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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!

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Dear Simone,

This is just a comment.

The answer to your question, if one exists, is surely that there exist universal polynomials $p_j^r$ ($1 \leq j \leq r = {\rm rank} E$) in the Chern classes of the vector bundle $E \to X$ such that $$ \int_X p_j{}^r(c_1(E), \dots, c_j(E)) \wedge \omega^{n-j} > 0 $$ for all $j$ implies that the bundle $E$ is ample. Here "universal" means that the polynomials in question only depend on the dimension $n$, but otherwise not on the variety $X$. For line bundles, these polynomials are known and are, as you wrote, $p_j{}^1(x) = x^j$ for all $j$.

The problem is that the condition you wrote for line bundles is a corollary of a more general theorem of father, one that characterizes Kahler classes amongst real $(1,1)$-classes. Our hope of approaching the problem should thus be to find suitable positivity criterion for higher degree classes. The ideal outcome would be a higher rank version of the Kodaira condition; so we'd know that if a "degree vector" $(u_1, \dots,u_r)$ of integral cohomology classes satisfies some conditions, then there exists an ample vector bundle $E$ of rank $r$ such that $c_j(E) = u_j$ (compare with $L$ ample iff $c_1(L)$ Kahler and integral). The trouble is that finding these conditions amounts to finding the universal polynomials $p_j{}^r$, and thus answering a much more general question.

In short, we have no idea what a "positivity condition" for a collection of cohomology classes $(u_1, \ldots, u_r)$ looks like. I agree with your approach of simplifying the problem and starting the search for these on complex tori. However, I think that if we knew the answer on complex tori, we'd know it on general complex manifolds too.

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Dear Gunnar, you are almost right... What you are saying already exists in some sense: it is a theorem by Fulton and Lazarsfeld which gives you necessary conditions for a vector bundle to be ample in terms of positive polynomials. Unfortunately, one knows thanks to an example by Mumford that a numerical criterion for ampleness of vector bundles cannot exists in general. But, on complex tori... Probably I should add this comment as a part of my question. – diverietti Jul 20 '12 at 6:52

Let $\mathscr E$ be a vector bundle on $X$ (not even necessarily a torus) and consider $P=\mathbb P(\mathscr E)$ the associated projective bundle with the corresponding natural relatively ample line bundle $\mathscr L:= \mathscr O_{\mathbb P(\mathscr E)}(1)$.

Then $\mathscr E$ is ample on $X$ if and only if $\mathscr L$ is ample on $P$ (it is always ample over $X$, but this means ample over the base field (or scheme)).

By the Nakai-Moishezon criterion $\mathscr L$ is ample on $P$ if and only if for any irreducible and reduced subvariety $Y\subseteq P$ of dimension $j$ for any $1\le j\le n=\dim X$, $$ c_{1}(\mathscr L)^j\cdot Y>0. $$

If $X$ is a torus, then you might be able to simplify this by limiting the list of $Y$ that you have to look at.

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Can we write down these criterion without classifying holomorphic vector bundles over complex tori? – Gunnar Þór Magnússon Jul 19 '12 at 18:15
Dear Sándor, I don't understand your answer. I am asking for a criterion in order to insure that a vector bundle on a complex torus is ample. The characterization I wrote for ample line bundles is valid only on tori. The projectivization of a vector bundle is not a torus. I am looking for a numerical criterion for ampleness of vector bundle on complex tori! – diverietti Jul 19 '12 at 18:21
Dear Simone, sorry, I was careless while reading and I assumed you wrote down the Nakai-Moishezon criterion... – Sándor Kovács Jul 19 '12 at 23:52
That's what it seemed to me, too! :) Thank you very much anyway! – diverietti Jul 20 '12 at 6:54

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