Moduli of cubic surfaces and Hodge theory

Arnaud Beauville

Description

Niveau: Supérieur, Doctorat, Bac+8
Moduli of cubic surfaces and Hodge theory [After Allcock, Carlson, Toledo] Arnaud BEAUVILLE Introduction This is a detailed version of three lectures given at the annual meeting of the Research Group “Complex Algebraic Geometry” at Luminy in October 2005. The aim was to explain, in a way as elementary as possible, the work of Allcock, Carlson, Toledo [ACT] which describes, using Hodge theory, the moduli space of cubic surfaces in P3 as a quotient of the complex ball in C4 . That work uses a number of different techniques which are quite basic in algebraic geometry: Hodge theory of course, monodromy, differential study of the period map, geometric invariant theory, Torelli theorem for the cubic threefold . . . One of our aims is to explain these techniques by illustrating how they work in a concrete and relatively simple situation. As a result, these notes are quite different from the original paper [ACT]. While that paper contains a wealth of interesting and difficult results (on the various moduli spaces which can be considered, the corresponding monodromy group, their description by generators and relations), we have concentrated on the main theorem and the basic methods involved, at the cost of being sometimes sketchy on the technical details of the proof. We hope that these notes may serve as an introduction to this nice subject.

• ?1 ?


• linear


• cubic surfaces


• v? ???


• defined modulo


• framed cubic


• group pgl

Sujets

• Beauville
• Organizational studies
• Linear

Informations

• Publié par: mijec
• Nombre de lectures: 51
• Langue: English

Extrait

Moduli of cubic surfaces and Hodge theory
[After Allcock, Carlson, Toledo]

Arnaud BEAUVILLE

Introduction
This is a detailed version of three lectures given at the annual meeting of the
Research Group “Complex Algebraic Geometry” at Luminy in October 2005. The
aim was to explain, in a way as elementary as possible, the work of Allcock, Carlson,
Toledo [ACT] which describes, using Hodge theory, the moduli space of cubic surfaces
3 4
inPas a quotient of the complex ball inC. That work uses a number of different
techniques which are quite basic in algebraic geometry: Hodge theory of course,
monodromy, differential study of the period map, geometric invariant theory, Torelli
theorem for the cubic threefold. . .One of our aims is to explain these techniques by
illustrating how they work in a concrete and relatively simple situation.
As a result, these notes are quite different from the original paper [ACT]. While
that paper contains a wealth of interesting and difficult results (on the various
moduli spaces which can be considered, the corresponding monodromy group, their
description by generators and relations), we have concentrated on the main theorem
and the basic methods involved, at the cost of being sometimes sketchy on the
technical details of the proof. We hope that these notes may serve as an introduction
to this nice subject.
In the next section we will motivate the construction by discussing a more
3
complicated but more classical case, namelyquarticsurfaces inP. In§2 we will
explain the main result; at the end of that section we will explain the strategy of
the proof, and at the same time the plan of these notes.

1. Motivation: the case of quartic surfaces
As announced, we start by recalling briefly the description of the moduli space
3
of quartic surfaces inP. References include [BHPV], [X], or [B2] for a short
introduction.
3
(1.1) A quartic surfaceS⊂Pis a K3 surface, which means that it admits a
unique (up to a scalar) holomorphic 2-formω, which is non-zero at every point.
2
The only interesting cohomology ofS isthe latticeH (S,Z) ,endowed with the
unimodular symmetric bilinear form defined by the cup-product. Moreover the vector
2 2
space H (S,C) = H(S,Z)⊗Cadmits a Hodge decomposition
2 2,0 1,1 0,2
H (S,C) = H⊕H⊕H,

1

2,0 2
which is determined by the position of the lineH =Cω(Sin H,C) (wehave
0,2 2,0 1,1 2,0 0,2 2
H =H ,and His the orthogonal ofH⊕The point is thatH (SH ).,C)
2,0
depends only on the topology ofS , while the position ofH =Cωdepends heavily
on the complex structure.
2
To be more precise, we denote byL alattice isomorphic toH (S,Z.all S) for
We fix a vectorh0∈L ofsquare 4 (they are all conjugate underO(L) ).Amarked
∼2
quartic surface is a pair(S, σan isometry) ofS anda quarticσ: L−→H (S,Z)
f
2
such thatσ(h0) =h, the class inH (S,Z) ofa plane section. We denote byM
the moduli space of marked quartic surfaces; it is not difficult to see that it is a
complex manifold. The groupΓ ofautomorphisms ofL whichfixh0acts on
f f
−1
Mbyγ∙(S, σ) = (S, σ◦γ) ;the quotientM:=M/the usual moduli spaceΓ is
0 3
of quartic surfaces, that is, the open subset ofP(H (P,OP(4)) parameterizing
3
smooth quartic surfaces modulo the action of the linear groupPGL(4) .
f
(1.2) The advantage of working withMis that we can now compare the
Hodge structures of different surfaces. Given(S, σ) , we extendσto an isomorphism
∼2
LC−→H (S,Cput) and

−1 2,0−1
℘˜(S, σ) =σ(H )=σ([ω])∈P(LC).

The map℘˜ iscalled theperiod map, for the following reason: choose a basis
∗22t−1
(e1, . . . , e22L, so that) of LCis identified withC. Putγi=σ(ei) ,viewed
as an element ofH2(S,Zthen) ;
Z Z
³ ´
21
℘˜(S, σ) =ω:. . .:ω∈P.
γ1γ22
R
The numbersωare classically called the “periods” ofω.
γi
R
Sinceωis holomorphic we haveω∧ω, and= 0ω∧¯ω >0 ;moreover, since
S
2
ωis of type (2,0) andhof type (1,1), we haveω.hin H (S= 0,C) . In other words,
℘˜(S, σΩ ofin the subvariety) liesP(LC) ,called theperiod domain, defined by

2
Ω ={[x]∈P(LC)|x=x.h0= 0, x.x¯>0}.

The action ofΓ on LC, and the mappreserves Ω
we have a commutative diagram:

f
M


y

M

℘˜
−−−→

℘
−−−→

2

Ω


y

Ω/Γ

.

℘˜ is Γ-equivariant.Thus