ALEX, DARIO. PRIMER ON HODGE THEORY
2011 TALBOT WORKSHOP
1. Definition: Hodge Structure
A pure, weight k Hodge structure (VZ , F · ) consists of a finitely generated Z-module
VZ , together with a descending filtration of its complexification
VC = VZ ⊗ C : · · · ⊃ F i ⊃ F i+1 ⊃ · · · ⊃ F ∞ = 0
such that for each p,
VC = F p ⊕ F k+1−p .
This all makes sense: although the bar operation doesn’t preserve complex subspaces, the image of a complex subspace is again a complex subspace.
We can convert this to something more familiar by defining
H p,q := F p ∩ F k−p
for q such that q + p = k. Then it’s a lemma that
VC = ⊕p+q=k H p,q
0
0
and H q,p = H p,q , so we recover the filtration F p = ⊕p0 ≥p H p ,k−p . In fact, the data
of the H p,q is equivalent to the original Hodge structure.
The original example is the Hodge structure on H k (X, C) for X compact Kahler,
loc
where H p,q consists of cohomology classes representable by closed forms in Ap,q =
{f dzi1 ∧ · · · ∧ dzip ∧ dz̄j1 ∧ · · · ∧ dz̄jq }. The filtration F · sweeps diagonally from
the upper-right edge of the Hodge diamond, and F · sweeps symmetrically from the
upper-left.
Finally, in the data of a Hodge structure we require a polarization: a hermitian
inner product on VC such that
(−1)p h, i > 0
on H p,q .
On cohomology, this should be just the L2 inner-product given by integrating the
pointwise product on forms induced by the metric.
The category of Polarized Hodge Structures is abelian and semi-simple.
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2011 TALBOT WORKSHOP
2. Kahler Manifolds
Definition 2.1. A hermitian manifold (X, h) is called Kahler if dω = 0, where
ω is the imaginary part of the hermitian metric h. Throughout, we assume X is
compact.
Equivalently, h can be written in local holomorphic coordinates near any point as
Id + O(2) (“h osculates to the Euclidean metric to order 2”).
Let L := (• ∧ ω), Λ = L∗ the adjoint w.r.t. the L2 inner product.
Two of the most important Kahler identities are as follows:
¯
∂ ∗ = − ∗ ∂∗ = i[Λ, ∂]
and
∂¯∗ = −i[Λ, ∂]
(This is the graded commutator.) These are proved by computing on Cn , then
observing that the identities involve only one derivative therefore hold pointwise on
any Kahler manifold. The same identities hold for the Chern connection on any
holomorphic vector bundle on a Kahler manifold and a bit more generally. This is
important for Goldman-Millson theory.
“Kahler” Identities are crucial for the following two big theories:
(1) The Riemannian Laplacian is proportional to the Dolbeault Laplacian:
∆d = dd∗ + d∗ d = 2∆∂¯ = 2∆∂ .
So we may define the harmonic forms Hp,q to be the kernel of ∆ in Ap,q . These sit
¯
¯∗
inside H∂p,q
¯ as the forms which are both ∂ and ∂ -closed. In fact there is at most
one harmonic representative in each class.
Theorem 2.1. (Hodge) The DeRham cohomology decomposes as a direct sum
H k (X, C) = ⊕p+q=k H∂p,q
¯ ,
and this is a Hodge structure.
This follows once one shows that the unique harmonic representative for each
cohomology class actually exists. The proof uses a Bochner-Kodaira formula to
show that ∆ is elliptic, and standard estimates.
The Hodge Theorem gives the horizontal symmetry of the Hodge diamond.
(2) The identity
[Λ, L] = ⊕(p + q − n)π p,q
suffices to define an sl2 action on the whole cohomology of X, with Λ and L as
raising and lowering operators, respectively. This gives you the Lefschetz decomposition, and the vertical symmetry of the Hodge diamond. There are many more
consequences (Hard Lefschetz Theorem, etc).
ALEX, DARIO. PRIMER ON HODGE THEORY
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3. Variations of Hodge Structure and the Period Map
Definition 3.1. A Variation of Hodge Structure over a complex analytic base B is a
local system VZ → B, together with a filtration of VZ ⊗C by holomorphic subbundles
F · such that each fiber is a Hodge structure (with smoothly varying polarization).
The filtration is further required to satisfy Griffiths transversality:
∇F i ⊂ F i−1 ⊗ ΩB .
Here ∇ is the flat connection for the local system (Gauss-Manin connection).
We now define the Period Map for a VHS, which contains all the holomorphic
information but “forgets” the lattice. (The discrete side of the story is encoded by
the Griffiths-Abel-Jacobi map to the Intermediate Jacobian).
If we pull back the VHS to the universal cover B̃ → B and fix a point b ∈ B̃, the
local system becomes trivial and we identify V˜b ∼
= Ṽb0 for all b0 ∈ B̃. Let D be the
flag manifold of subspaces of Vb with the dimensions of the filtration. We get the
following diagram
/
Ṽ
B̃
loc
V
/
v
B
v
v
v
/
v
/
v: D
D/π1 (B).
The lower-left map is the period map, although the co-domain (Period Domain) is
usually taken to be a closed/open analytic subset of D/π1 (B) defined by imposing
the Hodge-Riemann bilinear relations. These are satisfied by any “motivic” VHS, i.e.
one coming from an actual smooth family of Kahler manifolds. One also sometimes
mods out by the entirety of O(VZ,b ) rather than just the image of the monodromy
of the particular VHS over B.
Given X → B a smooth family of Kahler manifolds over B, we define a (putative)
VHS as the pair
(Rk f∗ C, F · )
where we put
0
0
Fbp = ⊕p0 ≥p H∂p¯ ,k−p .
Some harmonic theory tells you that the H p,q form smooth subbundles, but these
aren’t expected to be holomorphic—that’s why we keep these filtrations around.
Example 3.2. Families of curves of genus g, and H 1 .
The period domain for VHS’s coming from genus-g curves is the moduli space of
principally polarized abelian varieties of dimension g,
Ag = Hg /Sp(g, Z).
Here Hg = {(I, B), B = B T , ImB > 0} is the Siegel Upper-Half-Space, defined by
imposing the two Riemann bilinear relations on the normalized B-period matrix.
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2011 TALBOT WORKSHOP
The period map descends to a map of moduli spaces Mg → Ag , which is injective
according to the Torelli theorem.
The period map for a smooth family is in fact holomorphic and “horizontal” (i.
e. satisfies Griffiths transversality), therefore you do get a VHS. One shows this
by computing its differential, which later turns out to be just cup-product with the
Kodaira-Spencer class. Voisin does this painstakingly over 40pp, and I tried to sum
it up in the last 30 seconds. I won’t try again.
The period map is the “original version” of several things we’ve seen so far. If
you mod out further by a maximal compact of the symmetry group of D to get a
symmetric space, the period map becomes harmonic (see Aaron, Sushimita, Jesse’s
talks). It also has some key metric properties that Hitchin noticed are carried by
his map as well and lead to properness.