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Say we have projective variety $X$. Its Kodaira dimension $\kappa(X)$ is defined by the “growth exponential” of $P_d := \dim H^0(X,K_X^{\otimes d})$ with respect to $d$, i.e.

  • $\kappa(X) := -\infty$ if $P_d = 0$ for all $d > 0$,
  • $\kappa(X) := \min\{k \in \mathbb{Z}\mid \{P_d/d^k\} \text{ is bounded}\}$

I don’t understand how we are supposed to interpret this.

  1. From the literal definition, $P_d$ measures the space of sections of $K_X^{\otimes d}$, which in turns tell you how big a projective space can $X$ effectively embed in, “via the line bundle $K_X^{\otimes d}$." But what does this mean? And what does “growth of $P_d$" suppose to mean?

  2. Maybe my problem is, I don’t even understand the special role of $K_X$. Because, one could easily give a definition $\kappa(X;L)$ for any (holomorphic) line bundle over $X$ in a similar manner and compute it. (Is this a thing? And if so, how to understand it? E.g. what if we look at $L$ as the holomorphic tangent (or cotangent) bundle over $X$)

Thanks for your clarifications! I am an absolute beginner in this, so would appreciate any guidance.

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    $\begingroup$ A few comments: you can indeed define such an invariant for any holomorphic line bundle; this is called the Iitaka dimension of $L$. The special role of $K_X$ is that it is the unique line bundle we can associate to every smooth variety. And while it's not clear from this particular definition of $\kappa(X)$, what it actually gives you is the dimension of the canonical model of $X$, i.e. its image under $|nK_X|$ for $n >> 0$. What this does is generalize/refine the negative-zero-positive curvature trichotomy that one sees when classifying curves. $\endgroup$ Sep 23, 2020 at 10:02
  • $\begingroup$ Thanks for your reply! Could you please elaborate more on your last sentence about the curvature trichotomy? I mean, one could observe such trichotomy in the case of an algebraic curve (namely, dim 1 case). But, how to explain it beyond an adhoc coincidence? $\endgroup$ Sep 23, 2020 at 10:06
  • $\begingroup$ What are you claiming to be a coincidence? $\endgroup$ Sep 23, 2020 at 11:10
  • $\begingroup$ I meant the curvature trichotomy. That is, we can compute directly (separately) that (in the case of curves) has genus >= 2,1,0 || curvature <0,=0,>0 || kodaira-dimension 1, 0, -inf (where || here means corresponding to, in respective cases). Why is this the case? How do $K_X$ respond to curvature of $X$? And, how does this relationship generalize to higher dimension? Or more importantly, why/how would one come up with this definition of $\kappa(X)$ to start with? Surely, with the idea of "more refined version of curvature/genus dichotomy" in mind? $\endgroup$ Sep 23, 2020 at 12:49
  • $\begingroup$ I should say that I'm not a differential geometer, so I can't say too much about the curvature business; I was really talking about the three possible Kodaira dim's. But there are some things we can say: Fano varieties, such as projective spaces, are those varieties w/ $\kappa = -\infty$ which (by Calabi-Yau thm) are nice enough to admit metrics of positive curvature. Similarly, Calabi-Yau manifolds admit Ricci-flat metrics. It gets fuzzier when $\kappa > 0$ but I think the general idea is that varieties of maximal $\kappa$ (general type) admit metrics of negative curvature.... $\endgroup$ Sep 23, 2020 at 15:53

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In Question 1 you ask what "growth of $P_d$" means. Well, $P_d$ is a function of the natural number $d$, and (just like any such function) one can ask if is bounded above by a polynomial function of $d$. In this case one can prove that for any variety $X$ of dimension $n$, the function $P_d$ is bounded above by a constant multiple of $d^n$. But that is not necessarily optimal: there may be a smaller value of $k$ such that $P_d$ is in fact bounded above by a multiple of $d^k$. The "growth" of $P_d$ refers to the optimal such value of $k$. That is exactly what your definition encapsulates.

You also ask about the meaning of the phrase "via the line bundle $K_X^d$." Here it is not so clear to me where your confusion lies, but let me try to say something. Any line bundle $L$ on a projective variety $X$ such that $H^0(X,L) \neq \{0\}$ gives a rational map $X \dashrightarrow \mathbf P^N$ where $N=\operatorname{dim } H^0(X,L)-1$. If you're not familiar with how this works, it is explained for example in Hartshorne II.7 if I remember correctly. In particular you can apply this to the sequence of bundles $K_X^d$, and $\kappa(X)$ can be thought of as measuring the rate at which the dimension of the target projective space grows with $d$.

However, there is a perhaps more useful way to understand $\kappa(X)$, as follows. For each $d$, the closure of the image of the rational map $X \dashrightarrow \mathbf P^N$ described above is an algebraic subvariety, call it $Y_d$, of $\mathbf P^N$. As $d$ grows larger, the target spaces $\mathbf P^N$ get bigger and bigger, but the dimension of $Y_d$ must achieve a maximum value (since it can never be bigger than the dimension of $X$). And in fact (although this is a nontrivial claim) this maximum value of the dimension of $Y_d$ is equal to $\kappa(X)$.

Finally, you are right that the same definition makes sense for any line bundle $L$ on $X$. The corresponding invariant is called the Iitaka dimension of $L$, and denoted $\kappa(X,L)$. The reason that $K_X$ is important is that it is a canonically defined line bundle associated to any (say smooth) variety. Moreover it has the good property (not shared by its dual $K_X^\ast$) that the numbers $P_d$ are birational invariants. So from the point of view of birational classification of varieties, $K_X$ is the only natural choice of line bundle to consider.

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  • $\begingroup$ OK, I see now that Tabes Bridges said many of the same things more concisely in comments while I was slowly typing my answer. $\endgroup$ Sep 23, 2020 at 10:24
  • $\begingroup$ Sorry, that was a typo. I have corrected it to min. And secondly thanks for your reply. I understand what you are saying, and those are basically from the definition, as I described in my question. I guess I should rephrase the question to, how should one understand $\kappa(X)$ intuitively/geometrically? More precisely, how to see that the definition of $\kappa(X)$ lead to the curvature trichotomy described by Tabes Bridge as above? $\endgroup$ Sep 23, 2020 at 12:42
  • $\begingroup$ Well, one geometric interpretation is given by the varieties $Y_d$ that I talked about in my answer. The interpretation in terms of curvature is another one, but (to my mind) a bit more advanced (and not so easy to state in precise form).If you want to learn more in that direction, you can take a look at the freely available book "Complex Analytic and Differential Geometry" by J.-P. Demailly. $\endgroup$ Sep 23, 2020 at 13:00
  • $\begingroup$ I see. Thanks for the pointers! $\endgroup$ Sep 24, 2020 at 11:35
  • $\begingroup$ Dear @LazzaroCampeotti, in your third paragraph, how to understand the dimension of $Y_d$? Why it can never be bigger than the dimension of $X$? When we take the closure of the image, can't the dimension of it bigger than $\text{dim}(X)$? $\endgroup$
    – Steve
    Nov 24, 2020 at 14:16

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