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

Rubí E. Rodríguez
Anita M. Rojas

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 —plus a technical condition better explained in Figure 1—and that also share the property of being varieties.

Figure 1.

has an open covering and homeomorphisms such that for , one has either , or and are required to satisfy the condition displayed in this picture.

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

Remark 0.1.

The projective -space can be defined for any field , using a similar equivalence relation. When is a finite field, an interesting exercise is to compute the number of points in in terms of those in .

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 -dimensional manifolds in the one-dimensional case; hence we consider Figure 1 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.

Figure 2.

Topological construction of a compact Riemann surface of genus .

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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.

Definition 1.1.

Let and be two -linearly independent complex numbers, and consider the lattice generated by . The quotient is the complex torus of lattice (see Figure 3).

Figure 3.

Complex torus of dimension one.

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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:

called the analytic and rational representation of , respectively; they are -linear and -linear maps respectively. If 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).

Figure 4.

A morphism in local charts.

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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.

Definition 1.2.

Let be a compact Riemann surface and let .

acts on with signature if the quotient Riemann surface has genus and the quotient map ramifies over points with ramification indices , …, ; that is, the stabilizer in of each ramification point in the fiber of a branch point is of order .

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 -dimensional complex torus associated to , 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.

Definition 2.1.

A complex torus of dimension is the quotient of a -dimensional complex vector space by a discrete subgroup of maximal rank ; that is, a lattice in . A complex torus which is also a projective variety is called an abelian variety.

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.

Definition 2.2.

A polarization on a torus is a nondegenerate real skew-symmetric form on such that for all , and , where denotes a complex number with .

An abelian variety is a complex torus that admits a polarization, and a polarized abelian variety of dimension is a pair consisting of a complex torus and a polarization on .

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

Theorem 2.3 (Riemann Relations).

Let be a complex torus and a period matrix for it. Then is an abelian variety if and only if there exists a nondegenerate skew-symmetric integral matrix such that and is positive definite, where represents the transposed matrix.

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 .

Example 1.

Every compact Riemann surface of genus one satisfies the Riemann relations. Let us take with , . The real skew-symmetric form satisfies the requirements in Def. 2.2. Notice that we recover, in a different way as in the previous section, the fact that every Riemann surface of genus one is a variety.

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 -dimensional vector space : the dual of the space of holomorphic differential -forms, and the lattice , 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.

Remark 2.4.

By checking the Riemann relations (Thm. 2.3), one can verify whether a period matrix of a complex torus T corresponds to an abelian variety. Unfortunately, there is no practical method to identify when a matrix in corresponds to a Riemann matrix of a Jacobian variety. This is known as the Schottky problem.

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.

Theorem 2.5 (Poincaré’s complete reducibility).

Let be an abelian variety, then there are simple abelian subvarieties , …, of not isogenous to each other, and positive integers , …, , such that is isogenous to . This decomposition is unique up to permutation and isogenies of the factors.

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

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

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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 .

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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