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Let $X$ be a smooth, projective variety over $\mathbf{C}$ (to keep things simple) and let $\mathcal{F}$ be a vector bundle on $X$.

If $\mathcal{F}$ has a nowhere vanishing holomorphic section, then the top Chern class of $\mathcal{F}$ vanishes. Are there other (weaker) "natural" conditions on $X$ and/or $\mathcal{F}$ which guarantee the vanishing of the top Chern class?

Let $X$ be a smooth, projective variety over $\mathbf{C}$ (to keep things simple) and let $\mathcal{F}$ be a vector bundle on $X$.

If $\mathcal{F}$ has a nowhere vanishing holomorphic section, the top Chern class of $\mathcal{F}$ vanishes. Are there other (weaker, but "natural") conditions on $X$ and/or $\mathcal{F}$ which guarantee the vanishing of the top Chern class?

Let $X$ be a smooth, projective variety over $\mathbf{C}$ (to keep things simple) and let $\mathcal{F}$ be a vector bundle on $X$. If $\mathcal{F}$ has a nowhere vanishing holomorphic section, then the top Chern class of $\mathcal{F}$ vanishes. Are there other (weaker) "natural" conditions on $X$ and/or $\mathcal{F}$ which guarantee the vanishing of the top Chern class?

Let $X$ be a smooth, projective variety over $\mathbf{C}$ (to keep things simple) and let $\mathcal{F}$ be a vector bundle on $X$. 

If $\mathcal{F}$ has a nowhere vanishing holomorphic section, then the top Chern class of $\mathcal{F}$ vanishes. Are there other (weaker) "natural" conditions on $X$ and/or $\mathcal{F}$ which guarantee the vanishing of the top Chern class?

Let $X$ be a smooth, projective variety over $\mathbf{C}$ (to keep things simple) and let $\mathcal{F}$ be a vector bundle on $X$. It is known that if $\mathcal{F}$ has a nowhere vanishing section, then the top Chern class of $\mathcal{F}$ vanishes. Are there other (weaker) "natural" conditions on $X$ and/or $\mathcal{F}$ which guarantee the vanishing of the top Chern class?

Let $X$ be a smooth, projective variety over $\mathbf{C}$ (to keep things simple) and let $\mathcal{F}$ be a vector bundle on $X$. If $\mathcal{F}$ has a nowhere vanishing holomorphic section, then the top Chern class of $\mathcal{F}$ vanishes. Are there other (weaker) "natural" conditions on $X$ and/or $\mathcal{F}$ which guarantee the vanishing of the top Chern class?