Calculating Betti and Hodge number for product of a curve

Question

How can we compute the Betti numbers and Hodge numbers for $S=C\times C/\sigma$? (Where $\sigma$ is swapping the two factors.)

Answer 1

To make sense of it, I assume $C$ is Riemann surface of genus $g$. Then $S= {C\times C}/\sigma$ is an smooth algebraic surfaces, so we only need to care about $H^1$ and $H^2$ by Hard Lefschetz.

Let $\pi : C\times C\longrightarrow {C\times C}/\sigma$ be the projection map. Then we have pull back forms map $\pi^* : H^i(C\times C/\sigma,\mathbb{C})\longrightarrow H^i(C\times C,\mathbb{C})$ and also an isomorphism $\pi^* : H^i(C\times C/\sigma,\mathbb{C})\simeq H^i(C\times C,\mathbb{C})^\sigma$, where $H^i(C\times C,\mathbb{C})^\sigma$ is cohomology of $\sigma$ invariant forms.

Now denote $\alpha_1, \alpha_2,\ldots\alpha_g, \beta_1,\beta_2\ldots\beta_g$ is the standard symplectic homology base. i.e. the intersection number $\alpha_i\cdot\beta_j=\delta_i^j$. Then Let $\gamma^*=\alpha_i^*\cup\beta_i^*$ the cup product, check it is well defined. Then $H^*(C\times C,\mathbb{C})^\sigma\subseteq H^*(C\times C,\mathbb{C})=H^*(C,\mathbb{C})\otimes H^*(C,\mathbb{C})$ is generated by {$\alpha_i\otimes1+1\otimes\alpha_i,\beta_l\otimes1+1\otimes\beta_l, \gamma\otimes1+1\otimes\gamma$}. Thus the betti number $b_1=2g$ and $b_2=2g^2-g+1$, and the hodge number $h^{0,1}=h^{1,0}=g$

Similarly, We can compute Hodge numbers $h^{2,0}=h^{0,2}=g(g-1)/2$ and $h^{1,1}=g^2+1$, since we have decomposition $H^2(C\times C/\sigma,\mathbb{C})=H^2(C\times C,\mathbb{C})^\sigma=H^{2,0}(C\times C)^\sigma\oplus H^{1,1}(C\times C)^\sigma\oplus H^{0,2}(C\times C)^\sigma$.