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The answer is negative if the complex torus is not algebraic (or equivalently, not Moishezon). More generally, if $U$ is the analytification of a separated scheme of finite type over $\mathbf{C}$ and $Y$ is a proper complex-analytic space admitting an open immersion $j:U \hookrightarrow Y$ onto the complement of a nowhere-dense analytic set then $Y$ must be ...


The assertion (X) is false in any characteristic $p > 0$ for any $Y$ with genus at least 2. (It is true and easy for $Y$ of genus 1, and true and easy and uninteresting for $Y$ of genus 0.) To see this, we may and do choose a subgroup scheme $G \subset J := {\rm{Jac}}(Y)$ of the nonzero infinitesimal Frobenius kernel of $J$ such that $G$ is equal to ...


Yes, this is classic. A counterexample was found by Serre, and more classes of counterexamples are in Fulton's paper Ample Vector Bundles, Chern Classes, and Numerical Criteria (Inventiones, 1976).


Proposition 3.3 of Ullmo's paper "Hauteur de Faltings de quotients de J_0(N) " (American Journal of Math., 2000) seems to answer your question.


I think the following should give a counterexample. Let $\mathcal{O}$ be an order in an imaginary quadratic field $K$ and $\mathcal{O}_K$, the ring of integers. Then it's not too hard to find a (non-split) short exact sequence of $\mathcal{O}$-modules: $$0 \to \mathcal{O}_K \to \mathcal{O} \oplus \mathcal{O} \to \mathcal{O}_K \to 0,$$ e.g. if $1, ...

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