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

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Textbook: *Hodge Theory and Complex Algebraic Geometry I: Volume 1 (Cambridge Studies in Advanced Mathematics)* by Voisin.
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Recall that for a topological space $X$, we can assign invariants $H^i(X; \ZZ)$ which are abelian groups.
More generally, we can attach $H^i(X; F) \da H^i(X; \ZZ) \tensor F$ for a field $F$, and with reasonable finiteness assumptions on $X$ (e.g. compact manifolds), these will be finite-dimensional $F\dash$vector spaces.
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:::{.example}
For $X$ a genus $g$ Riemann surface,

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

This detects the fact that there are $g$ holes in $X$.
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Note that $H^*$ is functorial: for $f: X\to Y$ a continuous map, there is a pullback in cohomology $f^*: H^i(Y;\ZZ)\to H^i(X;\ZZ)$ 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 $\CC\dash$algebraic structure,

will induce extra structure on $H^*(X;\CC)$, i.e. a Hodge structure.
We will look in particular at complex manifolds $X$, i.e. those locally identifiable with $\CC^n$ for some $n$, which allows for a notion of holomorphic functions on $X$.
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Recall that a torus $X$ can be written as $\RR^2/\ZZ^2$, i.e. a quotient of the plane by the standard square lattice. 
We have $g=1$ and $H^*(X;\CC) = \CC \oplus \CC^2[1] \oplus \CC[2]$.
Putting a complex structure on $X$ amounts to replacing $\RR^2$ with $\CC$ and yields a complex manifold of complex dimension 1.
One can replace $X$ with $X_\tau = \CC/(\ZZ \oplus \tau \ZZ)$ for any $\tau \in \CC$.
It is a theorem that $X_\tau \cong X_{\tau'}$ as complex manifolds iff $\exist a,b,c,d\in \ZZ$ 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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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 $H^i(X; \CC) = \oplus_{p + q = i} H^{p, q}(X)$, 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 
\[
H^1(X;\CC) = H^{1,0}(X) \oplus H^{0, 1}(X)
\]
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 $H^i(X; \ZZ) \subseteq H^i(X; \CC)$.
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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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Next time: a word about complex analysis, complex/Kähler manifolds, the Hodge decomposition.
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