Let $f : X\rightarrow Y$ be a proper flat morphism of algebraic varieties 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/~idolga/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? 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 is not local on $Y$).

Even if there is no exact "formula" in general, what are some nontrivial results in this direction?