PDFLINK |

# A Fruitful Interaction Between Algebra, Geometry, and Topology: Varieties Through the Lens of Group Actions

Communicated by *Notices* Associate Editor Han-Bom Moon

Beautiful theories have emerged from mixing different areas of mathematics, when building *bridges* between areas provides the right paths to achieve the comprehension of a certain subject. Mathematicians know that if an object has several underlying structures, then the answer to some question considering one aspect of the object may come from another aspect of itself. A remarkable example of this is Descartes’s use of coordinates; this milestone was the key to translate questions from geometry to algebra. But there are many other examples: Galois theory, algebraic geometry, mathematical physics, and the list goes on and on.

In these notes we will discuss how the knowledge about group algebras and representation theory has proved fruitful in understanding fundamental questions about complex abelian varieties, compact Riemann surfaces, and their moduli spaces.

Let us begin by roughly introducing these objects. Abelian varieties and compact Riemann surfaces are *complex manifolds*; that is, (connected, Hausdorff) topological spaces that locally look like open subsets of for some a technical condition better explained in Figure 1—and that also share the property of being —plus*varieties*.

The word *variety* here is used to refer to a geometric object that can be described as the set of common zeros of polynomial equations in some appropriate space. Since we are considering compact Riemann surfaces and complex abelian varieties, the natural ambient space for them is the ,*(complex) projective -space*, where (complex) lines in through the origin are collapsed to points.

Formally, the complex projective is defined as the quotient of the complex space -space by the equivalence relation where and are related if and only if there is a scalar such that The usual notation is . .

For we talk about ,*the complex projective line* It is a compact topological space of genus zero. In fact, as (one-dimensional) complex manifolds, . is isomorphic to the Riemann sphere the complex plane compactified by adding one point. ,

For we have the projective space , Its points are denoted as . and they correspond to the lines , The set of zeros of a homogeneous polynomial . in three variables is well defined; a plane projective algebraic curve is the set of zeros in of a homogeneous polynomial. There is a rich theory of algebraic curves, so here begins a new approach to studying compact Riemann surfaces: Every compact Riemann surface corresponds to a plane projective algebraic curve, hence a variety, with a finite number of controlled singularities. In fact, the result is stronger: every compact Riemann surface can be represented as a smooth algebraic curve defined in some projective space.

On the other hand, an abelian variety of dimension is a complex torus: the quotient of a complex vector space of dimension by a discrete subgroup of maximal rank, hence a compact manifold (of dimension and an abelian group, that can be embedded in a projective space. ),

In what follows, we explain these concepts more rigorously, define their *parameters* or *moduli spaces*, explain how they are related, and illustrate current research and open questions in this field.

## 1. Compact Riemann Surfaces

In this section we elaborate a bit more the given definition for complex manifolds in the one-dimensional case; hence we consider Figure -dimensional1 with .

Let be a connected, Hausdorff, topological space locally homeomorphic to A complex atlas on . is a collection where , each , is an open set in and each is a homeomorphism, and either or the change of coordinates are holomorphic.

For every atlas there is a unique maximal atlas containing it, called a *complex structure* on .

Formally, a Riemann surface is a pair or, equivalently, a pair , Usually the complex structure (or the complex atlas) is omitted from the notation (if it is clear from the context), and one just says that a Riemann surface is a complex manifold of dimension one. .

The Uniformization Theorem FK92, a deep result in this area, states that the only simply connected Riemann surfaces (up to isomorphism) are the Riemann sphere the complex plane , and the upper half-plane in (which is isomorphic, as Riemann surfaces, to the open unit disc in ).

The Cauchy-Riemann relations for holomorphic functions imply that the change of coordinates are (when considered as real functions in and their Jacobian determinant is positive. Therefore, every Riemann surface is an orientable real surface. ),

Throughout this work, we will consider compact Riemann surfaces. Since Riemann surfaces are orientable differentiable real surfaces, topologically a compact Riemann surface of genus is a connected sum of one-dimensional complex tori, see Figure 2. Genus corresponds to the Riemann sphere, and its theory is slightly different from what we want to expose here, so we leave it out, and we will write about Riemann surfaces of genus Genus . is, again, a bit different from the rest; for instance, since genus one Riemann surfaces are also (abelian) groups, but this is not the case for higher genera.

### Genus one Riemann surfaces

We will see that each Riemann surface of genus is an elliptic curve, a one-dimensional complex torus and an abelian variety of dimension one. Since we are taking a *complex variables* approach to our exposition, we start with the first construction of genus one Riemann surfaces.

Observe that the sum of complex numbers descends to an operation on that makes it an abelian group; also the notion of analytic function between two tori is well defined (by considering the map in coordinates, see Figure 4). Thus the following definitions make sense: a homomorphism between two tori is an analytic map which is also a group homomorphism. Two tori are isomorphic if there is a bijective homomorphism between them.

Since is simply connected and it follows that a homomorphism , from to gives rise to two maps:

1.1called the analytic and rational representation of respectively; they are , and -linear maps respectively. If -linear is an isomorphism, then and are invertible maps.

Hence (modulo isomorphisms) it is enough to consider lattices of the form with a complex number with positive imaginary part, and two such tori and are isomorphic if and only if

Since a torus is compact, any holomorphic function from to is constant; nevertheless, it has non-constant meromorphic functions. In fact, its field of meromorphic functions is generated by two functions, the so-called *Weierstrass p-function* which depends on , and its derivative , Both functions are related by a cubic equation, giving a way of describing . as a plane algebraic curve, namely *an elliptic curve*. Therefore, every Riemann surface of genus one is an elliptic curve, hence an abelian variety: a complex torus that can be embedded in a projective space.

### Higher genus

As mentioned above, topologically a compact Riemann surface of genus is a connected sum of one-dimensional complex tori, see Figure 2. The classical theory of uniformization of (arbitrary) Riemann surfaces, see for instance FK92, allows us to understand compact Riemann surfaces of genus as (one-dimensional) complex manifolds.

In fact, a compact Riemann surface of genus is the quotient of the complex upper-half plane by a discrete torsion-free cocompact subgroup of Here cocompact means that the quotient . is compact. The subgroup is called a *surface Fuchsian group*.

Since we will study group actions on (compact) Riemann surfaces and their quotients, we need to enlarge the type of Fuchsian groups under consideration to groups including torsion elements.

In general, the action of a cocompact Fuchsian group on (not necessarily torsion-free) is (partially) captured by its *signature* where , denotes the genus of , …, , are the ramification indices in the associated projection and , is the number of *branch points* in Observe that the . are the orders of the (nontrivial) stabilizers of elements in fixing points in It is known that a Fuchsian group . of signature has a *canonical presentation* of the form

where stands for their commutator. A tuple of generators of satisfying the presentation 1.2 is called *a tuple of canonical generators* (with respect to ).

We refer to the beautiful work Oh22 for a clear discussion, with illuminating figures, about these kind of surfaces.

### Isomorphisms between Riemann surfaces

A continuous function between two Riemann surfaces is said to be *holomorphic* if for every chart in and in with the map is holomorphic (Fig. 4).

A bijective holomorphic map is an isomorphism, and if in addition we say that , is an automorphism of the group of automorphisms of ; is denoted by We say that a (finite abstract) group . acts on a Riemann surface if there is a monomorphism in this case we write ; The action of a group . as automorphisms of a compact Riemann surface is (partially) grasped by the *signature* of the cover as follows. ,

If is a surface Fuchsian group such that the condition on , acting on with signature is equivalent to the existence of a Fuchsian group together with a group epimorphism such that and In addition, . is called a surface-kernel epimorphism and the image under of a tuple of canonical generators (with respect to of ) is called a generating vector for .

Note that the sequence is exact and that .

A curve of genus is called *hyperelliptic* if the cyclic group of order acts on it with signature (with branch points).

Part of the geometry of the action of a group on is captured in the signature a bit more in the corresponding generating vector. There is a big community of geometers studying group actions on compact Riemann surfaces, classifying them up to equivalence, generating computer algorithms to work with them, etc. See ,LR22 and the references given there, for more on group actions, covers, examples, and applications.

### From compact Riemann surfaces to algebraic curves

To see that every compact Riemann surface of genus is also a variety, take a divisor on (a formal sum of points in and define ), as the space of meromorphic functions on whose order at poles are not worse than the corresponding coefficient in This vector space is finitely generated, say . so (when nontrivial) it provides a map , given by .

The Riemann-Roch theorem, a deep result that relates the degree of (roughly its number of points), the dimension of the canonical divisor and the genus of , implies that when the degree of , is then and therefore , is an immersion of into Then one still has to show that the image is a variety; this may be done is various ways. One is to appeal to another deep result, Chow’s theorem. .

Therefore every compact Riemann surface of genus is also a variety, an algebraic curve, but it is no longer a group. Nevertheless, Abel and Jacobi instead constructed a complex torus associated to -dimensional its ,*Jacobian variety* (see page 7). Next, we introduce the basics of complex tori and abelian varieties in higher dimension.

## 2. Higher Dimension

Instead of staying in (complex) dimension one and going from one-dimensional complex tori (that is, Riemann surfaces of genus one) to Riemann surfaces of higher genera, another way to move forward conceptually is to increase the dimension and stay in the world of complex tori, see Rod14 and the references given there.

Analogously to what was done for one-dimensional complex tori, where we defined elliptic curves as the quotient of by a lattice, one can build larger-dimensional tori in a very concrete way. Choose bases and of and respectively, and write each in terms of the that is, ; Then the matrix . encodes the relation between the real and the complex coordinate functions of its lattice and of its vector space respectively. It is called *a period matrix for* and it captures geometric information about , .

Since is a lattice, the rank of is hence we can normalize to see that there are bases for , and for which a period matrix for is of the form with , the identity matrix and a complex matrix with nonsingular imaginary part. This generalizes the simplification made for elliptic curves on page where , with .

In order for a complex torus to be an abelian variety, it needs to have enough meromorphic functions to be embedded into a projective space. There are several characterizations of when this happens; the following is the classical one, in terms of polarizations.

In terms of a period matrix for the torus the ,*Riemann relations* give necessary and sufficient conditions for to be an abelian variety BL04.

It is important to highlight that not every torus is an abelian variety.

For a polarized abelian variety of dimension with , Frobenius algorithm ,Lan82 gives a way of finding a basis for with respect to which is given by

where with natural numbers such that Such a basis for . is called *symplectic*, and the tuple is the *type of the polarization* If . , is called a *principally polarized abelian variety*, *ppav* for short.

If is an abelian variety, then it follows from Theorem 2.3 that there is a basis for and a symplectic basis for with respect to which a period matrix for has the form where , is the diagonal matrix reflecting the type of the polarization and , is a matrix in the Siegel upper space The matrix . is called *a Riemann matrix* for .

To define *homomorphisms* between abelian varieties, first consider their underlying complex tori structure. Hence homomorphisms between them correspond to the natural generalization to what was defined for dimension one. A homomorphism between two complex tori of dimensions and respectively, is an analytic map that is also a group homomorphism. If is any torus, a homomorphism is called an endomorphism of with , denoting the ring of endomorphisms of .

As in the one-dimensional case, see 1.1, a homomorphism gives rise to two representations and In terms of the period matrices . and for and respectively, is given by a matrix and by a matrix satisfying the *Hurwitz* equation

Interesting homomorphisms, which play a role in the problems we want to describe, are the *isogenies*. A homomorphism is an *isogeny* if it is surjective with finite kernel, or, equivalently, if it is surjective and The .*exponent* and the *degree* of the isogeny are the exponent and the order of its kernel, respectively. The degree corresponds to For every isogeny . of exponent there is a unique (up to isomorphism) isogeny , such that and are multiplication by on and respectively. Hence isogenies are the units of the endomorphism algebra Therefore, they establish an equivalence relation among complex tori. So, most classification theorems about complex tori and abelian varieties are given up to isogeny. .

In the following section, we go back to the last lines of Section 1; how to associate to every Riemann surface of genus an abelian variety.

### Jacobian varieties

Let be a compact Riemann surface of genus consider the ; vector space -dimensional the dual of the space of holomorphic differential : and the lattice -forms, where the injection of , in is given by

Then the complex torus is an abelian variety, since it admits a canonical principal polarization given by extending the geometric intersection number in the lattice This ppav . is called *the Jacobian variety associated to *. According to Torelli’s theorem, two Jacobian varieties are isomorphic if and only if the corresponding Riemann surfaces are. Therefore, there is a bijective correspondence between Jacobian varieties and Riemann surfaces.

### Some interesting properties for abelian varieties

Among the several properties of interest about abelian varieties, we focus on one about its geometry (*completely decomposable*) and another about its endomorphism algebra (*with complex multiplication*).

Let be a torus and be a subspace such that is a lattice in then ; defines a subtorus of Images and (the connected component containing . of kernels of homomorphisms between tori are examples of subtori. A complex torus ) is *simple* if its only subtori are itself and Since any subtorus . of an abelian variety is an abelian variety by restricting the polarization on to , is an *abelian subvariety* of thus an abelian variety is called ;*simple* if and only if its only abelian subvarieties are itself and the trivial one.

Clearly, one-dimensional abelian varieties are simple. Determining whether a given abelian variety is simple or not is not easy. There are several approaches to studying this property. For instance, see ALR17 where a criterion in terms of a period matrix is given, as well as examples of simple abelian varieties and references to go deeper in this subject.

The algebra of a simple torus is a skew-field of finite dimension over and any finite dimensional algebra is isomorphic to , for some torus The case is different for abelian varieties, as their endomorphism algebras are semisimple. This result is a consequence of Poincaré’s Reducibility Theorem, which says that every subvariety . of an abelian variety has a complementary abelian subvariety : is finite and the addition map is an isogeny. This leads to the fundamental decomposition theorem of this subject.

An abelian variety is *completely decomposable* if it is isogenous to a product of elliptic curves. A simple abelian variety has *complex multiplication* if is a number field of degree over In this case . is a CM-field; that is, a totally imaginary degree two extension of a totally real field. If is not simple, then it is said to be of CM-type if all of its simple factors (Theorem 2.5) have complex multiplication. We mention some open questions regarding abelian varieties with these properties in Section 5.

In what follows, we denote a polarized abelian variety (pav) by a single letter omitting the notation of the polarization. ,

## 3. Moduli Spaces

The natural next step after defining certain *objects* of study—for us, compact Riemann surfaces and principally polarized abelian varieties—and their notion of *equivalence*, here isomorphisms, is the *classification problem*: the objects are classified *up to equivalence*, and the set of equivalence classes is their *moduli space*.

It turns out that these moduli spaces, though defined as sets, can be endowed of a geometric (or algebraic) structure. The loci containing objects with nontrivial automorphisms complicate every approach.

In this work, we are interested in the moduli space of compact Riemann surfaces of genus , and , the moduli space of principally polarized abelian varieties of dimension , both can be given an orbifold structure. Also of interest are some loci in them, such as ;

- -
their

*singular locus*, which corresponds to the (isomorphism classes of) objects with nontrivial automorphisms (for in case of ).- -
the

*Jacobian (or Torelli) locus*which corresponds to the image by the injective ,*Torelli*map defined by assigning to each Riemann surface its Jacobian variety .- -
families in with complex multiplication (or of CM-type), or completely decomposable.

### The moduli space

Let and be two pav. A bijective homomorphism between the underlying tori is an isomorphism between pav if it preserves the polarization. In such a case, when bases are chosen so that and are the period matrices of and respectively, then an isomorphism preserves the polarization if and only if where ,

with and denoting the transpose of For the principally polarized case, where . the usual notation is , the symplectic group. ,

It follows from the Hurwitz’s equation 2.1 that acts on and the quotient is the moduli space of pav of type Therefore, for .*ppavs* one has