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 \(h\left(f_{1} \partial_{z}+g_{1} \partial_{\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu}, f_{2} \partial_{z}+g_{2} \partial_{\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu}\right)=f_{1} \mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu_{2}+g_{1} \mkern 1.5mu\overline{\mkern-1.5mug\mkern-1.5mu}\mkern 1.5mu_{2} .\)
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 \(J\curvearrowright{\mathbf{T}}{\mathbf{C}}\) by \({ \begin{bmatrix} {0} & {-1} \\ {1} & {0} \end{bmatrix} }\).
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 \(\star d z=i d \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu \quad \star d \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu=-i d z \quad \star(d z \wedge d \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu)=-2 i\)
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 \(\Delta f = 0\) on the torus are constants, so \(h^{0, 0} = 1\).
Compute \(\Delta(f d z+g d \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu)=\Delta f d z+\Delta g d \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu\) and \(\Delta(f d z \wedge d \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu)=\Delta f d z \wedge d \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu\), argue similarly for degrees 1 and 2.