The minimal model program attempts to classify algebraic varieties up to birational equivalence. For compact Riemann surfaces, Riemann's uniformization theorem tells us that the geometry of the curve is determined primarily by its genus. In particular, if $M$ is a compact Riemann surface of genus $g$, then $g=0 \implies M \cong \mathbb{P}^1$, $g =1 \implies M \cong \mathbb{T}^2$, and $g \geq 2 \implies M \cong \mathbb{H}/\Gamma$. Here, $\cong$ denotes biholomorphism (or isomorphism).


In higher dimensions, we forget the biholomorphic equivalence classes, even for smooth compact complex surfaces. I am convinced of the utility of this and understand that it can be very hard to identify algebraic varieties or complex manifolds up to biholomorphism. My question is:



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> Is there a nice very illustrative example that shows that for complex surfaces, you are essentially forced to proceed with birational equivalence and not biholomorphic equivalence?
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Note: This is an expository question, more than a mathematical question.



Example illustrating necessity of considering birational equivalence and not biholomorphic equivalence in MMP
This is a very long comment.


First I should point out that by GAGA, biholomorphic = isomorphic for smooth projective varieties so in this case, classification up to biholomorphism is the same as classification up to isomorphism. Note the non-compact case can be quite wild in higher dimensions (see [this example](https://mathoverflow.net/a/657/12402) or the fact that the unit ball and the unit polydisc in $\mathbb{C}^2$ are not biholomirphic) but I will only talk about the projective case here.


From the point of view of the minimal model program, the higher dimensional version of the classification of Riemann surfaces via uniformization is the classification of algebraic varieties into general type ($K\_X$ is big), Calabi-Yau ($K\_X \equiv 0$), and Fano ($-K\_X$ is ample). These correspond to negative, zero, and positive Ricci curvature in the same way that for Riemann surfaces, $g = 0,1$ and $\geq 2$ correspond to positively curved, flat, and hyperbolic respectively. General type varieties moreover have a unique birational representative, the *canonical model*.


One complication in higher dimensions is that not every variety fits into one of these categories. For example a variety can be positively curved in one direction and negatively in another, e.g. $\mathbb{P}^1 \times C$ where $g(C) \geq 2$ or $\mathrm{Bl}\_p(C \times D)$ where $g(C), g(D) \geq 2$.


The main goal of the minimal model program is to show that every algebraic variety can be decomposed (in an algorithmic way and involving well understood birational transformations) into building blocks which are of general type, Calabi-Yau, or Fano. This (conjecturally) reduces the classification of varieties to two parts: 1) understand the steps in the MMP which build up every variety from the basic building blocks, and 2) classify the basic building blocks.


The classification of the building blocks often involves producing moduli spaces (e.g. moduli spaces of canonical models which generalize $\mathcal{M}\_g$ to higher dimensions, and the $K$-moduli in the Fano case).


Finally there is the basic fact that in higher dimensions, there exist infinitely many non-isomorphic (and thus non-biholomorphic) but birational smooth projective varieties, e.g. a smooth projective surface $X$ can be blown up any number of times.


The MMP for surfaces then says that every surface is obtained as an iterated blowup (which has Fano fibers) of one of the following: 1) a minimal surface of general type, 2) a fibration by genus 1 curves (a Calabi-Yau fibration), 3) a Calabi-Yau surface, 5) a ruled surface (Fano fibration). Thus to classify smooth projective surfaces up to isomorphism = biholomorphism, we need to understand blowups, classify minimal surfaces of general type, and classify Fano and Calabi-Yau fibrations over a lower dimensional variety.


TLDR biholomorphism is the same as isomorphism for projective varieties by GAGA. The MMP posits that every smooth projective variety conjecturally can be decomposed, after a sequence of specific birational transformations, into the basic building blocks of general type, Calabi-Yau and Fano varieties. This is the higher dimensional generalization of the trichotomy for compact Riemann surfaces given by uniformization.


