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In continuation to my previous post: Question Regarding Riemann-Hurwitz Formula Proof

I'll be glad to receive some explanations regarding the following: 1) I know that when taking a sheaf $F$ , then the notation $f_* F $ corresponds to the direct image sheaf. In the answer Gunnar posted, he wrote: $f_* \mathbb{C} = \mathbb{C}^{ \oplus d} $ . What is the meaning of $f_* \mathbb{C}$ in this context? Does the notation $ \mathbb{C}^{ \oplus d}$ means the direct sum of d times $\mathbb{C}$ ?

2) Given the short exact sequence $ 0 \to f_* \mathbb{C} \to \mathbb{C}^{\oplus d} \to \mathcal{G} \to 0 $ .
I understand from the previous calculation, the map between $f_*\mathbb{C}$ and $\mathbb{C}^{\oplus d} $ is the identity. But what is the map between $\mathbb{C}^{\oplus d} $ and the skyscraper sheaf $ \mathcal{G}$ ?

3) Given a Riemann Surface $X$ , does the algebraic Geometry of its genus is $\chi(X) := \chi(X,\mathbb{C})$ ?

Thanks in advance !

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    $\begingroup$ 1) I should precise that $f_*\mathbb C = \mathbb C^{\oplus d}$ only on neighborhoods of images of non-ramified points on $Y$. Here $\oplus d$ is indeed the direct sum. 2) Note that the inclusion map $f_*\mathbb C \to \mathbb C^{\oplus}$ is absolutely not canonical! You need to construct it yourself, e.g. by choosing a fine enough covering of the surface. Then the other map is simply the quotient map. 3) $\chi(X)$ is the topological Euler characteristic of $X$. For a Riemann surface of genus $g$ it is equal to $2 - 2g$ by the Riemann-Roch theorem and Hodge theory. $\endgroup$
    – Gunnar Þór Magnússon
    Commented Jun 16, 2012 at 9:38
  • $\begingroup$ Thanks again Gunnar, regarding 1- But what do you mean by $f_* \mathbb{C} $ ? It seems not suitable for the direct image sheaf case regarding 3- I know the definition of the topological Euler char of $X$. But in your previous answers, you wrote: $\chi(\mathbb{C}^{\oplus d }) = d \chi(Y) $ . How did you get this result? Thanks ! $\endgroup$
    – jason mfash
    Commented Jun 16, 2012 at 13:41

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