---
title: "Hodge Theory"
subtitle: "Spring 2025  "
author-note: "Author note"
abstract: "Notes from a course on Hodge theory taught by Pierrick Bousseau in Spring 2025 at the University of Georgia."
---
























# 2025-01-06-10-30-25

::: remark
Textbook: *Hodge Theory and Complex Algebraic Geometry I: Volume 1 (Cambridge Studies in Advanced Mathematics)* by Voisin.
:::

::: remark
Recall that for a topological space \( X \), we can assign invariants \( H^i(X; {\mathbf{Z}}) \) which are abelian groups. More generally, we can attach \( H^i(X; F) \coloneqq H^i(X; {\mathbf{Z}}) \otimes F \) for a field \( F \), and with reasonable finiteness assumptions on \( X \) (e.g. compact manifolds), these will be finite-dimensional \( F{\hbox{-}} \)vector spaces.
:::

::: example
For \( X \) a genus \( g \) Riemann surface,

-   \( H^0(X;{\mathbf{Z}}) = {\mathbf{Z}} \),
-   \( H^1(X;{\mathbf{Z}}) = {\mathbf{Z}}^{2g} \),
-   \( H^2(X;{\mathbf{Z}}) = {\mathbf{Z}} \), and
-   \( H^i(X; {\mathbf{Z}}) = 0 \) for \( i\geq 3 \).

This detects the fact that there are \( g \) holes in \( X \).
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::: remark
Note that \( H^* \) is functorial: for \( f: X\to Y \) a continuous map, there is a pullback in cohomology \( f^*: H^i(Y;{\mathbf{Z}})\to H^i(X;{\mathbf{Z}}) \) which is a morphism of abelian groups. Over fields, this instead yields a linear map, which is generally easier to study.

A general theme: extra structure on \( X \), which can include

-   a complex structure,
-   a symplectic structure,
-   a Kähler structure,
-   a \( {\mathbf{C}}{\hbox{-}} \)algebraic structure,

will induce extra structure on \( H^*(X;{\mathbf{C}}) \), i.e. a Hodge structure. We will look in particular at complex manifolds \( X \), i.e. those locally identifiable with \( {\mathbf{C}}^n \) for some \( n \), which allows for a notion of holomorphic functions on \( X \).
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::: remark
Recall that a torus \( X \) can be written as \( {\mathbf{R}}^2/{\mathbf{Z}}^2 \), i.e. a quotient of the plane by the standard square lattice. We have \( g=1 \) and \( H^*(X;{\mathbf{C}}) = {\mathbf{C}}\oplus {\mathbf{C}}^2[1] \oplus {\mathbf{C}}[2] \). Putting a complex structure on \( X \) amounts to replacing \( {\mathbf{R}}^2 \) with \( {\mathbf{C}} \) and yields a complex manifold of complex dimension 1. One can replace \( X \) with \( X_\tau = {\mathbf{C}}/({\mathbf{Z}}\oplus \tau {\mathbf{Z}}) \) for any \( \tau \in {\mathbf{C}} \). It is a theorem that \( X_\tau \cong X_{\tau'} \) as complex manifolds iff \( {\exists}a,b,c,d\in {\mathbf{Z}} \) with \( ad-bc=1 \) such that \( \tau' = {a\tau + b\over c\tau + d} \). Thus these complex manifolds vary in continuous families, despite being identified as real manifolds.
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::: remark
We will examine the Hodge decomposition for compact Kähler manifold, which is a complex manifold with additional technical assumptions. This decomposition is of the form \( H^i(X; {\mathbf{C}}) = \oplus_{p + q = i} H^{p, q}(X) \), which implies \( \beta_i(X) = \sum _{p+q=i}h^{p, q}(X) \). The Hodge numbers \( h^{p,q} \) thus refine the Betti numbers, and may contain more information.
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::: example
For \( X \) a genus \( g \) Riemann surface, one has
\[
H^1(X;{\mathbf{C}}) = H^{1,0}(X) \oplus H^{0, 1}(X)
\]
where \( h^{1, 0}(X) = h^{0, 1}(X) = g \). Thus the Hodge numbers alone don't see the complex structure, since they are always \( g \). However, what will keep track of differences will be the interplay between the Hodge decomposition (as decompositions of vector spaces) as the integral structure of \( H^i(X; {\mathbf{Z}}) \subseteq H^i(X; {\mathbf{C}}) \).
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::: remark
Hodge structures will be related to period integrals \( \int_\gamma \alpha \), which is where calculus enters the picture. The proof of the Hodge decomposition uses real analysis, in particular elliptic PDEs, in a crucial way.
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::: remark
Next time: a word about complex analysis, complex/Kähler manifolds, the Hodge decomposition.
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