Ample vector bundles on complex tori - MathOverflow

Ample vector bundles on complex tori

2012-07-19T16:13:28Z

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!

Answer by Sándor Kovács for Ample vector bundles on complex tori

2012-07-19T17:49:57Z

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.

Answer by Gunnar Magnusson for Ample vector bundles on complex tori

2012-07-20T06:04:27Z

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.