Complex analysis on -minimal structures by Kobi Peterzil February–March 2022

1Topological fields

1.1Topological fields

Definition 1.1. A topological field is a field together with a topology on for which the sum, product, additive and multiplicative inverse functions are continuous.

Properties

• The space is Hausdorff.

• If is compact, then is finite.

• has no limit at .

• There exists a non-empty open proper subset of .

Example 1.2. The following are topological fields.

• Finite fields with the discrete topology.

• Linearly ordered fields with the order topology.

• If is any field, then the field of formal Laurent series with the valuation topology (in fact any valued field). Note that is closed, and that the induced topology on is the discrete topology.

• Let be a topological field, and consider an algebraic extension . Then as a vector space over . The product topology then induces a topological field.

1.2Differentiability in topological fields

We fix a topological field .

Definition 1.3. Let be open, let and let . Then is said differentiable if there exists a such that

The number is unique, and we write .

This is preserved by sums, products, quotients, composition and so on. Moreover differentiability implies continuity. We also have the Cauchy-Riemann equations.

We will next consider more specific contexts: -minimality in particular.

1.3The -minimal case

Let be -minimal and write for its algebraic closure . Any -rational function is differentiable.

Consider the case of . If converges in a neighborhood of zome , then there is some open such that is definable in .

On , the function is not definable in any -minimal expansion of . Fact: let be open such that is definable in some -minimal expansion of . Then the imaginary part of must be bounded. Indeed, defining on , and the set (project onto the -axis if you must) must be bounded by -minimality. Conversely, in , each such is definable whenever is definable. In particular is defined and injective on the strip of with . So is definable.

Question 1. Assume is open and is real-analytic and definable in some -minimal expansion of , then when can be extended definably to some which is complex-analytic, where is open? This is open for definable in .

Answer 1. Tobias Kaiser proved the result for unary functions definable in . Andre Opris shows in his PhD thesis that this is the case for a large class of functions called restricted --analytic functions, which is a proper subset of the set of definable functions in .

1.4Diverging series

Consider the field of formal Puiseux series over , where is a positive infinitesimal. The field is real-closed. Let denote its algebraic closure. For all (formal power series), we have a function . Write for the structure . Robinson and Lipshitz showed that this is -minimal. Any such function can be dedinably extended to .

2Topological analysis

We fix an -minimal expansion of a real-closed field , and write for the algebraic closure of .

2.1Winding numbers

Everything will be definable.

Definition 2.1. A definable closed curve in is a definable and continuous with . We sometimes write . The curve is called simple if is injective.

Example 2.2. . We fix some simple counter-clockwise semi-algebraic parametrization of .

Given any closed curve and a definable continuous map . We have maps

choosing so that . Write . For simplicity, assume that is not locally constant anywhere. The only possible discontinuities lie in , which by -minimality, is finite. Write be the discontinuities. We add as a point, in order to define the winding number of as follows. For , write

We define the winding number of as

xample

Example 2.3. Take . Then . Also , and .

Remark 2.4. If is a definable family of continuous functions . Then by -minimality the numbers corresponding to above are bounded, so the definition of can be done uniformly.

Let us give a few properties of those winding numbers.

1. If is not surjective, then .

2. Let be definably holomorphic. Assume that and are homotopy-equivalent, i.e. there is a continuous with and . Then . This relies on the facts that is dedinably connected in , and that the function is locally constant.

3. For continuous definable , we have .

4. If one reverses the parametrization of , then .

Definition 2.5. Let be definable and continuous, and let . We want to define the winding number of around . We define , and set .

2.2Winding number and -differentiability

Lemma 2.6. Let be definable, open and non-empty. Let be definable and continuous, and let such that is -differentiable at . If , then for all sufficiently small circles centered at , we have .

Proof. Write . Since , there is a with for all . So we can consider . Also set

Note that for all circles around . Since , for sufficiently close to , the element is close to . In particular, picking a sufficiently small circle around , the function is not surjective. So . We conclude that .

Write for the closed unit disk in , and for the unit circle, parametrized counter-clockwise.

Main Lemma. Let be definable and -differentiable on . Let . We have

Example 2.7. We give an example where ii.+iii. fail for a continuous but not -differentiable . Let . So is an upper arc in , and has one definably connected component, and has winding number around any element in the including those which lie in .

Proof of the Main Lemma. We first prove i. We can use an homotopy to shrink continuously, and as the radius of tends to , the curve is close to , so will not be surjective (it will only cover a small angle). So the winding number of is , hence the result.

Let us now prove ii. Fix a definably connected and open component of which contains . Since is definably connected (because is and is continuous and definable), the point is not isolated in , so is infinite. Consider the open set . We claim that the set

has dimention . Indeed assume for contradiction that has dimension . Then in particular has codimension . But has differential everywhere, so partitioning , we see that is locally constant, so takes only finitely many values. So is finite: contradicting the previous argument.

The statements of the main lemma are first-order, so we can move to a sufficiently saturated elementary extension, and consider a generic point in over . Then is at as an -function. Moreover is invertible (i.e. ), then the inverse function theorem for -minimal structures gives that contains an open set, whence in particular contains an open set.

Pick be generic, so . We claim that is finite and that is non-zero on this set. Assume for contradiction that is infinite. Let . Then since it contains the generic point . So is surjectve for all such infinite fibers, which is impossible since the dimentin of is . Assume for contradiction that there is with , and write for the set of such 's. Then again has dimension . By definable choice, there is a such that for all , there is a with and this yields a similar contradiction.

03-02: Lecture 5

Removal of singularities à la Riemann. Let be open and non-empty, let and let be definable, -differentiable and bounded. Then there is a unique such that the extension of to with is -differentiable.

Proof. Set for and . Since is bounded, we have , so is -differentiable on , so by the previous theorem, the function is -differentiable at , with . We then extend by continuity and see that is -differentiable at .

Using the previous result and the maximum principle, one can prove the following:

Theorem 2.8. If is definable and -differentiable, then is also -differentiable.

3Isolated singularities

We start with an -minimal fact:

Fact

Let be open and non-empty. Let be definable, let . Then there is a neighborhood of such that is not dense in .

Proof. Assume for contradiction that this is not the case. So for all there is a definable path converging to such that tends to . So tends to . So . But this is the frontier of a set of dimension , whereas has dimension : a contradiction.

We fix an open set , a point and a definable and -differentiable . Assume that is not constant around . We define the order of at as follows. Recall that for sufficiently large , the number is constant (where is the circle around of radius ). We then define to be that integer.

Case 1: is a removable singularity and

Write for the continuation on . We have , whence

by a previous theorem.

Case 2: is a removable singularity and

Again write for the continuation on . If , then we claim that

Indeed shrinking (hence ) sufficiently, we can obtain that lie outside of .

Case 3: is not removable

In particular must be unbounded near . We claim that

(3.1)

Let be a neighborhood of , let and such that

(3.2)

for all . Set

for all . The function is -differentiable, and bounded by (3.2). So is a removable singularity for . So . We conclude since is unbounded near that , whenc .

Let us show that

Set

on a [épointé] neighborhood of . By (3.1), we can extend to by setting . Now we know that , whence for all sufficiently small (i.e. whenever ).

Theorem 3.1. Set . There is a definable and -differentiable with such that

for all .

Proof. Let be sufficiently small, so . Note that , so setting on , we have . By the previous trichotomy, the function can be extended to with ; hence the result.

Corollary 3.2. Assume that is -differentiable at and that . Then where means that there is an analytic continuation of such that...

Corollary 3.3. For non-standard , “the” function cannot be defined as a -differentiable map on a neighborhood of .

4Taylor series

Let be a non-empty open definable set, and let .

From the proof of Theorem 3.1, we deduce:

Theorem 4.1. Let be definable and -differentiable. Then

If is a pole, then

for a fixed sequence .

Corollary 4.2. The map

from germs at of -differentiable and definable functions to power series in is injective.

Theorem 4.3. Every definable -differentiable function is a polynomial.

Theorem 4.4. If is definable and -differentiable, then is a polynomial.

5Some model theory

Proposition 5.1. Let be a definable family of -differentiable functions , then there is an such that each has degree .

Question 2. Let be a definable family of polynomial functions on . Is there a uniform bound on the degree of ?

Example 5.2. [by A. Piekosz] Let be an arbitrary sequence of complex numbers in with absolute value . Define

This is definable in , and for all , the function is a polynomial of degree . But for the function is not a polynomial, so this doesn't give a negative answer to the previous question.

Proposition 5.3. Let be a definable family of -differentiable functions on for open sets . Then there is an , such that for all , either is locally constant around or .

Proposition 5.4. Let be a definable family of -differentiable functions on for open sets . For , let

be the Laurent series associated to , where is as in the previous proposition. Then the function

is definable.

Let us now go back to the classical setting . Let be a simple closed curve and let have finitely many residues in . Recall that in the previous notations. We have .

So if is as above and is a definable family of simple closed curves, and is finite and uniformly definable, then the function

is also definable.

Theorem 5.5. If is definable and -differentiable, then is polynomial.

Proof. For all , the function is polynomial by the one variable corresponding result. By -minimality, the degrees of corresponding polynomials when ranges in are bounded by some . So . Loooking at , we can conclude by induction.

In fact, we have a result from Palais (1978) that for any uncountable field and which is polynomial in each variable, the function is in fact polynomial.

6Behavior at boundary points

Assume that is definable and -differentiable on a non-empty open set , and let such that is simply connected for some neighborhood of . Then exists in . Indeed recall that for definable, and non-constant and -differentiable on , then is finite on . Indeed assume that as infinitely many limit points around . Then sufficiently close to , one can also arrange that is injective and that it have non-zero derivative. So the inverse map of the restriction will be -differentiable on an open set. Then sends an infinite subset of to , so must be constant: a contradiction.

Question 3. Assume that a definable is holomorphic and bounded. Does extend to ? (preserving boundedness).

7Definable complex manifolds and analytic sets

We now work with , so . Apparently the results should still be valid in the more general context.

Definition 7.1. A definable -manifold is

Note that the transition maps are definable.

Example 7.2. Open subsets of , graphs of definable holomorphic maps , as well as the projective spaces are definable manifolds. If is a discrete sugroup of , then the quotient can be equipped with a -manifold chart by realizing this within

,

.

,

.

Fact: Every compact analytic manifold is definably biholomorphic (in the sense of manifolds) to a definable -manifold.

Definition 7.3. Let definable -manifolds. A definable holomorphic function is a definable function which is holomorphic through charts.

Definition 7.4. A definable submanifold is a definable subset which is an -submanifold, for which moreover the tangent space at each is -linear.

The only compact definale submanifolds of are the finite sets.

Theorem 7.5. Let be a definable -manifold. If is a definable submanifold, then it has a natural structure of definable -manifold, and the inclusion is a definable holomorphic function.

Definition 7.6. Let be a definable -manifold. A definable analytic subset of is a definable closed such that for all , there are a definale neighorhood of and finitely many definable and holomorphic functions with

It can be showed that in fact can be covered by finitely many such sets and functions.

Example 7.7. Algeraic varieties in are definable analytic subsets of . If is a definable compact -manifold for , then every analytic subset of is a definable analytic subset of (again in ).

(By Chow's theorem, every analytic subset of is an algebraic variety.)

8Removal of singularities

The basic problem: is a definable -manifold, we have a definable open , and a definable analytic subset of as per Definition 7.6. When is the closure of in an analytic subset of ? So did we “add singularities” by taking the closure?

Example 8.1. Let and take to be the unit disk. So is a definable analytic subset of itself. But the closed disk is not an analytic subset of .

8.1Main removal of singularities results

We recall a classical result of removal of singularities:

Remmert-Stein theorem. Let be a -manifold, let be a -analytic subset, and set . If is irreducible analytic subset, and , then is an analytic subset of .

A bunch of -minimal ROS results:

1. Assume that for all non-empty definable open . Then is an analytic susbet of .

2. Let be a definable analytic subset with . Then is an analytic susbet of .

A corollary of is that

Corollary 8.2. If is a definable family of subsets of a definable -manifold , then the set is definable.

Proof. Set for each . Each is a locally analytic definable subset of . The set is an analytic subset of if and only if is closed, if is dense in , and if for all non-empty definable open .

It follows that we have a:

Definable Chow theorem. If is a definable analytic subset, then is algebraic.

Corollary 8.3. If is a definable family of subsets of , then is definable.

9Moduli spaces of elliptic curves

Assume that we have a lattice where is an -basis of . Write with its definable structure of -manifold. Then we saw that is isomorphic to an elliptic curve .

We may assume that where . Recall that and are isomorphic if and only if there is a with (where is the standard action of on ). Moreover, there is a holomorphic and transcendental surjective map with

where the isomorphism is as abelian varieties, analytic manifolds. The function is called the invariant.

9.1Fundamental domain for

Set

Then each orbit of has excatly one representative in . It follows that is still surjective (and injective).

Theorem 9.1. The function is definale in .

Proof. Consider the function . Then is definable in by previous results. Now on any bounded part of , the function is definable on in . For , we have , and we see that is the punctured disk centered on . Recall that in particular for all , and as well. So we can factor by and get an analytic map with

Fact: , so , i.e. is a pole of , and we can write where are analytic, definable in . So is definable in .