Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Hey Damiano, in fact I think that a the family of line bundles associated to the exceptional divisors of the blow-up at point of $X$ is bounded since $X$ is compact, isn't it? But the argument must be a little more subtle since $\sigma^*A$ is no more ample...
J.C. I think my question is more of metric nature: I am asking for positivity, not for ampleness. They are the same, after Kodaira's embedding theorem, but this question is motivated by the proof itself of that theorem, thus it should be better to do not confuse ampleness with positivity.
Dear Sándor, I am not essentially asking about Seshadri constant. I don't want to measure locally the positivity of A and I do know the example of Miranda. My question (apart from the first, which is obviously false, as I do have also pointed out), after a moment of reflection, was about smooth hermitian metrics construction on the (line bundle associated to) exceptional divisor of blow-ups nearby the point already blown-up. Of course the bound on the multiple of $A$ may depend on $X$: my question was not universal w.r.t the line bundle $A$ but w.r.t the point on $X$ I am blowing-up.
Ciao Damia'!! Thank you for your answer... It is absolutely close to what I want: my question came from the following on positivity which I post here below.
You are definitely right. In any case, this condition on the degree seems to be enough in order to control the intersection products in the Nakai-Moishezon criterion, right?
I do not need more. I was just curious... Anyway Voisin's examples won't work for the surface case: she gives constructions starting from dimension 4 I guess...
In the same paper Hironaka says that the same question is open for a family of 2-dimensional manifold. By a paper of our big brother Dan Popovici, such an example cannot be of algebraic nature, since he proved that in a holomorphic family where all fibers are projective except at most the central one, all fibers are in fact projective.
Thanks little brother, I had just found it! I haven't read the paper neither... Apparently he proves more: he constructs a family $V_t$ of 3-dimensional abstract non-singular algebraic varieties such that $V_t$ is projective for $t\ne 0$ but $V_0$ carries a positive $1$-cycle, algebraically equivalent to zero.
It is a conjecture by Kobayashi (open starting from dimension three) that projective hyperbolic manifolds have ample canonical bundle. In particular, conjecturally you won't be able to find such examples by finding hyperbolic manifolds with non maximal Kodaira dimension.
