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.

*

*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?


*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: 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.
