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In "Basic algebraic geometry 2", Shafarevich finds a relation between the Euler characteristic and the genus of the curve. At page 139 he says that there's no analogue for varieties of dimension $>1$.

Is there some other relation connecting these objects in higher dimension, like a Riemann-Hurwitz type formula or something else?

At which point are researches on this problem?

Can you give me also some reference?

Using tropical geometry, is it possible to achieve some result in this direction?

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See this question for Riemann--Hurwitz in higher dimensions.

But there cannot be a simple relation between (topological) Euler characteristic and "genus" in higher dimensions, for any definition of "genus" that is a function only of the Hodge numbers $h^{i,0}=h^0(X, \Omega^i)$. (Remember that for a curve, $g=h^{1,0}$.) The reason is that the Hodge numbers are birational invariants of smooth varieties. In particular, they don't change under blowing up. But the Euler characteristic does: for example, blowing up a point on a smooth surface increases the Euler characteristic by 1.

On the other hand, as Shafarevich goes on to say in the next paragraph, for any smooth projective variety $X$, there are some relationships between these Hodge numbers and the topology of $X$. For example the first Betti number of $X$ – that is, the rank of $H_2(X)$ – is $b_1(X)=2h^{1,0}$.

If you want a more precise relationship, you need to take into account all the Hodge numbers of $X$. (Recall that these are defined as $h^{i,j}=H^j(X,\Omega^i)$, so they involve not just holomorphic differentials, but higher cohomology of the sheaves of differentials. Don't ask me why $i$ and $j$ get flipped – read @GeorgesElencwajg's comment below!)

Then the Hodge Decomposition Theorem says that for each $k$, we have

$$ H^k(X,\mathbb C) \cong \bigoplus_{i+j=k} H^j(X, \Omega^i) \ ;$$

taking the alternating sum for $k$ from $0$ to $\operatorname{dim} X$ we get finally an expression for the Euler characteristic in terms of (cohomology groups of sheaves of) differentials on $X$:

$$e(X) = \sum_k (-1)^k b_k = \sum_{i,j} (-1)^{i+j} h^{i,j}.$$

To complete the picture, note that the only nonzero Hodge numbers on a curve are $h^{0,0}=h^{1,1}=1$ and $h^{0,1}=h^{1,0}=g$, so in this case the formula reduces to the familiar $e(X)=2-2g$ .

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    $\begingroup$ Dear Asal: excellent answer as usual. Here is why $i$ and $j$ get flipped: the sheaf $\Omega^i$ has a Dolbeault resolution $(...\to\mathcal E^{ij}\to...)_j$ by fine sheaves which calculate its cohomology. So $H^j(X,\Omega ^i)$ is calculated by taking closed forms in $\mathcal E^{ij}(X)$ modulo exact forms . Since the forms in $\mathcal E^{ij}(X)$ are written as $\Sigma f_{KL}dz^K \wedge d\bar z^L$ with $i$ factors $dz^k$ and $j$ factors $d\bar z^l$ this explains (I think) the flipped terminology $h^{ij}(X)$ . $\endgroup$ – Georges Elencwajg Jan 20 '15 at 14:46
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    $\begingroup$ @GeorgesElencwajg: thanks for that explanation. This was something that always bothered me a bit, but never quite enough to track down the answer! $\endgroup$ – user64687 Jan 20 '15 at 14:49
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    $\begingroup$ @Asal Thank you! Can you suggest me some references on this topic? $\endgroup$ – TheWanderer Jan 20 '15 at 16:51
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    $\begingroup$ @Federico: I am sure all this is discussed in great detail in Huybrechts' book Complex Geometry, in Griffiths and Harris' Principles of Algebraic Geometry, and in Demailly's freely available book _Complex Analytic and Differential Geometry online at www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf. I don't have more precise references, I am afraid. $\endgroup$ – user64687 Jan 20 '15 at 16:57

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