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I'm reading some paper recently. I find a notion which I can not find the exact definition. What is the numerically positive cone in the Neron-Severi group?

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It could mean several things. What is the paper you read it in? –  Artie Prendergast-Smith Sep 19 '13 at 8:22

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One of the definitions would be the set of real (1,1)-cohomology classes $\alpha$ such that for any analytic cycle $Y\subset X$ of dimension $\dim Y=d$ one has $\int_Y\alpha^d>0$. A result of Demailly-Paun shows that the Kähler cone is nothing but a connected component of the numerically positive one.

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Now, let $X$ be a non-singular projective surface over $\mathbb{C}$ and $L$ be a line bundle on $X$. The cone is defined by $c_1(L)^2 >0$ and $c_1(L)\cdot H >0$ in the Neron-Severi group. Is this cone and K$\ddot{a}$hler cone coincide? Where H denotes the hyperplane bundle. –  SWalker Sep 20 '13 at 4:26
    
Yes, if $X$ is projective algebraic, then the two coincide (see Demailly). –  Anton Fonarev Sep 20 '13 at 12:34

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