BUNTES Variations of Hodge Structures (Sachi)

Section 5.5 Variations of Hodge Structures (Sachi)

Subsection 5.5.1 Review of Hodge Theory

\(X\) complex manifold \(X\subseteq \PP^N\) which is \(m\)-dimensional. For each \(n\) associate to \(X\)

\begin{equation*} H_\ZZ = H^n_{\mathrm{sing}}(X, \ZZ) / \tors \end{equation*}

\begin{equation*} H_\CC = H_\ZZ \otimes \CC = H^n_\dR(X) \end{equation*}

we have a bilinear pairing

\begin{equation*} Q\colon H_\ZZ \times H_\ZZ \to \ZZ \end{equation*}

\begin{equation*} Q(\alpha,\beta) = \int_X \alpha \cup \beta \cup \omega^{m-n} \end{equation*}

where \(\omega\) is a generator of \(H^2( \PP^N , \ZZ)\) restricted to \(X\text{.}\) This gives us the set-up of \(X\) as a differentiable manifold. Now say something about complex structure. We have a decomposition of differential forms on \(X\)

\begin{equation*} A^n(X) = \bigoplus_{p+q =n} A^{p,q} \end{equation*}

degree \(n\) forms decomposing as a combination of type \(p,q\) forms.

Hodge theorem descends to a decomposition on cohomology

\begin{equation*} H^n_\dR(X) = \bigoplus_{p+q = n} H^{p,q} \end{equation*}

\begin{equation*} H^{p,q} = \overline{H^{q,p}} \end{equation*}

\begin{equation*} Q(H^{p,q}, H^{p',q'}) = 0 \end{equation*}

unless \(p+p' = q+q' = n\text{.}\)

A hodge structure of weight \(n\) is the data \((H_\ZZ, Q)\) satisfying the Hodge decomposition, Bilinearity

Questions:

To what extent does the HS of \(X\) determine \(X\text{?}\) (Torelli problem)
To what extent can we read off the geometric data of \(X\) from its Hodge structure.

Subsection 5.5.2 Variations of Hodge structures:

Let \(Y \subseteq X\) be codimension \(k\text{,}\) this gives a class in

\begin{equation*} H^{k,k}(X) \subseteq H^{2k}(X,\CC) \end{equation*}

what about the converse?

For each cohomology class \(\gamma\) in \(H^{2k} (X,\CC)\) is \(\gamma\) a rational linear combination of classes of subvarieties. (Hodge conjecture).

Subsubsection 5.5.2.1 Hodge theory for curves

\((H_\ZZ, Q)\text{,}\) \(H^{1,0} \oplus H^{0,1}\) have the period matrix

\begin{equation*} H^{0,1}/\Lambda \cong \Jac(C) \end{equation*}

\begin{equation*} y^2 = x(x-1)(x-\lambda) \end{equation*}

\begin{equation*} \lambda \in \PP^1 - \{0,1,\infty\} \end{equation*}

each \(E_\lambda \leftrightarrow H^{1,0} \oplus H^{0,1}\) so can ask as \(\lambda\) varies we can ask how \(H^{1,0}\) is situated inside of \(H^{1,0} \oplus H^{0,1}\text{.}\)

\begin{equation*} \omega = \frac{\diff x}{y} \in H^0(X,\Omega_X) \end{equation*}

pairing with \(H_1(X)\)

\begin{equation*} \int_\gamma \omega\text{.} \end{equation*}

For \(B\) a variety \(\{X_b\}\) are varieties with Hodge structures for each \(b\in B\text{.}\) Locally we can identify

\begin{equation*} H_\ZZ = H^n(X_b , \ZZ)/\tors \end{equation*}

and \(H_\CC \) with that of \(X_{b_0}\text{.}\)

Then consider

\begin{equation*} H^{n-k, k}(X_b) \end{equation*}

or the associated

\begin{equation*} F^k = \bigoplus_{l=0}^k H^{n-l,l}(X_b) \end{equation*}

subspaces of \(H_\CC\text{.}\)

Question: What is a moduli space of linear subspaces?

Answer: The grassmanian!

\begin{equation*} {\mathrm{Gr}}(k,V) \end{equation*}

of \(k\)-dimensional subspaces of a fixed vector space \(V\text{.}\) What is the tangent space to the Grassmanian at a point \(W\subseteq V\text{?}\)

\begin{equation*} \Hom(W, V/W) \end{equation*}

if we take the complementary subspace \(W \oplus C = V\) given another subspace

\begin{equation*} W' \cap C = \{ 0 \} \end{equation*}

have \(\pi_{W'}, \pi_C\)

\begin{equation*} {\mathrm{Gr}}(k,V) = \{\text{all\,}W'\} \end{equation*}

\begin{equation*} \cong \Hom(W,C) \end{equation*}

by \(\pi_C \circ (\pi_W|_{W'})\inv\text{.}\)

Fact 5.5.1.

\(\phi\colon B\to {\mathrm{Gr}}\) mapping \(b \mapsto F^k(X_b) \subseteq H_\CC\) is holomorphic.
In terms of identifying the tangent space of the grassmanian to the hom set, the image under
\begin{equation*} \diff \phi_k = \delta_k \end{equation*}
of any tangent vector of \(B\) at \(b_0\) carries \(F^{k} \) to \(F^{k+1}/F^k\) so we have maps
\begin{equation*} \delta_k \colon T_{b_0} B \to \Hom(H^{n-k,k} (X), H^{n-k-1, k+1}(X)) \end{equation*}
satisfying
\begin{equation*} \delta_{k+1} (V) \circ \delta_k(W) = \delta_{k+1}(W) \circ \delta_k(W) \end{equation*}
for all \(v,w\in T\text{.}\)

Since \(F^k(X_b)\) satisfy

\begin{equation*} Q(F^k, F^{n- k - 1} ) = 0 \end{equation*}

for all \(b\text{.}\)

\begin{equation*} Q(\delta_v(v)(\alpha), \beta) + Q(\alpha, \delta_{n-k-1}(v)(\beta)) = 0 \end{equation*}

for all \(\alpha\in H^{n-k, k}(X)\text{,}\) \(\beta \in H^{k+1, n-k - 1}(X)\) for \(v \in T\)

Definition 5.5.2. Infinitesimal variation of Hodge structures.

An infinitesimal variation of Hodge structures is

\begin{equation*} (H_\ZZ, Q, H^{p,q}, T, \delta_q\colon T \to \Hom(H^{p,q}, H^{p-1, q+1})) \end{equation*}

Two observations:

Remark 5.5.3.

Variations of hodge structures are often computable, e.g. for hypersurfaces in \(\PP^N\text{.}\)

\begin{equation*} X\subseteq \PP^{n+1} \end{equation*}

let \(X = \{ f = 0 \}\) of \(\deg d\text{.}\)

Lefschetz implies the only interesting cohomology is in the middle dimension.

\begin{equation*} H^n(X) \end{equation*}

\begin{equation*} H^{n,0}(X) \end{equation*}

Poincaré residues of \(n+1\) forms of \(\PP^{n+1}\) with poles along \(X\)

\begin{equation*} \frac{ \Res_\omega g(z_0, \ldots, z_{n+1}) \Omega} f = \frac{g \widetilde \Omega}{\sum \partder[f]{z_i}}\text{.} \end{equation*}

\begin{equation*} \CC[z_0, \ldots, z_{n+1}]/\text{Jacobian ideal} \end{equation*}

graded parts \(H^{p,q}(X)\)

Problem 5.5.4.

Identify \(H_\ZZ\) inside of \(H^n\)

Solution: VHS \(\delta_k\) maps turn out to be polynomial multiplication \(d\ge n+1\text{.}\)

Theorem 5.5.5. Noether-Lefschetz.

A surface \(S \subseteq \PP^4\) of degree \(d \ge 4\) having general moduli contains no curves other than complete intersections \(S \cap T\) with other surfaces \(T\text{.}\)