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date: 2022-04-24 18:43
modification date: Sunday 24th April 2022 18:43:19
title: "examples of computations of Hodge numbers"
aliases: [examples of computations of Hodge numbers]
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# examples of computations of Hodge numbers


# Projective space

![](attachments/Pasted%20image%2020220424184327.png)

# Torus


Hodge diamond for the torus:
![](attachments/Pasted%20image%2020220424182331.png)

How to compute:

- Define the Hermitian inner product on vector fields $h\left(f_{1} \partial_{z}+g_{1} \partial_{\bar{z}}, f_{2} \partial_{z}+g_{2} \partial_{\bar{z}}\right)=f_{1} \bar{f}_{2}+g_{1} \bar{g}_{2} .$
- Take the real part $g\left(f_{1} \partial_{x}+g_{1} \partial_{y}, f_{2} \partial_{x}+g_{2} \partial_{y}\right)=f_{1} f_{2}+g_{1} g_{2}$
- Let $J\actson \T\CC$ by $\matt 0 {-1} 1 0$.
- Compute the symplectic form $\omega\left(f_{1} \partial_{x}+g_{1} \partial_{y}, f_{2} \partial_{x}+g_{2} \partial_{y}\right)=f_{1} g_{2}-g_{1} f_{2}$
- Compute the stars $\star d z=i d \bar{z} \quad \star d \bar{z}=-i d z \quad \star(d z \wedge d \bar{z})=-2 i$
- For $f$ a global function compute $\Delta f=\star d \star f=-\left(\partial_{x x} f+\partial_{y y} f\right),$ 
- Argue that the only solutions to $\laplacian f = 0$ on the torus are constants, so $h^{0, 0} = 1$.
- Compute $\Delta(f d z+g d \bar{z})=\Delta f d z+\Delta g d \bar{z}$ and $\Delta(f d z \wedge d \bar{z})=\Delta f d z \wedge d \bar{z}$, argue similarly for degrees 1 and 2.