# How should I understand Kodaira dimension?

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.

• 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. Sep 23, 2020 at 10:02
• 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? Sep 23, 2020 at 10:06
• What are you claiming to be a coincidence? Sep 23, 2020 at 11:10
• 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? Sep 23, 2020 at 12:49
• 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.... Sep 23, 2020 at 15:53

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.

• OK, I see now that Tabes Bridges said many of the same things more concisely in comments while I was slowly typing my answer. Sep 23, 2020 at 10:24
• 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? Sep 23, 2020 at 12:42
• 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. Sep 23, 2020 at 13:00
• I see. Thanks for the pointers! Sep 24, 2020 at 11:35
• 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 $\text{dim}(X)$? Nov 24, 2020 at 14:16