### Birational geometry and the abundance conjecture

The main objects of interest of classical algebraic geometry are smooth algebraic varieties. Given their omnipresence in the world of mathematics, there is nothing more an algebraic geometer could dream of than to develop a complete classification. Obtaining such a classification up to birational equivalence, the task that captivated many generations of mathematicians despite looking nearly insurmountable, is the ultimate goal of the branch of algebraic geometry called birational geometry.

Throughout this article I suppose that the reader knows the basic concepts of algebraic geometry. For simplicity, all the varieties here are defined over $\mathbb{C}$ and assumed to be projective.

The first thing which I should explain is what I mean by “birational equivalence”.

Definition. Two algebraic varieties $X$ and $Y$ are birationally equivalent if there exist two open subsets, in the Zariski topology, $U \subseteq X$ and $V \subseteq Y$ which are isomorphic.

Informally, one could say that two varieties are birationally equivalent, if one can be obtained from another by making a “small surgery”, by removing and adding subvarieties of smaller dimension.

A natural question to ask is, why the classification should be made up to birational equivalence. The basic example of a birational equivalence is taking the blow-up $\mathrm{Bl}_p(X)$ of a point $p$ on a variety $X$, which topologically is homeomorphic to the connected sum $X \# \mathbb{P}^n$. Here, $n$ is the dimension of $X$. It is quite clear, that by performing such an operation, the geometry of our variety becomes more intricate. The good standing point for the sought-for classification is to simplify our variety, so that it cannot be constructed from other varieties by some basic operations like a blow-up. Such a birationally-equivalent most simplified version of our variety is called a minimal model. Given how easily we can construct different varieties by performing blow-ups, it seems impossible to present a classification up to isomorphism.

## Curves

How does such a classification look like for smooth curves, that is compact Riemann surfaces? Fortunately, two projective curves are birationally equivalent if and only if they are isomorphic, hence there is no need to care about constructing a minimal model. The classification goes in the following way. Let $C$ be a smooth projective curve and $g$ be its genus.

• If $g=0$ then $C$ is isomorphic to $\mathbb{P}^1$ (see [4, Section IV.1]).
• If $g=1$, then $C$ is an elliptic curve. Elliptic curves can be classified by a $j$-invariant (see [4, Section IV.4] or [1]).
• If $g \geq 2$, then $C$ is called a curve of general type.

Even the most basic case of algebraic curves reveals a crucial problem — the class of varieties of general type. Due to the impenetrable complexity of algebraic varieties, it is impossible to give a classification in which all the varieties would have an explicit description. Anyway, I will argue later, that it is not really caused by the lack of our intellectual capabilities, but rather more by the fact that there are just no universally defined properties that can be associated with those varieties.

## Surfaces and Kodaira dimension

Now, let me consider the case of surfaces (see [4, Section V.6]), the birational classification of which has been the greatest achievement of the famous Italian school of algebraic geometry. First, I will discuss minimal models. For this, I need to define exceptional curves.

Definition. Let $C$ be an algebraic curve on a surface $S$. The curve $C$ is exceptional if it is both isomorphic to $\mathbb{P}^1$ and a $(-1)$-curve that is

$\deg \mathcal{O}_S(C)|_{C} = -1$,

where $\mathcal{O}_S(C)$ is the line bundle on $S$ associated to $C$.

Exceptional curves are exactly what is added to a variety upon performing a blow-up. The natural map $\mathrm{Bl}_p(S) \to S$ is the contraction of an exceptional curve to the point $p$. Further, any exceptional curve comes from a blow-up of some smooth surface, i.e. it can be contracted (see [4, Theorem 5.7 (Castelnuovo)]).

Definition. A surface $S$ is minimal, if there are no exceptional curves on it.

In other words, $S$ is minimal, if it cannot be obtained from another smooth surface by blowing-up a point. After repeating a finite number of contractions of exceptional curves, any surface can be made minimal.

Before writing down the glamorous classification of surfaces, I need to introduce a fundamental invariant of algebraic varieties called the Kodaira dimension.

Definition. The Kodaira dimension of a variety $X$ is defined as

$\kappa(X) := \lim_{m \to \infty} \frac{\ln h^0(X, K_X^{\otimes m})}{\ln m}$,

where $h^0(X, \cdot)$ is the dimension of the space of global sections of a line bundle and $K_X$ is the canonical bundle, that is the top antisymmetric product of the tangent bundle of $X$.

Although the definition may seem vague at first glance, this is a natural way of measuring how the number of global sections of the line bundles $K_X^{\otimes m}$ change approximately. The Kodaira dimension $\kappa(X)$ is uniquely determined by the approximation

$h^0(X, K_X^{\otimes m}) = O(m^{\kappa(X)}).$

One can prove that the Kodaira dimension $\kappa(X)$ is either $-\infty$, or a natural number satisfying $0 \leq \kappa(X) \leq \dim(X)$. The variety $X$ is said to be of general type, if $\kappa(X) = \dim(X)$.

Having defined the Kodaira dimension, I may finally present the milestone classification. Let $S$ be a minimal smooth projective surface.

• If $\kappa(S) = -\infty$, then $S$ is isomorphic to $\mathbb{P}^2$ or a ruled surface (a locally trivial fibration over a curve with fibres isomorphic to $\mathbb{P}^1$).
• If $\kappa(S) = 0$, then $S$ is an abelian surface, a $K3$ surface or an Enriques surface.
• If $\kappa(S) = 1$, then $S$ is an elliptic fibration, that is a fibration over a curve with fibres being elliptic curves.
• If $\kappa(S) = 2$, then $S$ is of general type.

The classification is, at the same time, stupendous and immensely unsatisfactory. It describes explicitly the surfaces $S$ with $\kappa(S) < 2$, whereas nothing is said about surfaces of general type. Thus, it is compelling to ask, whether this classification is of any importance.

The answer is — with no doubt — yes, it is of great importance, both from a philosophical and a practical point of view. Notice, that if you are given an algebraic variety, the only line bundle that is naturally defined on it is the canonical bundle. A variety $S$ being of general type, means that the fibration associated to the canonical bundle, the Itaka fibration, goes to a variety of the same dimension, and hence this gives no real insight into the geometry of $S$. To sum up, in the case of varieties of general type, some more explicit, refined structures need to be pointed out in order to bring out more information about the variety — the universal constructions just do not convey any essential information. In this sense, the classification somehow exhaust everything that is possible to say.

Furthermore, miraculously, the birational classification captures various, seemingly unrelated, geometric and arithmetic properties of varieties. For instance, the notion of varieties of general type occurs in the Faltings’ theorem and the Bombieri-Lang conjecture.

Bombieri-Lang conjecture. Let $X$ be a smooth variety of general type over a number field $k$. Then $k$-points of $X$ are not Zariski dense in $X$.

Notice, how the general theory of rational points fits into the framework of the classification of surfaces. The behaviour of rational points depends strongly on the type of a surface, e.g. $k$-points on an abelian surface sometimes are dense, sometimes not, in contrast to $k$-points on a $K3$-surface which conjecturally are always dense.

As for the practical side, the classification is indispensable in proving various theorems for surfaces. One can proceed in the proof case by case, considering all the types of varieties occuring in the classification. The effectivness of this method depends strongly on the problem, the bottleneck being the varieties of general type. Anyway, in innumerable situations, supposing that the considered variety is of general type — which is a strong assumption on the canonical bundle — makes the problem abundantly easier.

## General theory

It took mathematicians many years to develop general appropriate definitions for the aforementioned notions. Although, at first glance they may seem as of a ”deus-ex-machina” type, they turn out to be quite natural.

A variety $X$ is said to be minimal, if its canonical bundle $K_X$ is nef.

Definition. A line bundle $\mathcal{L}$ on a variety $X$ is nef, if for any algebraic curve $C \subseteq X$ it holds that $\mathrm{deg}(\mathcal{L}|_{C}) \geq 0$.

A curve $C \subseteq X$ is exceptional, if it prevents $K_X$ from being nef, that is $\mathrm{deg} K_X|_{C} < 0$.

There are a few problems that pop out. Contracting exceptional curves does not take the smoothness of a surface away. Unfortunately, this is not the case for higher dimensional varieties. It forces us to work within the category of varieties with some mild singularities, called terminal singularities. But even so, only exceptional curves which come in families of codimension one can be contracted. Contracting lower dimensional subvarieties does not preserve the $\mathbb{Q}$-factoriality of a variety, which is definitely undesirable.

Further, it may happen that there is no codimension one exceptional family that can be contracted, but the variety is still nonminimal. In such case, one needs to consider another operation called a flip. It is a kind of a ”surgery” of codimension two on a variety. Intuitively, it removes an exceptional family and glues it back to the same place but in a different way.

The minimal model program claims that any variety can be simplified to a minimal model by performing a sequence of operations of the above type. The case of dimension three was proved by Kollar et al., but the general theory remained open. And it was not until 2006, twenty years after the threefolds case had been solved, that any significant progress was achieved. But that year, news broke out and they electricized the community of algebraic geometers. Four mathematicians, Birkar, Cascini, Hacon and McKernan went down in history cracking the problem for varieties of general type (see [2]). This breakthrough result, often called the greatest achievement of algebraic geometry of the last 30 years, was a major step in the completion of the minimal model program, entailing various magnificent corollaries. Nonetheless, despite its significance, the ultimate goals of birational geometry are still far from being reached.

The core of the theory, implying many other theorems, is the abundance conjecture. It states that under certain assumptions on singularities of a variety $X$, the canonical bundle $K_X$ is nef if and only if it is semiample. Without going over the definition of semiampleness, let me state the following direct corollary of the abundance conjecture.

Corollary. Suppose that $X$ is minimal, i.e. $K_X$ is nef. Further, assume that $X$ if not of general type. Then, $X$ is a fibration over a lower-dimensional variety

• with Fano fibres, if $\kappa(X)=-\infty$, or
• with Calabi-Yau type fibres, if $0 \leq \kappa(X) < \dim(X)$.

A variety $Y$ is said to be Fano, if $-K_Y$ is ample, and of Calabi-Yau type, if $mK_Y$ is trivial for some $m > 0$.

There is nothing more fulfilling for a mathematician than to classify objects they devote themselves to examine so thoroughly. The proof of this corollary and of the existence of minimal models would bring the study of the geometry of algebraic varieties to a completely new dimension, enabling to show various theorems using the most basic reasoning in mathematics, induction. First, one would consider the case of varieties of general type. Subsequently, the other cases could be deduced from lower dimensions by applying properties of Fano or Calabi-Yau type varieties. Although, a significant work still needs to be undertaken in order to gain a holistic understanding of those special varieties, it seems that such a theory lies within the range of our possibilities, especially in the light of the recent progress in mathematics like the development of mirror symmetry.

Literature. The most fundamental reference on birational geometry is [3]. Readers interested in getting an easy introduction into the main concepts of the minimal model program and birational geometry are encouraged to consult the beautiful book [5].

## References.

1. J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics.
2. C. Birkar, P. Cascini, C. D. Hacon and J. McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc.
3. J. Kollar and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics.
4. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics.
5. O. Debarre, Higher-dimensional algebraic geometry, Universitext.