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Is there a version of Noether's formula for the Euler characteristic of a surface for Real algebraic surfaces?

Specifically, given $X$ a real algebraic compact smooth surface, what is the relationship between the Euler characteristic of $X$ as an algebraic scheme and the Euler characteristic of $X(\mathbb{R})$ as a topological subspace with the Euclidean topology$?

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    $\begingroup$ Could you clarify what you mean by "the Euler characteristic of $X$ as an algebraic scheme"? $\endgroup$
    – abx
    Commented Feb 23 at 6:14
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    $\begingroup$ But $X$ is an affine scheme, so this doesn't make much sense. $\endgroup$
    – abx
    Commented Feb 23 at 15:29
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    $\begingroup$ @abx Why not? In the complex case we have that if $X$ is a complex nonsingular projective variety of dimension $n$ then $\chi_{top}(X(\mathbb{C}))=12 \chi(X) - K_X^2$. I am interested in an analogue of this over the real numbers. I clarified the statement now. $\endgroup$
    – Serge the Toaster
    Commented Feb 23 at 16:32
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    $\begingroup$ For an affine scheme $X$, $H^{i}(\mathscr{O}_X)=0$ for $i>0$, and $\dim H^0(\mathscr{O}_X)=\infty$. $\endgroup$
    – abx
    Commented Feb 23 at 20:22
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    $\begingroup$ @abx Please take a look at en.wikipedia.org/wiki/Riemann%E2%80%93Roch_theorem_for_surfaces $\endgroup$
    – Serge the Toaster
    Commented Feb 23 at 21:33

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