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Winnie_XP
Winnie_XP
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I am wondering if the following argument is true: Let

Let $X$ be a $\dim n$ compact projective complex manifold, let $\alpha\in H^{2k}(X,\mathbb{Q})$$\alpha\in H^{2n-2}(X,\mathbb{Q})$ be a cohomology class. If for any ample line bundle $L$, we have $c_1(L)\cup \alpha=0$, can I argue that $\alpha=0$?

I am wondering if the following argument is true: Let $X$ be a compact projective complex manifold, let $\alpha\in H^{2k}(X,\mathbb{Q})$ be a cohomology class. If for any ample line bundle $L$, we have $c_1(L)\cup \alpha=0$, can I argue that $\alpha=0$?

I am wondering if the following argument is true:

Let $X$ be a $\dim n$ compact projective complex manifold, let $\alpha\in H^{2n-2}(X,\mathbb{Q})$ be a cohomology class. If for any ample line bundle $L$, we have $c_1(L)\cup \alpha=0$, can I argue that $\alpha=0$?

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Winnie_XP
Winnie_XP
  • 287
  • 1
  • 8

Cup Product with Ample Line Bundles

I am wondering if the following argument is true: Let $X$ be a compact projective complex manifold, let $\alpha\in H^{2k}(X,\mathbb{Q})$ be a cohomology class. If for any ample line bundle $L$, we have $c_1(L)\cup \alpha=0$, can I argue that $\alpha=0$?