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Suppose you have a projective variety $\tilde{Y}$ on $\mathbb{P}^n$ defined by the zero locus of homogeneous polynomials $F_1,\dots,F_m$ (with all the hypotheses as you wish). And denote by $Y$ its affine cone as a subvariety of $X=\mathbb{C}^{n+1}-0$.

Also consider the following known exact sequences for the sheaf-theoretical restriction to $Y$ (of $(\mathcal{O}_X)|_Y-modules)$: \begin{align}\label{int1} \nonumber 0\rightarrow I_Y \rightarrow &\mathcal{O}_X \rightarrow (i_Y)_*(\mathcal{O}_Y) \simeq \mathcal{O}_X/_{I_Y} \rightarrow 0\\ 0\rightarrow (I_Y)|_Y \rightarrow &(\mathcal{O}_X)|_Y \rightarrow ((i_Y)_*(\mathcal{O}_Y))|Y \simeq \mathcal{O}_Y \rightarrow 0 , \end{align}

Question: When can I say that the global sections in $H^0(Y,(\mathcal{O}_X)|_Y)$ coincides with the resctriction to $Y$ of an element in $H^0(X,\mathcal{O}_X)$ ?

Maybe this is equivalent to ask the same for the global sections in $H^0(Y, \mathcal{O}_Y)$.

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    What is $i_Y$? And what do you mean by $\vert_Y$? – 2017-01-18
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    $i_Y$ denotes the inclusion from $Y$ to $X$. And $|_Y$ denotes the restriction of sheaves. This operator conserves the same stalks but its only supported on the points of $Y$ . In general, if you have a sheaf $\mathcal{F}$ of $\mathcal{O}_X$-mod, then $\mathcal{F}|_{Y}$ is a sheaf on $Y$ of $(\mathcal{O}_X)|_Y$-mod. The question can be resumed by: what can be said for the global sections $H^0(Y,(\mathcal{O}_X)|_Y)$? – 2017-01-18
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    Sorry, I was confused, all the notation was wrong. I'm going to edit the entire question. Thanks. – 2017-01-18
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    Anyway, $O_X\vert_Y \cong O_Y$, and a suffucient condition for the morphism $H^0(X,O_X) \to H^0(Y,O_Y)$ to be an isomorphism is $H^0(X,I_Y) = H^1(Y,I_Y) = 0$ (it doesn't matter here what are $X$ and $Y$, the only important thing is that $Y$ is a closed subscheme in $X$). – 2017-01-18
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    Are you sure that $\mathcal{O}_X|Y \simeq \mathcal{O}_Y$?, if think they have different stalks. If $y\in Y$ then: $$\mathcal{O}_{Y,y} \ne (\mathcal{O}_X|Y)_{y} = \mathcal{O}_{X,y}$$. I'm using the definition for $|_Y$ established at the first section of the second chapter of Hartshorne's book. – 2017-01-18
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    I've found a good discusion about the difference between the intuitive restriction and the sheaf-theoretical's one. It can be consulted the page 20 of "Hans Grauert Reinhold Remmert Coherent Analytic Sheaves" – 2017-01-18
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    You are confused between two notions of a pullback (restriction is a particular case of it). The one used for sheaves of abelian groups and the one for coherent sheaves. In the latter case the pullback is the composition of the sheaf-theoretic pullback with an extension of scalars. – 2017-01-18
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    I'll think about that... thanks! – 2017-01-18

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