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A line bundle over a complex manifold is called positive is if its Chern class is the fundamental form of a Kaehler manifold. For vector bundles of higher rank, the Chern class is no longer in general a $2$-form, but a sum of forms of different degree.

How does the notion of positivity extend to higher rank vector bundles? An obvious idea is to ignore the higher classes and to define positivity in terms of the first Chern class just as for line bundles. In this case, for example, the standard global proof of Kodaira vanishing (see Huybrechts for example) appears to carry over.

Since this is not the definition that appears in a quick google search of positive line bundles, I guess that that this is not a good way to do things, however I can't see why. Is this definition too restrictive to be of use?

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    $\begingroup$ Given a vector bundle $V$ on a Kähler manifold $X$, call $V$ ample (or nef or...) when the (dual of the) tautological line bundle $\mathcal{O}_{\mathbf{P}(V)}(1)$ on the projectivized vector bundle $\mathbf{P}(V)$ is ample (or nef or...). See the first chapter of Lazarsfeld's "Positivity in Algebraic Geometry II" for more. For a more analytic geometric perspective, see also section 6 of Demailly's "Complex Analytic and Differential Geometry". $\endgroup$
    – Raymond Cheng
    Commented Jan 21, 2018 at 1:19
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    $\begingroup$ It is incorrect to claim that Kodaira vanishing depends only on the first Chern class. Just consider a direct sum of two holomorphic invertible sheaves on a complex projective space. If one of the two sheaves has negative degree, yet the other has positive degree such that the sum of the degrees is positive, then the first Chern class of the direct sum is positive. Yet Kodaira vanishing fails for this direct sum. $\endgroup$
    – Jason Starr
    Commented Jan 21, 2018 at 1:40
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    $\begingroup$ You are not the first one to ask this question, there has been a flurry of activity around this 50 years ago. Lazarsfeld's Positivity in Algebraic Geometry, II gives an excellent survey of the subject. $\endgroup$
    – abx
    Commented Jan 21, 2018 at 5:36

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