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Let $X$ be a $n$-dimensional complex torus and $\omega$ a Kähler form on $X$. Then, it is well known that a real $(1,1)$-class $[\alpha]\in H^{1,1}(X,\mathbb R)$ is a Kähler class if and only if for all $1\le j\le n$ one has $$\int_X\alpha^j\wedge\omega^{n-j}>0.$$ In particular, if $L\to X$ is a holomorphic line bundle, then $L$ is ample (and hence $X$ is an abelian variety) if and only if $$c_{1}(L)^j\cdot[\omega]^{n-j}>0,\quad 1\le j\le n.$$

Question. Is it known any similar numerical criterion for higher rank holomorphic vector bundle on complex tori?

I would be interested also in weaker forms of this question. For instance, adding the hypothesis of semi-stability for the vector bundle, or possibly just requiring the vector bundle to be big instead of ample.

Any hint or comment is very welcome!

2 added 10 characters in body

Let $X$ be a $n$-dimensional complex torus and $\omega$ a Kähler form on $X$. Then, it is well known that a real $(1,1)$-class $[\alpha]\in H^{1,1}(X,\mathbb R)$ is a Kähler class if and only if for all $1\le j\le n$ one has $$\int_X\alpha^j\wedge\omega^{n-j}>0.$$ In particular, if $L\to X$ is a holomorphic line bundle, then $L$ is ample (and hence $X$ is an abelian variety) if and only if $$c_{1}(L)^j\cdot[\omega]^{n-j}>0,\quad 1\le j\le n.$$

Question. Is it known any similar numerical criterion for higher rank holomorphic vector bundle?

I would be interested also in weaker forms of this question. For instance, adding the hypothesis of semi-stability for the vector bundle, or possibly just requiring the vector bundle to be big instead of ample.

Any hint or comment is very welcome!

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Ample vector bundles on complex tori

Let $X$ be a $n$-dimensional complex torus and $\omega$ a Kähler form on $X$. Then, it is well known that a real $(1,1)$-class $[\alpha]\in H^{1,1}(X,\mathbb R)$ is a Kähler class if and only if for all $1\le j\le n$ one has $$\int_X\alpha^j\wedge\omega^{n-j}>0.$$ In particular, if $L\to X$ is a holomorphic line bundle, then $L$ is ample (and hence $X$ is an abelian variety) if and only if $$c_{1}(L)^j\cdot[\omega]^{n-j}>0,\quad 1\le j\le n.$$

Question. Is it known any similar criterion for higher rank holomorphic vector bundle?

I would be interested also in weaker forms of this question. For instance, adding the hypothesis of semi-stability for the vector bundle, or possibly just requiring the vector bundle to be big instead of ample.

Any hint or comment is very welcome!