Algebraic differential equations from covering maps
Let be a complex algebraic variety, an action of an algebraic group on , a complex submanifold, a discrete, Zariski dense subgroup of which preserves , and an analytic covering map of the complex algebraic variety expressing as . We note that the theory of elimination of imaginaries in differentially closed fields produces a generalized Schwarzian derivative (where is some algebraic variety) expressing the quotient of by the action of the constant points of . Under the additional hypothesis that the restriction of to some set containing a fundamental domain is definable in an o-minimal expansion of the real field, we show as a consequence of the Peterzil-Starchenko o-minimal GAGA theorem that the prima facie differentially analytic relation is a well-defined, differential constructible function. The function nearly inverts in the sense that for any differential field of meromorphic functions, if then if and only if after suitable restriction there is some with .
As is well-known, the complex exponential function admits a local analytic inverse, the logarithm function, but the logarithm cannot be made into a globally defined analytic function. The ambiguity in the choice of a branch of the logarithm comes from addition by an element of the discrete group . Hence, if we regard the logarithm as acting on meromorphic function via , then while the operator is not well-defined, the logarithmic derivative, is. Of course, more is true in that the logarithmic derivative is given by the simple differential algebraic formula .
That the logarithmic derivative is a differential rational function admits various proofs ranging from a direct computation, to Kolchin’s general theory of logarithmic differentiation on algebraic groups [Kolchin-DAAG], to the techniques we employ in this paper. In Section LABEL:examplessect we discuss the algebraic construction of the logarithmic derivative in detail.
The purpose of this paper is to show that under very general hypotheses, differential analytic operators constructed by inverting analytic covering maps and then applying differential operators to kill the action of the constant points of some algebraic group are actually differential algebraic. Let us describe more precisely what we have in mind. We are given an algebraic group over , a complex algebraic variety , a regular action of on , a complex submanifold of , a Zariski dense subgroup for which preserves and an analytic covering map expressing the complex algebraic variety as the quotient . Because is a covering map, the inverse is a many-valued analytic function, well-defined up to the action of . Using the theory of elimination of imaginaries in differential fields, we produce a differential constructible function, which we call a generalized logarithmic derivative associated to , (where is an algebraic variety) so that for any differential field having field of constants and points one has if and only if there is some some with . Hence, the differential analytic operator gives a well defined function for any field of meromorphic functions.
Under a mild hypothesis on , namely that there is an o-minimal expansion of in which there is a definable subset for which the restriction of is definable and surjective onto , we then deduce from a remarkable theorem of Peterzil-Starchenko that the a priori differential analytically constructible function is in fact differential algebraically constructible. This definability hypothesis holds in many cases of interest, such as for the covering maps associated to moduli spaces of abelian varieties and of the universal families of abelian varieties over these moduli spaces.
Our work on this problem was motivated by our attempt to understand Buium’s construction of differential rational functions on moduli spaces of abelian varieties whose fibres are finite dimensional differential varieties containing the isogeny classes encoded by such moduli points [Buium]. His construction is algebraic in the style of Kolchin’s construction of logarithmic derivations, though much more sophisticated. Buium’s maps are differential rational, meaning that they have a non-trivial indeterminacy locus; ours are defined everywhere, though they are intrinsically differential constructible, meaning that they are only piecewise given by differential regular functions. We seek an algebraic interpretation, for example in terms of a variant of the notion of -Hodge structure, for our maps on Buium’s indeterminacy locus. More generally, knowing that the map is differential algebraic, one expects a direct algebraic construction. We speculate about this in Section LABEL:cq.
This paper is organized as follows. We begin in Section 2 by recalling some of the basics of differential algebra, complex analysis and especially the Peterzil-Starchenko theory of o-minimal complex analysis. In Section 3 we state precisely and prove our main theorem. In Section LABEL:theorygld we develop some of the basic theory of generalized logarithmic derivatives. In Section LABEL:examplessect we discuss specific examples of covering maps to which our main theorem applies. In particular, we note the existence of differential constructible functions whose fibres are the Kolchin closures of isogeny classes in moduli spaces of abelian varieties. We also discuss the problem of extending our main theorem to the context of Picard-Fuchs equations associated to families of varieties and to the analytic construction of Manin homomorphisms for nonconstant abelian varieties coming from families of covering maps. We close in Section LABEL:cq with some natural questions.
As our main theorem involves a comparison of three different kinds of structures, namely, complex manifolds, differential algebraic functions, and o-minimally definable sets, it should come as no surprise that sometimes the same object may be considered differently in each of these domains. In this section we establish our notation and explain how these theories interact with each other.
2.1. Jet spaces
We shall use a construction of higher order tangent spaces which goes variously under the names of spaces of jets, arc spaces, and prolongation spaces. We begin by recalling the differential geometric jet spaces modified slightly for the complex analytic category. The reader can find details of the space of jets construction in Chapter 12 of [Bourbaki], though the discussion there covers only jets of maps from one manifold to another while we shall consider jets of germs. The extension to our case is routine.
We begin with our notation for balls and polydiscs.
We denote the unit disc in the complex plane by
For a natural number , the polydisc
is the Cartesian power of . More generally, for we denote the disc of radius centered at the origin by and its Cartesian power by .
Let us recall the construction of the space of jets.
If is a complex manifold and and are two analytic maps into from polydiscs of some radii and and is a natural number, then we say that and have the same jets up to order if there is a coordinate neighborhood with so that for each and each multi-index with one has . We write for the equivalence class of with respect to the equivalence relation of having the same jets up to order . Note that depends only on the germ of at the origin.
The set of equivalence classes of maps from -dimensional polydiscs into may be given the structure of a complex manifold, , and comes equipped with an analytic map making into an affine bundle over . The construction is functorial in that if is a map of complex manifolds, then there is an induced morphism given by .
The space of jets construction has an algebraic geometric counterpart with the notion of arc spaces. For an introduction to arc spaces see [DenefLoeser] in which what we call arc spaces are called jet spaces. For details of these constructions at the level of generality we use in this paper, see [MoosaScanlon].
If is a scheme over , then for each and the functor from -algebras to sets given by is represented by a scheme . In the literature, this construction is usually limited to the case of for which is the arc bundle of .
We shall use the more general to discuss partial differential equations. To ease readability, when no confusion would arise we shall suppress the subscript .
In general, even if is an algebraic variety, it may happen that is non-reduced. However, when is a smooth variety, one has and we will need the arc bundle construction only in the case of smooth varieties.
2.2. Differential algebra
Let us recall some of the basics of differential algebra for which the book [Kolchin-DAAG] is the standard reference. For an introduction to the model theory of differential fields, see [MMP].
A differential ring is a (commutative) ring given together with a finite sequence of commuting derivations. If is a field, then we call a differential field. Generally, we write for the tuple . When , we call an ordinary differential field and otherwise is a partial differential field. In what follows we shall work only with differential fields of characteristic zero and the phrase differential field shall mean differential field of characteristic zero. A map of differential rings is given by a map of rings which respects the derivations in the sense that for all . By a --algebra we mean a differntial ring given together with a -algebra structure for which the map is a map of differential rings.
A standard example of a differential field is given by taking a connected open domain and setting where the field of meromorphic functions on where are the standard coordinates on . The Seidenberg embedding theorem asserts that every finitely generated differential field of characteristic zero may be realized as a subdifferential field of for some connected domain . In fact, a little more is true.
Theorem 2.6 (Seidenberg [Seidenberg1, Seidenberg2]).
If is a finitely generated differential subfield of the differential field of meromorphic functions on some connected domain and is a simple differential extension of (meaning that is a differential field extending which is generated as a differential field by and the single element ), then there is a connected domain and an embedding of differential fields compatible with the embedding .
To speak of differential equations on an algebraic variety, we need to generalize the arc space construction.
Consider a differential field. For each there are two natural -algebra structures on . First, there is the standard structure given by . Secondly, using the derivations, we have an exponential map given by
Note that the standard map is itself an exponential map with respect to the trivial derivations.
If is an algebraic variety over , then the functor from -algebras to defined by
is representable by a scheme called the prolongation space of . Note that when the derivations are trivial, then and .
From the description of as a functor, one sees that there are natural projection maps for corresponding to the reduction maps and that canonically. On the other hand, corresponding to the exponential map one has a differentially defined map . That is, for any --algebra we have a map of sets corresponding to the map
coming from .
Using the prolongation spaces, one can make sense of the notions of differential regular (respectively, rational or constructible) functions on the algebraic variety . That is, a differential regular (respectively, rational or constructible) function (where is another algebraic variety over ) is a map from to , considered as functors from the category of --algebras to which is given by a regular (respectively, rational or constructible) function for some in the sense that for any --algebra and point one has .
A differential subvariety of is given by a subvariety for some . For any --algebra , the set of -points on is
Sets of the form are called Kolchin closed (or, sometimes, -closed) and as the name suggests are the closed sets of a noetherian topology on . If is an irreducible differential subvariety of the algebraic variety , then the set of differential rational functions on forms a differential field denoted by . If is finite, then we say that is finite dimensional and define .
An important class of Kolchin closed sets comes from the constants.
If is a differential ring, then
is the ring of constants. We sometimes write for .
If is a differential field and is an algebraic variety over the constants , then as we noted above , canonically, and there is a natural regular section of the projection map corresponding to the standard map . We define to be the differential subvariety of given by . At the level of points, if is --algebra, then . Note that if is an algebraic variety over the constants, then its dimension as an algebraic variety is equal to the dimension of as a differential variety.
Just as algebraically closed fields play the role of universal domains for ordinary algebraic varieties, differentially closed fields serve as universal domains for differential algebraic geometry. Here a differentially closed field is an existentially closed differential field in the sense that if is an algebraic variety over and is a differential subvariety for which there is some differential field extension with , then . The theory of differentially closed fields (of characteristic zero in commuting derivations) is axiomatized by a first-order theory in the language of rings augmented by unary function symbols to be interpreted as the distinguished derivations. For us, the crucial facts about are:
eliminates quantifiers. Geometrically, this means that if is a differentially closed field, is a differential subvariety of an algebraic variety over and is a differential rational function, then is a differentially constructible subset of . That is, it is a finite Boolean combination of Kolchin closed sets.
It follows from Theorem 2.6 that if is a finitely generated differential field of meromorphic functions on some domain , then there is a differentially closed field containing which may be realized as a subfield of a differential field of germs of meromorphic functions at some point in .
Finally, eliminates imaginaries.
The precise content of elimination of imaginaries is given by the following theorem of Poizat in the case of and of McGrail in general.
Theorem 2.8 (Poizat [Poizat], McGrail [McGrail]).
The theory of differentially closed fields of characteristic zero with commuting derivations eliminates imaginaries. More concretely, if is a differentially closed field, is a definable set and is a definable equivalence relation, then there is a differential constructible function so that for one has . Moreover, may be defined over the same parameters required for the definitions of and .
Prima facie, Theorem 2.8 applies only to differentially closed fields, but the map may be taken to be defined over the parameters required to define and and then from the constructibility of it is easy to see that for any differential field over which everything is defined, if , then . The proof of Theorem 2.8 passes through the corresponding elimination of imaginaries theorem for algebraically closed fields (which is also proven in detail in [Poizat]) and for our applications, we shall need to unwind this connection between elimination of imaginaries for differentially closed fields and for ordinary algebraically closed fields.
Finally, we shall make use of the theory of o-minimality for which the book [vandenDries] is a good introduction.
An o-minimal structure on the real numbers is given by the choice of a distinguished set of functions , where may depend on , having the property that in the first-order structure
every definable subset of is a finite union of points and intervals.
With the usual definition of an o-minimal structure, any expansion of the language of ordered sets is allowed. However, it follows from the existence of definable choice functions that every o-minimal expansion of an ordered field is bi-definable with one obtained by expanding the language of ordered rings by function symbols.
That interesting o-minimal structures exist at all is highly non-trivial. For us, the most important o-minimal structure is in which the set of distinguished functions consists of the real exponential function and local analytic functions. That is, for each real analytic function defined on some open neighborhood of the unit box we are given the function defined by if and otherwise. O-minimality of is established in [vdDM].
O-minimality implies many strong regularity properties of the definable sets in any number of variables. One result which we shall use is the existence of definable choice functions. If is a definable (in the o-minimal structure ) surjective function, then there is a definable right inverse (see Proposition 1.2 in [vandenDries]).
Since o-minimality is fundamentally a theory of ordered structures, it does not directly apply to complex analysis. However, by realizing as via the real and imaginary parts, one may interpret complex analysis within an o-minimal structure and in the series of papers [PS1, PS2, PS3, PS-cag, PS4, PS5] Peterzil and Starchenko do just this. For us, the most important result from their work is the following strengthening of Chow’s Theorem.
Theorem 2.11 (Peterzil-Starchenko, Corollary 4.5 of [PS-cag]).
Let be some o-minimal structure on the real numbers. Let be a quasiprojective algebraic variety over . If is an -definable, closed complex analytic set in , then is algebraic.
The statement of Corollary 4.5 in [PS-cag] takes , but the proof uses only the fact that embeds into a projective space.
Note that in Theorem 2.11 there is no hypothesis that be projective nor that satisfy any kind of growth condition towards the boundary of in some compactification. On the other hand, one cannot completely avoid such considerations in that to establish that the relevant analytic sets are definable it may be necessary to study their boundary behavior.
The proof of Theorem 2.11 requires much of the Peterzil-Starchenko theory of o-minimal complex analysis, but the basic idea is clear. Many standard theorems in the theory of complex analysis assert that if some closed subset of a complex manifold is generically analytic in some precise sense (for example, with respect to some reasonable dimension) and the ambient space is sufficiently nice, then that set must be analytic (see the theorems of Bishop [Bishop], Remmert and Stein [RemmertStein], and Shiffman [Shiffman]). Sets definable in o-minimal structure enjoy a very smooth dimension theory as do all of the sets naturally associated to them through standard geometric and elementary analytic constructions because these are also definable.
3. Main theorem
With our preliminaries in place, we flesh out the sketch of our main theorem from the introduction. Throughout this section, we fix a natural number and when we speak of a differential field we mean one with distinguished derivations. Likewise, when speaking of arc and jet spaces and differential fields we suppress the index . That is, we write for and for .
In what follows we shall make the following hypotheses.
is a fixed o-minimal structure on the reals.
is an algebraic group over .
is a complex algebraic variety and is a regular function expressing a faithful action of on .
is a complex submanifold of the -points of .
is a Zariski dense, discrete subgroup of .
Via the restriction of the action of on to , acts as a group of automorphisms of .
is a complex analytic covering map of the algebraic variety expressing as the quotient .
is an open -definable subset of for which the restriction is -definable and surjective onto .
Let us note that for each , the set is definable since the action of on is algebraic and hence a fortiori definable. Thus, we may cover by a set of definable sets indexed by .
It follows from the existence of definable choice functions in o-minimal expansions of ordered fields that there is a definable so that the restriction of to is definable and a bijection between and . It is convenient for us to take to be open in which case we cannot assume that the restriction of to is one-to-one.
From these data we construct a differential analytic map on which nearly inverts . We shall call this resulting map the generalized logarithmic derivative associated to .
We begin with two lemmata on jet spaces, the first showing that jets of covering maps are themselves covering maps and the second showing that jets of definable functions are also definable.
For each natural number , there is a natural action of on and with respect to this action, expresses as .
Define the action of on by . By the functoriality of the space of jets construction, we see that for each we have . That is, is invariant under precomposition with the action of . Let us check now that if and are two points of having the same image under , then there is some with . Taking the images in , we see that their images and in have the same image under . Hence, there is some with . As is a covering map, we can find a neighborhood on which is biholomorphic. By functoriality of , is also biholomorphic. In particular, because , we must have . ∎
With our second result on jets we note that the jet of a definable function is itself definable.
Fix an o-minimal expansion of the real field. Let be a complex algebraic variety, an open subset of , a complex algebraic variety, a natural number and a definable analytic function. Then is a definable analytic function.
For the sake of legibility, we suppress the subscripts from and . Let us write for the natural projection map.
Analyticity of is a general feature of the jet of an analytic function (see [Bourbaki]). Definability of follows from the facts that for any definable complex analytic function its complex derivatives are definable and the jet space is definable as a subset of . Indeed, the usual limit definition of a complex derivative may be naturally expressed using the formula defining and the field operations. As is analytic, it follows that all of its derivatives of all orders are definable. Read in coordinates, the map is given by the usual Faà di Bruni formula which is polynomial in the derivatives of up to order and the coordinates on the jet space.
One could generalize Lemma 3.4 to the case that is merely a complex submanifold by noting that the jet space of at a point as a submanifold of may be identified by iterating the standard definitions of tangent spaces from calculus.
The relation on defined by if and only if is an equivalence relation which on -points identifies everything, but when interpreted in a differential field with field of constants is a nontrivial definable equivalence relation. In particular, when is a differentially closed field with field of constants , Theorem 2.8 says that there is a differential constructible function defined on having the property that for one has if and only if . Since is differential constructible, if is some intermediate differential field, it is still the case that for the equality holds if and only if .
In the case that and acting by fractional linear transformations, then the function may be identified with the classical Schwarzian derivative. Similar maps, sometimes with explicit formulae, appear in the literature for other quotients of algebraic varieties by the constant points of an algebraic group (see, for example, the generalized Schwarzians for and in [Buium] or [SasakiYoshida]). In analogy with the classical Schwarzian, we shall call the map a generalized Schwarzian derivative.
Our generalized Schwarzian admits a more algebraic description. Being a differential constructible function, there is some and a constructible function defined on for which . The algebraic group acts algebraically on for precisely the same reason that acts on . While quotients in the category of algebraic varieties with regular maps as morphisms do not always exist, they do always exist constructibly. That is, there is some constructible function on expressing the quotient . We aim to show that for one may take so that .
To this end we first prove two basic lemmata on differential algebraic geometry, one result about definable equivalence relations and a second about the connection between the Kolchin topology on a variety and the Zariski topology on its prolongations.
Let be a differentially closed field. If is a differential variety over and is a definable equivalence relation on , then for and elements of , one has if and only in where we write for the Kolchin closure of the set .
Certainly, if , then so that . Conversely, by quantifier elimination in differentially closed fields we know that the sets and are differentially constructible. Hence, there is a (Kolchin) dense open in set and a dense open in set . If , then is a dense open subset of . In particular, it is nonempty so that is non-empty implying that . ∎
The Kolchin topology and the Zariski topology are related through the prolongation spaces. With the next lemma we show how to compute a Kolchin closure via Zariski closures in prolongation spaces.
Let be a differentially closed field and be an algebraic variety over and let be a set of -points on . Then
Moreover, this set may be realized as for sufficiently large.
Each of the sets is a Kolchin closed set containing . Hence, is contained in .
Before embarking on the proof of the other inclusion let us note that the intersection defining the righthand side of our purported equation is actually a descending intersection. That is, for each we have . Indeed, the prolongation spaces form a projective system and the maps respect this system in the sense that if is the projection map from level to level , then . Thus, if is a Zariski closed set containing , then is a Zariski closed subset of containing so that
For the inclusion of the righthand side in the left, consider a Kolchin closed set which contains . By the definition of the Kolchin topology, there is some and a Zariski closed with . Since , we have . Thus, so that . Clearly, for any if we set , then . Thus, for all we have . On the other hand, by the observation above, we have .
The moreover clause is an immediate consequence of the noetherianity of the Kolchin topology. ∎
We employ the above lemmata to relate the quotient of an algebraic variety by the constant points of an algebraic group to algebraic quotients.
Let be a differentially closed field, an algebraic group over the constant field , an algebraic variety over and an action of on over . For -points the following are equivalent.
There is some with .
For all there is some with .
Moreover, there is some natural number so that these conditions are equivalent to
There is some with .
For this proof, we write for the Zariski closure of a set and for its Kolchin closure.
Define the equivalence relation on by
and let be given by
Note that is a definable equivalence relation while prima facie is merely type-definable. Clearly, for if satisfies , then for each we may take (which belongs to the image of the zero section as the constant points are precisely those whose image under agree with the zero section) to witness that there is some with .
On the other hand, we observe that for each the Zariski closure of is the Zariski closure of the -orbit of . Indeed, consider the algebraic group defined as the stabilizer of in :
As is a homogeneous space for under the natural action of on , we see that . As is Zariski dense in and is closed, we conclude that . Thus, . On the other hand, since , we must have equality.
Therefore, if , then for each by definition over we have that so that . As this equality is true for every , . By Lemma 3.7, .
The moreover clause follows by the compactness theorem. ∎
It follows from elimination of imaginaries in algebraically closed fields that if the algebraic group acts on an algebraic variety , then the quotient may be realized constructibly. In fact, while the map expressing the quotient cannot be taken to be regular, it may be assumed to have some regularity properties.
Let be an algebraically closed field, an algebraic group over , an algebraic variety over , and an action of on , also defined over . Then there are
a chain of closed algebraic varieties ,
an algebraic variety , and
regular maps for each positive
each is -invariant and
for one has .
We work by noetherian induction on with the case that being immediate. By elimination of imaginaries we find a constructible function to some algebraic variety expressing . (Note: we are not claiming that .) As is a constructible function, we can find a dense open for which is regular. Set which is again a dense open subset of . We claim that is regular. Indeed, we may cover with charts of the form where is an open affine in and . By the -invariance of , we see that on , satisfies . That is, agrees with via the isomorphism . Thus, is regular and therefore is regular. Set . By induction, admits the requisite chain, of length , say. Let . ∎
We are now in a position to prove an algebraic counterpart of our main theorem.
With the notation and hypotheses as introduced at the beginning of this section, if is a constructible function from to some algebraic variety expressing the quotient as in Lemma 3.10, then defined by (for any choice of a branch of ) is a constructible function.
For this proof, by a differential field we mean one with field of constants .
Consider the following set.
Let us observe that the set is definable. Indeed, because is -invariant and is -invariant (hence, also -invariant), in the definition of we may restrict to . That is, we have
As is definable, this expression presents as a definable set.
Not only is the set definable, but it is the graph of a function. Indeed, using definable choice in , we see that there is a definable function which is a right inverse to . Again using the -invariance of we see that is the graph of . Let us write for the function whose graph is . Let us note that while this definition of expresses its definability, there are other ways it could be presented. Indeed, from the -invariance of for any we have where is any branch of near .
By Lemma 3.10, there is a sequence of a closed subvarieties so that each is -invariant and the restriction of to is regular. For each , define . Note that . We show by induction on that is constructible. The case of is trivial. Let us consider now the case that . Let us define . Since is a constructible function, is a constructible subset of . Consider . The set being the intersection of two definable sets is definable. Moreover, our second presentation of shows that is the graph of an analytic function. Indeed, if , then fix a branch of near . We have and, in fact, near , is the graph of . As and and are -invariant, necessarily . Thus, near , is the graph of the analytic function . By the Peterzil-Starchenko o-minimal GAGA Theorem 2.11, is algebraic. By induction, is algebraically constructible and the graph of is simply the union of and the graph of . Hence, is itself algebraically constructible.
Taking , we see that is algebraically constructible. ∎
We deduce our main theorem from Proposition 3.11.
With the notation and hypotheses as introduced at the beginning of this section, if is a generalized Schwarzian for the action of on , that is, it is a differential constructible function from to some algebraic variety expressing the quotient , then defined by (for any choice of a branch of ) is a differential constructible function, which we shall call the generalized logarithmic derivative associated to .
By Proposition 3.9 for there is a constructible function so that takes the form and for any differential field with field of constants one has that
With this choice of , we may realize our generalized logarithmic derivative as as in the following diagram.