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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 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. In general I am also interested in anything interesting you can say about $\chi(X,\mathcal{O}_X)$ given $Y$ and information about $X$ which is local on $Y$.

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 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 $\chi(X,\mathcal{O}_X)$.

Let $f : X\rightarrow Y$ be a proper flat morphism 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 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 is not local on $Y$).

Even if there is no exact "formula" in general, what can be said about $\chi(X,\mathcal{O}_X)$ using information about $Y$ and local information near the fibers of $f$? I'm happy to assume $X$ is smooth (over $\mathbb{C}$).

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 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. In general I am also interested in anything interesting you can say about $\chi(X,\mathcal{O}_X)$ given $Y$ and information about $X$ which is local on $Y$.

Let $f : X\rightarrow Y$ be a proper flat morphism 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 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 is not local on $Y$).

Even if there is no exact "formula" in general, what can be said about $\chi(X,\mathcal{O}_X)$ using information about $Y$ and local information near the fibers of $f$?

Let $f : X\rightarrow Y$ be a proper flat morphism 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 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 is not local on $Y$).

Even if there is no exact "formula" in general, what can be said about $\chi(X,\mathcal{O}_X)$ using information about $Y$ and local information near the fibers of $f$? I'm happy to assume $X$ is smooth (over $\mathbb{C}$).