Let $f : X\rightarrow Y$ be a proper flat morphism (of schemes) with connected fibers over a smooth projective curve $Y$ over $\mathbb{C}$. Let $X\_{y\_0}$ denote a smooth fiber over $y\_0\in Y$.


If $f$ is smooth, then the topological Euler characteristic is multiplicative: $\chi(X) = \chi(Y)\cdot\chi(X\_{y\_0})$. In general, there is a [formula](http://www.math.lsa.umich.edu/%7Eidolga/sbore72.pdf) for the topological Euler characteristic which includes some correction terms for the singular fibers:
$$\chi(X) = \chi(Y)\chi(X\_{y\_0}) + \sum\_{y\in Y}(\chi(X\_y) - \chi(X\_{y\_0})$$


Does there exist an analogous formula for the holomorphic/algebraic Euler characteristic? I'm in particular interested in the case where $X\rightarrow Y$ is a family of curves over $Y$ (so $\dim X = 2$). To be precise, I'm looking for a formula for $\chi(X,\mathcal{O}\_X)$ in terms of geometric invariants of $Y$ and some fibral data. By fibral data I mean geometric properties which are local on the base $Y$ (e.g. geometric invariants of fibers, monodromy around singular fibers acting on the cohomology of a smooth fiber...etc).


As a nonexample, Riemann-Roch relates this Euler characteristic to the intersection number of $X\_{y\_0}$ with $K - X\_{y\_0}$ (which can be understood locally on $Y$) plus $\chi(X,\mathcal{O}\_X(X\_{y\_0}))$ (which does not appear to be local on $Y$).


**Added in edit:** To state a more precise question -- Is the holomorphic Euler characteristic $\chi(X,\mathcal{O}\_X)$ determined by $Y$ and data which is local on $Y$? Stated another way, do there exist maps $f : X\rightarrow Y$ and $f' : X'\rightarrow Y$ (satisfying the above conditions) such that


1. there is a covering in the etale topology $\{U\_i\rightarrow Y\}\_{i\in I}$ and isomorphisms $\phi\_i : X\_{U\_i}\rightarrow X'\_{U\_i}$, and
2. $\chi(X,\mathcal{O}\_X)\ne \chi(X',\mathcal{O}\_{X'})$.


Here I'm happy to assume that $X,X'$ are smooth over $\mathbb{C}$ (but I still want to allow singular fibers of $f,f'$). If it changes the answer, I'm also interested in the variation where $\{U\_i\}$ is instead an open covering in the analytic topology. I am also interested in results where we assume that $X,X'$ are moreover general type. If the euler characteristic is not determined by $Y$ and data local on $Y$, then I'm also interested in any contraints such data might impose on $\chi(X,\mathcal{O}\_X)$.



Relating the holomorphic Euler characteristic of a family of algebraic varieties to properties of the base and fibers
I'm not entirely sure what would constitute an answer. But here a few simple observations. Let me focus on what you seem be interested in, namely a projective family of connected curves over smooth projective curve $f:X\to Y$. Let me also assume $X$ smooth. Let $h$ be the genus of $Y$, and $g$ the genus of the general fibre. Then from the Leray spectral sequence, Riemann-Roch and Grothendieck duality
$$\chi(\mathcal{O}\_X)= (1-g)(1-h) - \deg R^1f\_\*\mathcal{O}\_X=\chi(\mathcal{O}\_Y)\chi(\mathcal{O}\_{X\_{y\_0}})+\deg f\_\*\omega\_{X/Y}$$
A theorem of Fujita then gives an inequality
$$\chi(\mathcal{O}\_X)\ge \chi(\mathcal{O}\_Y)\chi(\mathcal{O}\_{X\_{y\_0}})$$
which is something.
To say more, one would need to compute $\deg f\_\*\omega\_{X/Y}$. But I think this depends on more than local information at the bad fibres, which is what I think you are asking. The reason I say this, is that this degree can be positive and variable for Kodaira surfaces, which have no bad fibres at all. [**Added Comment** I guess there are various things called Kodaira surfaces. The ones I have in mind are of general type, and hence algebraic. See page 220 of *Compact Complex Surfaces* by Barth, Hulek, Peters and Van de Ven for further details.]


