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Q:

  • How to show that on the smooth projective variety $X$, any divisor $D$ can be represented by the difference of two effective divisors, i.e. $D=D_1-D_2$.

  • If $X$ is just any complex manifold, does above also hold?

DLIN
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    Isn't this true by definition of a divisor? A divisor is just a formal $\mathbf{Z}$-linear combination of codimension one subvarieties of a variety $X$. So you just express $D$ as the part of the linear combination with positive coefficients, minus the part with negative coefficients. Depending on what you mean by a divisor on a complex manifold, the same argument works. – Takumi Murayama Feb 12 '17 at 07:13
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    A more subtle question is whether you can choose $D_1,D_2$ to be smooth divisors. See this other question for how you can ensure this version, at least on a smooth projective variety over an infinite field (note it is written for surfaces, so just replace "curve" with "divisor" to make the proof work in arbitrary dimension). – Takumi Murayama Feb 12 '17 at 07:14

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