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First let's fix some definitions:

Definitions:

  • Complex manifold (of dimension n): Is a locally ringed space $(X,\mathscr F)$, where there is an open cover $\bigcup_{i\in I} U_i=X$ such that $(U_i,\mathscr F_{X|U})\cong (V, \mathscr O_V)$ (as locally ringed spaces). Here $V\subseteq\mathbb C^n$ is an open set and $\mathscr O_V$ is the sheaf of holomorphic functions. This definition is equivalent to the usual "chart definition".

  • Complex Variety (of dimension n): Is a locally ringed space $(X,\mathscr G)$, where there is an open cover $\bigcup_{i\in I} U_i=X$ such that $(U_i,\mathscr G_{X|U})\cong (V, \mathcal O_V)$. Here $V\subseteq\mathbb C^n$ is an open set and $\mathcal O_V$ is the sheaf of regular functions.

  • Holomorphic $1$-form on a manifold: Is a holomorphic section of the cotangent bundle. In particular a global holomorphic form can be written as $\omega= fdz$ where $f\in\mathscr F(X)$

  • Algebraic $1$-form on a variety: If $A$ is a $k$-algebra, one defines the Kahler module of differentials $\Omega_{A/k}$. Then the $\mathcal O_X$-module of algebraic differentials on $X$ is the sheaf $$\Omega^1_X(U):=\Omega_{\mathcal O_X(U)/\mathbb C}$$ An element of $\Omega^1_X(X)$ is a global algebraic $1$-form.

  • Meromorphic $1$-form on a manifold: Is a meromorphic section of the cotangent bundle. In particular a global meromorphic form can be written as $\omega= fdz$ where $f$ is meromorphic.

My questions:

  1. Clearly an algebraic $1$-form is also an holomorphic $1$-form, but the converse doesn't hold. Are there some particular cases where every holomorphic $1$-form is also algebraic?

  2. What are the algebraic correspondents of meromorphic $1$-forms? Maybe one should define the sheaf $$\mathcal M(U):= \Omega_{\mathbb C(X)/\mathbb C}$$ where $\mathbb C(X)$ is the field of rational functions.

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    $\begingroup$ For (1), check out Serre's GAGA theorem. If your variety is projective then anything holomorphic is actually algebraic. $\endgroup$ – Gunnar Þór Magnússon Nov 13 '13 at 16:16
  • $\begingroup$ The algebraic counterpart of meromorphic 1-forms are elements of the stalk of the sheaf of algebraic 1-forms at the generic point. $\endgroup$ – Zhen Lin Nov 13 '13 at 17:22

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