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Let $f: X \rightarrow Y $ be a Galois cover of with $X$ and $Y$ algebraic curves over $\mathbb{C}$. I want to compute the dimension of the subspace of $G$-invariants in $H^{0}(X,\omega^{\otimes2})$ (or by Serre duality $H^{1}(X,T_{X})$). The point is that I always get this dimension to be zero, which I know is not true, because at least I knew a proof that for $Y=\mathbb{P}^{1}$ and $G$ cyclic it must be equal to $s-3$, where $s$ is the number of branch points. My argument is as follows:

We have the exact sequence of tangent bundles $0\rightarrow T_{X/Y}\rightarrow T_{X}\rightarrow f^{*}T_{Y} \rightarrow0 $. The associated long exact sequence gives: $...\rightarrow H^{1}(X,T_{X/Y})^{G}\rightarrow H^{1}(X,T_{X})^{G}\rightarrow H^{1}(Y,f_{*}\mathcal{O}_{X}^{G}\otimes T_{Y})=H^{1}(Y, T_{Y})$. The last assertion because of the projection formula and $f_{*}\mathcal{O}_{X}^{G}= \mathcal{O}_{Y}$. But for $Y=\mathbb{P}^{1}$, $H^{1}(Y,T_{Y})=0$ and since $T_{X/Y}$ is with finite support (the ramification points) we have that $H^{1}(X,T_{X/Y})=0$ , this forces $H^{1}(X,T_{X})^{G}=0$. Can someone tell me where I am making a mistake. You can always assume that $Y=\mathbb{P}^{1}$ and that $G$ is cyclic.

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2 Answers

up vote 2 down vote accepted

Your short exact sequence is wrong. Unless the morphism $f$ is etale (which it is not when $Y$ equals $\mathbb{P}^1$), then the induced morphism $T_X\to f^*T_Y$ is not surjective.

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But Hartshorne, chapter IV , prop 2.1 says that if the cover is finite and separable (the dual) of this exact sequence is true. –  Darius Math Jul 27 '13 at 11:58
    
Yes, but there is an Ext^1 that occurs when you "dualize" the short exact sequence of sheaves of relative differentials. That Ext^1 is the cokernel of this (non-surjective) morphism. –  Jason Starr Jul 27 '13 at 12:03
    
Dear Jason, thanks alot. Do you know a reference for the computation of $H^{1}(X,T_{X})^{G}$? I desperately need to know the answer for $G$ abelian. For $G$ cyclic I know that this dimension is $s-3$ (s: number of branch points) does this also hold for $G$ abelian? –  Darius Math Jul 27 '13 at 13:23
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You may get what you need from various papers on the equivariant Riemann-Roch theorem applied to your G-linearized sheaf $T_X$. For example, look at the papers of Borne and/or Ellingsrud-Lonstead.

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Thank you. It would have been more useful if you gave a more exact reference. I will search this on net. –  Darius Math Jul 27 '13 at 16:32
    
Searching MathSciNet may help you to discover even more useful paper. The difficult problem is the case when the order of the group is divisible by the characteristic of the field, and there is much research done on this problem. But, I guess you are only interested in the complex case. –  user37622 Jul 28 '13 at 17:11
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