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It was quite difficult writing this question...I hope you can understand the goal of that. I want to ask it in order to clarify my ideas. I'm studying Hyperkahler geometry (principally on works by Huybrechts, Markman and O'Grady) and I am wondering why one wants to look at the structure of a particular cone in order to get geometric properties about the object he is studying.

I can understand that if you want to study problems regarding projectivity then it is useful to look at the ample cone. And if you are interested in the Kahler geometry then you can look at the Kahler cone.

Anyway, for example, I cannot see the following: one of the claims (or one of the conseguences, if you want) of the Global Torelli Theorem for Hyperkahler manifolds states that a marked pair $(X,\eta)\in\mathfrak{M}_{\Lambda}$ (where $X$ is an Hyperkahler, $\eta:H^2(X,\mathbb{Z})\to\Lambda$ is a marking and $\mathfrak{M}_{\Lambda}$ is the moduli space of such pairs) is an Hausdorff point in $\mathfrak{M}_{\Lambda}$ if and only if the positive cone $\mathcal{C}_X$ coincides with the Kahler cone $\mathcal{K}_X$, i.e. $\mathcal{C}_X=\mathcal{K}_X$. Why did the author decide to look at the Kahler cone and not at some other cones?

In the last period, I am been introduced in a lot of cones (apart from positive, ample and Kahler, also the Mori amd bimeromorphic ones) but I cannot fairly see the geometric meaning of them. Can you suggest me some results (expecially on Hyperkahler manifolds) obtained by the study of one of these cones, specifying which is the geometric intuition behind the choise of that cone?

Thank you very much!

P.S.: any suggestions, remarks, etc...you want to give me will be very welcome!

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It is not so much that the study of cones is important. It is the study of the objects (ample, effective, nef, movable, divisors or cones of differentials like the Kahler cone, etc) that is important. But these objects, and many others, often naturally form a cone so it is convenient to use this fact when studying these objects.

For example, let us say we are interested in birational geometry of a projective variety $X$ (say smooth for simplicity). How do we construct birational projective models of $X$? By understanding rational maps to projective and picking out which ones are birational with the image. Rational maps to projective space correspond to picking a line bundle $L \in \operatorname{Pic}(X)$ and some global sections of $L$. The fact that $L$ has global sections means that it corresponds (under the isomorphism between $\operatorname{Pic}(X)$ and the divisor class group) to an effective divisor class. The effective divisors by definition form a cone, the $\textit{effective cone}$, in the vector space $\operatorname{Pic}(X) \otimes \mathbb{R} = V$. Thus starting with the problem of understanding rational maps to projective spaces, we are naturally led to a special cone in a vector space $V$.

If we want to study maps to projective space that are embeddings, we are naturally led to the $\textit{ample cone}$ which is a subset of the effective cone and consists of all divisors $D$ so that a sufficiently large multiple $nD$ gives an embedding into projective space. It is easy to see these divisor classes do form a cone so if we want to study embeddings into projective space, we want to study divisors that lie in this cone.

More generally, one can hope to decompose the whole effective cone into smaller cones or $\textit{chambers}$, so that within each chamber, every divisor gives essentially the same rational map on $X$, and so the same birational model of $X$ in the case when the rational map is birational onto the image. So in some sense, this decomposition of the effective cone into these smaller cones breaks it up into simpler pieces based on what information these divisors give from the point of view of birational geometry.

In fact when $X$ is a $\textit{Mori dream space}$ which means the effective cone has a chamber decomposition as above that is as nice as possible, then we can understand all of the birational geometry of $X$ from the chambers in this cone. Furthermore, it turns out that all of the birational models come from geometric invariant theory in this case.

So the point is that we are interested in some geometric properties (e.g. birational geometry), these properties are understood by studying some objects (e.g. various types of divisor classes) which are naturally assembled into cones and we exploit this fact because the study cones in a vector space is in general much more tractable than the algebraic geometry of $X$.

I hope that helps and that I didn't ramble too much!

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  • $\begingroup$ Dear Dori, thank you very much for your long and detailed answer. I completely agree with you about that. So, if I am right, the feeling for my question about the GTT for HK is that there is no longer a 'deus ex machina' geometric intuition, but just an observation: one studies Kahler cone, because we are dealing with Kahler manifolds, and then he discovers geometry properties apparently not related (how it often happens..). Is it? I was worried about the fact that there was something more deep I couldn't see.. $\endgroup$
    – User3773
    Commented Mar 9, 2015 at 9:37
  • $\begingroup$ Also, since you talked about the wall and chamber decomposition: is there a general approach to find wall divisors? I mean, sometimes it seems to me that the definition of such divisors is quite random. Of course it is not, but most of the time it is very difficult to understand the geometric intuition behind them. For sure, such definition depends from case to case and from what you want they are useful for..but I think there should be a general leading line to write down the right definition. Anyway, I have never heard about the Mori dream space, I'll take a look at that, it seems useful. $\endgroup$
    – User3773
    Commented Mar 9, 2015 at 9:43

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