ag.algebraic geometry - Deformations of hypersurfaces

Let's assume that we are working over $mathbb{C}$.

First of all, hypersurfaces in $mathbb{P}^n$ are unobstructed, so their first-order deformations always correspond to small deformations (deformations over a disk).

As a general fact, when you consider a smooth variety $X$ with a finite group $G$ acting $holomorphically$ on it, the invariant subspace $H^1(X, T_X)^G$, it parametrizes those first-order deformations that preserve the holomorphic $G$-action. This essentially comes from the fact that, being the action of $G$ holomorphic, if you take $sigma in G$, then $sigma_*$ commutes with $bar{partial}$ and the Green operator $boldsymbol{G}$, so if $varphi(t)$ solves the Kuranishi equation

$varphi(t)=t + frac{1}{2}bar{partial}^* boldsymbol{G}[varphi(t), varphi(t)]$

for $t$, then $sigma_*varphi(t)$ solves the Kuranishi equation for
$sigma_*t$, and $sigma_{*} varphi(t) = varphi(sigma_*t)$.

Example Let us consider a quintic Fermat surface $X subset mathbb{P}^3$ of equation

$x^5+y^5+z^5+w^5=0$.

It admits a free action of the cyclic group $mathbb{Z}_5$ given as follows: if $xi$ is a primitive $5$-th root of unity, then

$xi cdot (x,y,z,w)=(x, xi y, xi^2 z, xi^3 w)$.

The quotient $Y := X/mathbb{Z}_5$ is a Godeaux surface (i.e. a surface of general type with $p_g=q=0, K^2=1$ ) with fundamental group $mathbb{Z}_5$. M. Reid proved that, conversely, every Godeaux surface with fundamental group $mathbb{Z}_5$ arises in this way and that, moreover, the corresponding moduli space is generically smooth of dimension $8$. Then in this case we have

$dim H^1(X, T_X)=40$

$dim H^1(X, T_X)^G=H^1(Y, T_Y)=8$,

since the number of moduli of quintics keeping the free $G$-action equals the number of moduli of the Godeaux surface $Y$ (well, Horikawa showed that the deformations of quintic surfaces are complicated enough, anyway $40$ is the right number).

Actually, one can say more and check that for every irreducible character $chi$ of $G$ one has

$dim H^1(X, T_X)^{chi} = 8$,

but I do not know any easy interpretation of these eigenspaces in terms of the deformations of the quintic.

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