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For each 2-dimensional complex torus $T$, we construct a compact complex manifold $X\left(T\right)$ with a ${ℂ}^2$-action, which compactifies ${\left({ℂ}^{*}\right)}^4$ such that the quotient of ${\left({ℂ}^{*}\right)}^4$ by the ${ℂ}^2$-action is biholomorphic to $T$. For a general $T$, we show that $X\left(T\right)$ has no non-constant meromorphic functions.

We establish new results on weighted $L^2$-extension of holomorphic top forms with values in a holomorphic line bundle, from a smooth hypersurface cut out by a holomorphic function. The weights we use are determined by certain functions that we call denominators. We give a collection of examples of these denominators related to the divisor defined by the submanifold.