finally
we define the Neron-Severi group of X.by
where
Divs
andDiv,
are asabove,
and we define the traceto be the
homomorphism
inducedby
03B4*.5. The
multiplicative
groupPm
The
multiplicative
groupPm
has anotherinterpretation,
which we men-tion for its usefulness in the
applications.
Define a lattice(the " triple-point lattice")
and observe that the standard
spectral
sequence for thehomology
of asemisimplicial
space defines a naturalisomorphism
Elements of this lattice are
given by special zero-cycles
Dual to
T( X.)
is themultiplicative
groupWe claim that there is a natural
isomorphism
To construct
it,
consider avariety Y,
ameromorphic
functionf,
an azero-cycle e
whosesupport
isdisjoint
from thesupport
of(f).
Define apartial pairing by
where ny
is themultiplicity of y
ine.
Iff
is constant oncomponents and e
hasdegree
zero on eachcomponent, then ~f,03BE~
= 1. Nowconsider the
pairing
given by
Because
M*03B4(X1)
is annihilatedby T( X.). Indeed,
if8* f
=8*g
with g~ C*(X1),
then
The natural
pairing
between functionson Xl
andzero-cycles on X2
therefore descends to a
pairing
and this
pairing
induces therequired
map.That this map is an
isomorphism
followsby composition
of thefollowing
sequence ofisomorphisms
6.
Semisimplicial Spaces (Cohomology)
To compare the trace
homomorphism
with theHodge motif,
we shall usethe
cohomology theory
of sheaves on asemisimplicial
space. In this section we review the rudiments of thetheory,
which isessentially
thesame as for
simplicial
spaces[D,
section5].
To
begin,
recall that asemisimplicial object
in thecategory KA
ofcomplexes
of A-modules isby
definition a sequence ofcomplexes
(Kp, dp)
connectedby morphisms
which
satisfy
Define
and note that 8* o 8* = 0. Denote the
q-th graded piece
of Kpby (Kp)q,
and define the shifted
complex Kp[r] by
Set
and define
The
object ( sK, D)
is thesingle complex
associated to thesimplicial object
K: Thesimplicial
structure defines adecreasing
filtrationand a
spectral
sequenceOn
H"( sK )
there is acorresponding increasing
filtrationfor which
GrWlHn(sK)
is asubquotient
ofH’(Kn-1).
This construction suffices to define the
singular cohomology
of asemisimplicial
space: If C is thesingular
cochainfunctor,
then theobject
is a
simplicial object
inKA,
so that one definesIn the same way one defines the
homology
ofX.,
and one obtains theanalogues
of the usual universal coefficient theorems. In either case, thecohomology
comesequipped
with thesimplicial filtration,
whereIf X. is
augmented
towardsY,
there is an induced mapwhich is an
isomorphism
whenever E is a resolution.When X. is
smooth,
the same constructionapplies
with C ’replaced by A ;
the de Rham functor. Thus de Rhamcohomology
is definedby
If X. is a
semisimplicial object
in thecategory
ofcomplex manifolds,
theneach
complex A(Xp)
carries aHodge
filtration F : As aresult,
there areinduced filtrations on both the de Rham
complex
and the de Rhamcohomology of X..
If in addition X. iscompact
andKâhler,
thesimplicial (weight) spectral
sequencedegenerates
atE2
and theHodge
filtrationinduces a
Hodge
structure ofweight
1 onGrWlHnDR. Consequently,
thecomplex
andsemisimplicial
structuresjointly
define a mixedHodge
structure
[D,
sections 8.12through 8.15].
A
sheaf
on asemisimplicial
space consists of a sheaf Sp on eachXp
and of
morphisms
satisfying
the commutation relations(*).
Note that asemisimplicial
sheafis filtered
by weight:
theobjects
define
subsheaves,
and thegraded quotients
are
ordinary
sheaves.Typically examples
are the sheavesZ i W i
and 0 *which fit
together
togive
anexponential
sheaf sequence forsemisimpli-
cial spaces:
Sheaf
cohomology
in thesemisimplicial setting
is definedby choosing
f-acyclic
resolutionsC(Sp)
whichsupport morphisms
satisfying
the normal commutation rules. Theobject FC*(S*)
is thensemisimplicial
inKA,
so that one may setwith
Thus
semisimplicial
sheafcohomology
comesequipped
with aweight
filtration and a
spectral
sequencewhich relates it to
ordinary
sheafcohomology.
We remark that if G. denotes the constant sheaf of G-valued func-
tions,
thenCONVENTIONS: If S is a sheaf on
X.,
we shall use the notationswhere the last group is
ordinary
sheafcohomology.
Theexplicit
calcula-tions of later sections will be carried out
using Cech cohomology
forsemisimplicial
sheaves. TheCech complex
is thenwith differential
on
Xp.
HereU(Xp)
refers to a suitable cover ofXp.
7. The Cartier motif
To show that the trace and the
Hodge
motif arenaturally isomorphic,
weintroduce two versions of what we shall call the Cartier
motif,
This motif is of
independent
interest because of thefollowing:
DEFINITION: A line bundle on X. is a sheaf S such that each SP is invertible.
REMARK: If X. is
augmented
towards Y and if L is a line bundle onY,
then L’= E*L is a line bundle on X..THEOREM D: An element Z E
NS( X.)
arisesfrom
a line bunlde on X.if
and
only if k(Z)
vanishes inCA(X.).
(Thus k(Z)
is the obstruction whichdistinguishes
Cartier divisorsîrom Weil
divisors.)
THEOREM E: Let X be a normal
crossing surface,
X. its canonical resolu- tion. An element Z ENS( X.)
comesfrom
a line bundle on Xif
andonly if k(Z)
vanishes inCA(X.).
PROOF
(E):
Because of theorem D it suffices to show that the map below is anisomorphism:
To this
end,
consider theexponential
sheaf sequences for both X and X. :The induced map on
Z-cohomology
is anisomorphism
because theaugmentation
is a resolution(section 6).
For theanalogous
statement forO-cohomology,
weapply
theorem 3 of[S].
The five-lemma thenimplies
the
required isomorphism
forO*-cohomology. Q.E.D.
To construct the first version of the cartier
motif,
consider onceagain
the
exponential
sheaf sequence for X.. Because this sequence is filteredby
weight,
one can defineFigure 1
To construct the motivic
homomorphism
consider the commutative
grid
obtainedby filtering
theexponential
sheafsequence
by weight (Figure 1),
and consider itsresulting cohomology grid (Figure 2).
Make the identification(justified below)
and observe that
DC-1
defines acorrespondence
fromNS( X.)
toH2((O0)
whose
image
lies in theimage
of E. Since the setDC-1(x)
is a coset ofthe
image
ofDF,
thecorrespondence
defines ahomomorphism k’,
asdesired.
To
justify
the identification of the Neron-Severi group of X. with ker A n kerB,
we note first that the kernel of B is the Neron-Severi group ofXo :
it is the group ofintegral cohomology
classes of dimensionH3(W1Z.)
Figure
2.
2 and
type (1, 1). Next,
we claim thatH3(W1Z)
isjust H2(XI, Z).
Begin
with the relationsand then
apply
thespectral
sequence for the last group:where the last
equality requires
the fact that X. isdistinguished.
Onetherefore has
where
A,
as thecoboundary
in the filtration sequence forZ.,
is identified with8*,
the difference of the two maps inducedby
the inclusions ofXl
in
Xo.
Thiscompletes
therequired justification.
To construct the second Cartier
motif,
make the identificationand observe that
dc-’
defines acorrespondence
fromNS(X.)
toH2(W1O*)
whoseimage
is in the kernel of e. Since the setdc-1(x)
is acoset of the
image
offg,
there is a well-definedhomomorphism
where
As we show in section
10,
the two versions of the Cariter motif are in fact the same:PROPOSITION F: There is a canonical
and functorial isomorphism from
k’ tok"
The
proof
of Theorem D followsdirectly
from the definitions: on theone
hand, c-1(x)
lifts toH1(O*)
modulo theimage
ofH1(O0)
if andonly
if itsimage
under dvanishes,
but on the otherhand, dc-’
defines k"The
strategy
of theproof
of Theorem A is now to establish the sequence ofisomorphisms
8. The
isomorphism of u
and k’Grothendieck noted in
[GA]
that since the Poincaré lemma holds in the Zariskitopology
for the mapof sheaf
complexes,
that one has analgebraic
de Rham theorem on the level ofhypercohomology:
Deligne
shows that ananalogous algebraic
de Rham theorem holds in thesemisimplicial setting [D]. Thus,
theobject
is a
semisimplicial complex
ofsheaves,
and it admits acompatible
resolution
(Cech, injective, etc.):
Taking global
sections one obtains asemisimplicial object
inKA:
Thus one can define
algebraic
de Rhamcohomology
for X.by
The
underlying semisimplicial complex
of sheavessupports
two filtra-tions,
onecoming
from thealgebraic
and onecoming
from thesimplicial
structure:
These filtrations
give
a mixedHodge
structure onH(X.)
which isisomorphic
to the one definedusing
forms[D].
We now claim that
GrF
commutes with H:To see that this is so, observe first that
by [D, 8.1.9.v],
that thespectral
sequence for
H(X., 2 *)
definedby
theHodge
filtrationdegenerates
atEl . By Deligne’s degeneration
criterion[D, 1.3.2]
this isequivalent
to the factthat the differential for the
complex
K =s0393sC03A9( X.)
isstrictly compati-
ble with the
Hodge
filtration. Strictnessimplies
that the exact sequence ofcomplexes
passes to an exact sequence in
cohomology:
It follows that FP and H commute:
To
complete
theargument,
consider thediagram
below:The
top
row is exactby
definition and the bottom row is exactby
strictness of the differential with
respect
to F.Commutativity
of thediagram
and the existence of the indicatedisomorphisms
forces thedesired
isomorphism
on the level ofGrF.
Since the source and the
target
of the map(a)
are filteredby weight,
there are induced maps
Proposition
7.2.8 of[D]
asserts that(c)
is anisomorphism,
from which itfollows that
(b)
is as well.To construct the asserted
isomorphism between q
andk’,
we mustconstruct a commutative
diagram
To construct the left-hand
isomorphism,
consider thefollowing
di-agram, induced from the
homomorphism
Z.-+ 03A9 :Because of the identifications
and
and because p and v are
essentially
the maps " tensorwith C ", v
inducesa
surjective
map from NS toL1H2
whose kernel is the torsionsubgroup.
To construct the
right-hand isomorphism,
recall thatNow
so that
The
required isomorphism
now results from the commutativediagram
To show that the
diagram
which compares q and k iscommutative,
recall that the motivic
homomorphism q
is constructed from sections si and sF ofover
L1H2(X)
which preserve,respectively,
the lattice andHodge
filtra-tion. Consider the exact commutative
diagram
Let
C-1
be a section of C overL1H2(X),
set sz = D’ 0C - 1,
and writeProjection
toH2(O) yields
as
required.
9. The
isomorphism
of k’ and k"Referring
onceagain
toFigure 2,
we see that thepath Ef-1
induces amap from CA" to CA’:
The exactness of
Figure
2implies
thatEf-1
is anisomorphism,
so itremains
only
to show that the abovediagram
commutes.To this
end, let e
be anarbitrary
element ofNS( X.),
and let{c03B103B203B3}
bea
cocycle
inC2(Z0)
whichrepresents e. By hypothesis, {c03B103B203B3}
lifts to acocycle
where
Therefore
To
compute
arepresentative
ofk"(03BE),
notethat e
is the Chern class ofa line bundle L on
Xo.
Let{g03B103B2}
be acocycle
inC1(O*0) representing
this line bundle and observe that
perhaps
aftermodifying {c03B103B203B3} by
acoboundary.
ThusAs a cochain
03B4*{g03B103B2}
lifts toH2(W1O) according
to the formulaNote, however,
thatso that this lift
is
notnecessarily
acocycle. Nevertheless,
theimage
03B4*{c03B103B203B3}
inH2(W1Z)
vanishesby hypothesis,
so there is a cochainsuch that
The element
is then the desired
lifting
to acocycle.
Sinceit remains to show that
vanishes in
H2(O)
modulo theimage
ofH2(W1Z).
Because of the relationthe
right
hand side of(*)
can beexpressed
aswhich is
manifestly
in theimage
ofH1(W1Z).
Thiscompletes
theproof
of the
commutativity.
10. The
isomorphism
of k" and TTo compare k" and T we use the divisor sequence
for a
semisimplicial
space. This makes sense when the local sections of themultiplicative
sheaf M*pon Xp
are ingeneral position
withrespect
to the
images 03B4i(Xp+1).
To
begin,
we claim thatP(X.)
has thefollowing
sheafcohomology description:
To see
this,
consider the exact commutativegrid
of sheavesanalogous
tothat of
Figure
1 which comes from the divisor sequence and the filtration sequenceThere results an exact commutative
grid
ofcohomology
groups, the relevantportion
of which isdisplayed
below:Here we have used the reductions
as well as the fact that
e 2
= 0 for surfaces. Thediagram
thenyields
theisomorphisms
Under the above
identification,
thesubgroup P
issimply
the kernel of thecomposition
and the trace is induced
by
thecoboundary
mapwhich comes from the
weight
filtration of the divisor sheaf. We cantherefore compare T with k"
using
thediagram
below:It then suffices to show that
modulo
We will show
separately
that the upper and lower squares are com- mutative.First, let {f03B1}
be asystem
ofmeromorphic
functions repre-senting
adivisor 03BE~H0(D0).
Thecomposition
8* o div is then repre- sentedon 03BE by
thecocycle
while the
composition
div isrepresented on 03BE by
Since 8 and 8* commute, so does the upper square.
Second, let t
be a class inNS(X.),
and let c ={c03B103B203B3}
be arepresenta-
tivecocycle
inC2(Z0).
Since itsimage
inH2(O0)
is zeroby
definition of the Neron-Severi group, there is a cochain{f03B103B2} ~ C1(O0)
such thatThe
cocycle
represents k’(03BE)
inH2(W1O),
and soSince t = [{ c,,,,,8.y 1 ]
is also in theimage
ofH1(O*0),
there is acocycle { g03B103B2 1
EC1(O*0)
such thatwhere
ka8
is inC1(ZO).
ThenCombining (*)
and(**),
we haveTherefore
where
Exponentiating
andapplying 8*,
we find thatThus the lower square is commutative up to the action of exp
203C0i03B4*H1(O0),
as desired.11. Polarizations and
primitive cohomology
We shall now construct bilinear forms on each of the
graded pieces
ofH2(X.),
where X. is adistinguished semisimplicial
resolution. These bilinear formsplay
a crucial role in theapplications
of ourtheory.
WEIGHT 2: Since
the second
graded piece
inherits the form onH2(X0) given by cup-prod-
uct followed
by
evaluation on the fundamental class. The latter isby
definition the sum of the fundamental classes of the
components.
WEIGHT 1: Because the resolution X. is
distinguished, X2 is
zero-dimen-sional,
and so theweight (simplicial) spectral
sequencegives
the firstgraded piece
as thequotient
of aHodge
structureby
asub-Hodge
structure:
Cup-product
defines anondegenerate
skew form onH1(X1)
whichremains
nondegenerate
upon restriction to03B4*H1(X0).
Restriction of this form to theorthogonal complement
of03B4*H1(X0)
defines therequired
form on the first
graded piece.
WEIGHT 0: Since the zeroth
graded pieces
ofhomology
andcohomology
are dual
by
the universal coefficient theorem modulotorsion,
a bilinearform on one induces a bilinear form on the other. Now the
homological object
has a naturalgeometric interpretation
asthe "triple-point
lattice"(section 5):
Since
X2
is a finite set ofpoints,
its zerothhomology
carries the obvious bilinear form definedby
Since this form is
positive
definite onH0(X2),
so is the induced form onT( X.).
Primitive
cohomology
is defined in the usual way: Let Y. be asubobject
of X. such thatYp
is anample
divisor on everypositive-dimen-
sional
component
ofXp.
The divisor Y. willnecessarily
beempty
on zero-dimensionalcomponents.
Letbe the
primitive subspace.
We claim that the bilinear forms defined abovepolarize
theprimitive cohomology
in the sense that relative tothem the
graded pieces
arepolarized Hodge
structures: theprimitive
cohomology is
To seethis, a
itpolarized
suffices to mixed establish theHodge
structure.relationsConsider therefore the
diagram below,
which relates the"Mayer
Vietorissequence"
of X. to that of Y. :Since
dim Yp=1-p, W1H2(Y.),
which is asubquotient
ofH1(Y1)~
H0(Y2),
must be zero. Thisfact, coupled
with the exactness of thediagram implies
that(**)
is true.Moreover,
since p induces aninjection
of the left-hand side of
(*)
into theright-hand side,
it suffices to establishsurjectivity. Thus,
let a be aprimitive
element ofH2(X0)
which isannihilated
by 03B4*,
and let a be any lift toH2(X.).
Then 1*à is annihilatedby
q, and so is in theimage
ofW1 H2(y.).
Since this latter groupvanishes,
i* =0,
and so â isprimitive,
as desired.12. A Torelli theorem
We shall now prove a
slightly
weakened version of the Torelli theorem mentioned in the introduction.THEOREM G: Let X be the
union in P3 of a
smooth cubicsurface
A and aplane
Bmeeting transversely
in a curve C. Then X is determined up toisomorphism by
thepolarized
mixedHodge
structure onH2(X)0.
The essence of the
proof
is to show that both X and its mixedHodge
structure
correspond
to aplane elliptic
curve with six markedpoints (denoted (P, C, P2) below).
We
begin by recalling [M]
that the latticeL1H02
isisomorphic
to thatof the
Dynkin diagram E6,
with choices ofpositive primitive
rootsystems
in one-to-onecorrespondence
with choice of a skew-six of lineson A. Observe first that because the
hyperplane
section of X is homolo-gous to C on
A,
thatView A
as P2-blown
up at sixpoints Pl,,,.,P6, let L1,...,L6 be the corresponding exceptional
curves, and let03BB1,...,03BB6
be theircohomology
classes in
H2(A).
Let h be thecohomology-class
of ahyperplane
sectionof
A,
and setThe intersections of these vectors follow the
diagram
below.The
weights
attached to the nodesrepresent
the self-intersectionnumber,
while distinct vectors have intersection one or zero
according
to whetheror not the
corresponding
nodes arejoined by
anedge.
Observe next that a choice of a skew six on X determines a
triple
where C is a cubic curve in
P2
and where is a set{p1,...,p6} of points
on C. Thepair (C, P2) is
thepair (C, B )
definedby X,
while thepair (9, C )
is that definedby p;
=L,
~C,
with the intersection taken in A.Conversely,
such atriple
determines X: Blow upP2 along 9
to obtaina cubic surface A’ with a
distinguished plane
sectionC’, namely
thatobtained as the proper transform of C. Let B’ be the
unique plane
whichcuts out C’ on
A’,
and construct thequartic
surfaceThe
given
surface X and the constructed surface X’ areclearly
isomor-phic.
The
proof
of the theorem therefore rests on the construction from theHodge-theoretic
dataH2(X0)
of atriple (9h, Ch, P2)
which is isomor-phic
to some naturaltriple (P, C, P2).
To construct the abstract curveCh simply
takeand observe that
Ch
= C(use
aMayer-Vietoris sequence).
To constructthe correct
projective imbedding,
use a basis of sections forLô,
whereLo
is the line bundle defined
by
the divisor D = zero element of the groupstructure.
To construct the
point
set-9h
inCh,
fix an ordered set ofprimitive positive
roots,{e0,...,e5}
and consider thesystem
ofequations
onCh
defined in the Jacobian
by
Elimination of P2 and P3 from the first three
equations yields
so that p
is
determined up to translationby points
of order three onCh.
The
equations
for el, ... , e5 determine theremaining points
modulo thesame translation. Choose such a solution set
Yh,
and consider theresulting Hodge-theoretic triple (Ph, Ch , P2).
Because the group of three-divisionpoints
onCh
acts onLô,
alltriples
constructed from afixed root
system
areprojectively equivalent.
Finally,
consider the skew-six{L1...L6}
definedby
thegiven
rootsystem.
Then thepoints
p, =L, -
C which occur in the naturaltriple
solvethe
equations
inPic0(C)
definedby
From the
isomorphism
of u with T, we see that the naturaltriple
arisesas a solution set to the same set of
equations
as does theHodge-theoretic triple.
Thiscompletes
theproof.
13. Two surf aces
meeting
in a curveThe
preceding
theorem holds even when thecomponents
of thequartic
surface are
tangent.
Togeneralize
theproof,
it suffices toidentify
thegroup
P(X).
PROPOSITION H: Let X be the union
of
two smoothsurfaces
A and B.Suppose
that the intersection curve C has at mostordinary singularities.
Then
where the
right-hand
group is thegeneralized
Picardvariety of
Rosenlicht.PROOF:
Repetition
of the sequence ofoperations "desingularize,
thenform the
mapping
cone over thenonhomeomorphism
locus" leads to apolyhedral
resolution of an n-dimensionalvariety
in at most nsteps ([C3]).
For surfaces of thetype considered,
such a resolution isgiven by
the
semisimplicial
space below:Here
T+, T_,
andTo
denotecopies
ofT,
thesingular
locus ofC,
whilej4
is the lift of T to
C,
the normalization of C.Because Xl equals C
up tospaces of dimension zero,
Divh(X1)
is the group of divisors onC
whichhas
degree
zero on each irreduciblecomponent:
The
compact part
ofP(X)
is therefore the Picardvariety
of divisors ofdegree
zero on eachcomponent
ofC:
To determine the structure of P
itself,
write ameromorphic
functionf on Xl as
and observe that
f
is inMs
if andonly
ifSuch a function lies in the
subgroup Ms
if andonly
if the relationholds as well. But
and
so that the
defining
condition forMs
reduces toIf
Û
isconnected,
this reduces still further toso that
f
is constantalong
the fibers of the normalization map. If C hasordinary singularities,
these last two relations(GP
orGP’)
are thedefining
conditions for thespecial meromorphic
functions used in the construction of thegeneralized
Picardvariety
of C.REMARK:
According
to the results of section5,
themultiplicative part
ofP(X)
isgiven by
the group of characters on the latticeIn the
present
case, this lattice isgiven by divisors t
on Csubject
to thefollowing
restrictions:1) 03BE
issupported
in7"
2) 03BE
is ofdegree
zero on eachcomponent
ofC
3) 03BE
is ofdegree
zero on each fiber of gr, the normalization map.Proof
that Theorem G holds in thetangent
caseLet A be a smooth cubic
surface,
and let B be aplane
which istangent
to A at a
single point
qo. Thus qo is a nodalpoint
of the intersectioncurve C. Set
and let v = q+ - q -
generate T(X). Let e; = li+1 - 1,
be aprimitive
rootvector for
L1H2(X),
and consider its motivic valueSince
C
isrational, P(X)
has trivialcompact part,
so thatT( e; )
takesvalues in
Pm ( X ).
To
compute
thisvalue,
observe that the divisoris the divisor of a
meromorphic
functionf
on C. Via thesimplicial coboundary, f yields
a character on the latticeT( X.),
and this character is the motivic valueof e;
viewed under the identification ofPm(X)
with(X).
In the case athand, T(X)
isgenerated by
v, so that the motivic value is determinedby
thesingle complex
numberIf we choose a coordinate t on
~P1
so thatt( p, ) = 0, t(pi-1)=
oo,then we can take