RESEARCH STATEMENT
MICHAEL WAN
1. Model theory and almost complex manifolds: a general
introduction
In my thesis, I study the model theory of almost complex manifolds, generalizations of complex manifolds that are used throughout symplectic and differential
geometry. Briefly, an almost complex manifold is an even-dimensional real smooth
manifold M , equipped with a family of smooth linear maps Jp : Tp M → Tp M
varying smoothly over p, and satisfying (Jp )2 = − idTp M .
In my work, I define the notion of a pseudoanalytic subset of a real analytic almost
complex manifold, a generalization and analogue of a complex analytic subset of a
complex manifold. I prove results, like the following theorem, that show that these
more general subsets behave in much the same way. Let me note for model theorists
that definability here is relative to the o-minimal theory Ran , and the real analytic
requirement ensures local Ran -definability. For non-logicians, I provide background
shortly.
Theorem 1 (identity principle for pseudoholomorphic maps). Suppose f : M → N
is a definable, real analytic map of definable, real analytic, almost complex manifolds, and A ⊂ M is a connected pseudoanalytic set of real dimension 2d. If f is
constant on a subset of A of real dimension 2d − 1, then f is constant on all of A.
My results point toward a deeper model-theoretic classification theorem for compact almost complex manifolds, generalizing the well-known classification theorem
for compact complex manifolds—namely that they form Zariski geometries in the
sense of Hrushovski and Zilber.
Model theory analyzes structures from the perspective of mathematical logic,
and has recently been applied in several key ways to prove conjectures in algebraic
geometry and number theory. A mathematical object M is studied by equipping
each of its cartesian powers M n with a special class of subsets, called definable
sets. The definable sets are required to be closed under certain logical operations,
like finite intersections, finite unions, and complements (corresponding to boolean
operations), and also under coordinate projections (corresponding to existential
quantification). In this document, we will always work in the setting where M
(and N , etc.) is a real analytic manifold, and the definable sets on M n is the class
of subanalytic sets, roughly, sets which are formed by taking boolean combinations
and coordinate projections of local solution sets to real analytic equations and
inequations (f (x) ∕= 0). For instance, if M = R, then the unit circle {(x, y) ∈ R2 |
x2 + y 2 ≤ 1} is a definable (i.e. subanalytic) subset of R2 . We say that functions
are definable if their graphs are, and that manifolds are definable if there exist finite
atlases of definable coordinate charts.
Date: December 16, 2016.
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MICHAEL WAN
In a landmark paper from 1996, Ehud Hrushovski and Boris Zilber defined the
notion of a Zariski geometry, essentially a model-theoretic structure augmented
by topological and geometric data, including a robust notion of dimension [HZ96].
Zariski geometries are fundamental in geometric model theory: the classification
of one-dimensional Zariski geometries by Hrushovski and Zilber plays a key role
in Hrushovski’s proof of the geometric Mordell-Lang conjecture and other recent
applications to number theory and algebraic geometry [Hru96].
It is natural to ask whether a given geometric structure can be viewed as a Zariski
geometry. For instance, Zilber proved that a compact complex manifold equipped
with its complex analytic topology and complex analytic dimension forms a Zariski
geometry. The big picture goal of my work is to generalize this result to almost
complex manifolds.
Goal 1 (Zariski geometries). Let M be a compact real analytic almost complex
manifold, equipped with topologies on each M n generated by the pseudoanalytic
subsets and with dimension given by half of the real analytic dimension. Show that
the resulting structure is a Zariski geometry.
This was partially realized in earlier work by Kessler, but under strong hypotheses on M , and with a different topology [Kes11]. My goal is to prove it in the
general case, with the more natural topology described above. This would immediately imply the classification result of Hrushovski and Zilber in the almost complex
setting. It could also lead to finer model-theoretic classifications of almost complex
manifolds, and broaden the reach of geometric model theory.
However, almost complex manifolds are significantly more general than complex
manifolds, and as a result, real work is needed to show that almost complex maps
and their level sets behave anything like their holomorphic counterparts. Laying
down this groundwork is the main occupation of my thesis. For instance, I prove:
Theorem 2. Pseudoanalytic sets have even real analytic dimension.
Thus, the dimension defined in Goal 1 takes on integer values. To get further towards this goal, we need to find almost complex versions of results from
classical complex geometry. The identity principle, Theorem 1, is one step down
this path. Indeed, it is a direct almost complex analogue of a theorem of Peterzil and Starchenko, who have an extensive theory of complex analytic geometry generalized to o-minimal theories, including our Ran (subanalytic) setting
[PS01, PS03, PS08, PS09]. Almost complex analogues of their later theorems would
directly imply some of the Zariski geometry axioms.
In the next section, I introduce the main tool for bridging the complex/almost
complex divide. I then outline how this tool is used to prove the theorems stated
above, and finally, I conclude with a section on future plans.
2. Pseudoholomorphic curves, and adapting them for model theory
Pseudoholomorphic curves are almost complex analogues of holomorphic curves
inside complex manifolds. They play an important role in symplectic topology and
string theory, and are a fundamental tool for my main results. In this section, I
will show how known facts about pseudoholomorphic curves can be adapted to the
real analytic setting amenable to model-theoretic analysis.
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Definition 1. An almost complex manifold M is an even dimensional real
differentiable manifold (also called M ) together with a real bundle automorphism
JM : T M → T M satisfying (JM,p )2 = − idp for p in M . A real manifold map
f : M → N between almost complex manifolds is called pseudoholomorphic if its
real differential df : T M → T N exists and JN ◦ df = df ◦ JM . (Earlier these maps
were called “almost complex maps”.)
Complex manifolds are examples of almost complex manifolds, with JM,p given
by multiplication by i at all points p.
Definition 2. A pseudoholomorphic curve is a pseudoholomorphic map u :
Σ → M from a one-dimensional complex manifold Σ to an arbitrary almost complex
manifold M .
In the literature, pseudoholomorphic curves are not always given as differentiable
functions. Instead, they often belong to the class W k,p of Sobolev functions, i.e.
measurable functions whose weak derivatives of order up to k have finite Lp norm,
where k, p ≥ 2. Luckily, we can convert these curves to real analytic functions, as
required by model theory:
Proposition 1 (elliptic regularity). Let u : Σ → M be a function of class W k,p
satisfying du ◦ i· = JM ◦ du (i.e. a “W k,p pseudoholomorphic curve”). Then if M
(including JM ) is real analytic, u is as well.
This can be proved by using various “elliptic regularity” results from the literature, especially [MS12, Theorem B.4.2]. With a bit more work, we can regularize
more complicated objects as well, as in the following theorem.
Theorem 3 (existence of pseudoholomorphic curves). Suppose that M is a real
analytic almost complex manifold of real dimension 2m, and let p ∈ M . Then for
sufficiently small 󰂑 > 0 there exists a real analytic map
Φ : B󰃕2m → N ω (B1 , M )
where B󰃕2m ⊂ R2m is the ball of radius 󰂑, and N ω (B1 , M ) is the space of all real
analytic pseudoholomorphic curves from the unit ball B1 ⊂ C to M , such that for
each w ∈ B󰃕2m , u = Φ(w) is a real analytic pseudoholomorphic curve with u(0) = p
and ∂z u(0) = w.
The idea of the proof is to apply the infinite dimensional inverse function theorem
to the evaluation operator ev : u 󰄳→ (u(p), ∂z u(p)). This is done in [Wen14] in the
W k,p setting, and we can translate this into the real analytic setting by combining
elliptic regularity (Proposition 1) with some real analytic Banach manifold theory.
This existence theorem is critical in what follows, but we’ll need other results as
well. In most cases, some version of elliptic regularity is used to ensure that we can
work in the real analytic setting.
3. A model-theoretic approach to almost complex analytic geometry
In this section, I outline my work in developing an almost complex analogue to
complex analytic geometry, aided by model theory.
I employ a branch of model theory called o-minimality, which generalizes semialgebraic and subanalytic geometry. O-minimality plays a key role in recent applications to number theory, including Pila’s celebrated proof of the André-Oort
conjecture in the case of products of modular curves [Pil11, PW06].
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MICHAEL WAN
As mentioned earlier, definability is always relative to the o-minimal structure
Ran , the real field equipped with restrictions of all real analytic functions Rm → R
to the closed boxes [−1, 1]m ⊂ Rm . The reason we require all of our entities—
manifolds, maps, subsets—to be real analytic is so that they are all locally definable
in Ran .
As touched on earlier, those unfamiliar with model theory can take the adjective
definable to mean that the entity in question lives in the “category of subanalytic
geometry”, which itself can be construed loosely as meaning “built out of real
analytic equations and inequations”.
Here is a precise, strengthened version of Theorem 1 from the introduction.
Theorem 4. Suppose that M and N are definable almost complex manifolds with
2m := dimR M , and U ⊂ M is a definably connected open set. Let f : U → N be a
definable real analytic pseudoholomorphic map, and let c ∈ N .
Extend f by continuity to a subset Û of the topological closure of U where possible:
fˆ : Û → N . Let Z := fˆ−1 (c). Then if dimR Z ≥ 2m − 1, then Z = Û .
This is an analogue of a theorem of Peterzil and Starchenko [PS09, Theorem
3.1], who develop an extensive o-minimal version of complex analytic geometry.
The proof is analogous to theirs, with the main novelty being that of complex
linear subspaces of Cm are replaced by images of pseudoholomorphic curves. The
main challenges are (i) the existence of these curves in all directions, particularly
in the real analytic setting, which is handled by the existence theorem from the
previous section, (ii) getting the images of the curves to behave like complex lines,
and (iii) as the proof is by induction on the dimension of M , proving the base case,
where f is itself a pseudoholomorphic curve. O-minimality is used throughout.
Results like Theorem 4 and some of its corollaries show that level sets of pseudoholomorphic maps behave similarly to complex analytic subsets of complex manifolds. These level sets will allow us to define the topologies required by the Zariski
geometries set-up, and the hope is that these results point towards proofs of specific
Zariski geometry axioms, as outlined in the next section. For now, let’s make the
precise definition.
Definition 3. A definable subset A of a definable almost complex manifold M is
called pseudoanalytic if for every p ∈ M there exists a definable pseudoholomorphic map f : U → N from a neighbourhood U ⊂ M of p to a definable almost
complex manifold N such that
U ∩ A = f −1 (c)
for some c ∈ N .
Theorem 2 mentioned in the introduction states says that pseudoanalytic sets
have even real dimension, which is an imporant step towards Goal 1. It follows
from a basic result relating the almost complex geometry with the underlying real
analytic geometry:
Proposition 2. If A is a pseudoanalytic set in a definable almost complex manifold
M , then the points in A with neighbourhoods that are real analytic submanifolds of
M also have neighbourhoods that are almost complex submanifolds of M , that is,
regR A = regac A.
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This implies Theorem 2 because the real dimension of a pseudoanalytic set A
is the real dimension of regR A = regac A, and almost complex submanifolds are
automatically even-dimensional.
4. Future work: towards Zariski geometries
In this final section I describe plans for continuing my research in the model
theory of almost complex manifolds. I am also interested in working in other areas
of geometric model theory and o-minimality, but I restrict my comments here to
my current, concrete research task.
The main hope is that by continuing to develop almost complex versions of
Peterzil and Starchenko’s o-minimal complex analytic geometry, we can prove the
Zariski geometries result, Goal 1. One important axiom for Zariski geometries is
the semi-properness/quantifier elimination condition, which would follow from the
following analogue of Peterzil and Starchenko’s proper mapping theorem:
Goal 2 (proper mapping theorem). Suppose f : M → N is a definable real analytic map between two real analytic almost complex manifolds. Prove that if A is
pseudoanalytic, and f (A) is closed, then f (A) is pseudoanalytic.
A key to the proof of the proper mapping theorem, as well as many other results
in analytic geometry, is an understanding of the singular points of an analytic set.
Thus, a major stepping stone would be:
Subgoal 1. Show that if A is pseudoanalytic, then so is singac A := A \ regac A.
I have made progress towards proving this, by analyzing the singular points of
a well-presented A = f −1 (c) via the Jacobian condition
codimp A
p ∈ singac A ⇐⇒
󰁬
dfp ≡ 0.
In the almost complex setting, the right hand side does not automatically describe a level set. However, I am working on a result about almost complex vector
bundles that will allow the right hand side to be interpreted as saying that p is
in the set of points which take the same value under two given pseudoholomorphic functions. This would expand our notion of pseudoanalytic from level sets to
“equalizers”, but there is some indication that this is not too costly.
In addition to proving an almost complex proper mapping theorem, there is
other foundational work that needs to be done. Part of the strategy has been to
work in the definable/real analytic categories, and piggyback on basic results about
irreducibility and dimension. However, sometimes these results do not translate
into the almost complex setting for free. For instance, perhaps surprisingly, the
following does not seem to be trivial:
Goal 3. Show that a pseudoanalytic set A is irreducible as a pseudoanalytic set if
and only if it is irreducible as a real analytic set.
This is true with “pseudoanalytic” replaced with “complex analytic” throughout,
but even then it is not entirely trivial. Unfortunately, the proof does not seem to
generalize to pseudoanalytic sets.
Difficulties like this suggest another condition, that we restrict ourselves to pseudoanalytic sets which are given as level sets to pseudoholomorphic maps to C. Since
C is a ring, this gives us an algebraic approach to studying pseudoanalytic sets.
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MICHAEL WAN
Goal 4. Develop an algebraic theory of level sets of pseudoholomorphic maps to C.
Relate ideals of pseudoanalytic sets in this setting to the corresponding real analytic
ideals.
Some concrete progress has been made in this area. Specifically, in special cases,
converting “almost complex ideals” to real ideals and back is harmless: (I R )ac = I.
There are two promising approaches to getting this to work in all settings, one via
partial differential model theory, and another via analytic geometry.
Results along any of these lines would advance model theory’s understanding
of almost complex manifolds, and hence expand its range of geometric specimens.
Looking further out, the hope is that model-theoretic techniques could ultimately
be brought to bear on questions in symplectic and differential geometry.
References
[Hru96] Ehud Hrushovski. The Mordell-Lang conjecture for function fields. J. Amer. Math. Soc.,
9(3):667–690, 1996.
[HZ96] Ehud Hrushovski and Boris Zilber. Zariski geometries. J. Amer. Math. Soc., 9(1):1–56,
1996.
[Kes11] Liat Kessler. Holomorphic shadows in the eyes of model theory. Trans. Amer. Math.
Soc., 363(6):3287–3307, 2011.
[MS12] Dusa McDuff and Dietmar Salamon. J-holomorphic curves and symplectic topology, volume 52 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, second edition, 2012.
[Pil11] Jonathan Pila. O-minimality and the André-Oort conjecture for Cn . Ann. of Math. (2),
173(3):1779–1840, 2011.
[PS01] Ya’acov Peterzil and Sergei Starchenko. Expansions of algebraically closed fields in ominimal structures. Selecta Math. (N.S.), 7(3):409–445, 2001.
[PS03] Ya’acov Peterzil and Sergei Starchenko. Expansions of algebraically closed fields. II.
Functions of several variables. J. Math. Log., 3(1):1–35, 2003.
[PS08] Ya’acov Peterzil and Sergei Starchenko. Complex analytic geometry in a nonstandard
setting. In Model theory with applications to algebra and analysis. Vol. 1, volume 349
of London Math. Soc. Lecture Note Ser., pages 117–165. Cambridge Univ. Press, Cambridge, 2008.
[PS09] Ya’acov Peterzil and Sergei Starchenko. Complex analytic geometry and analyticgeometric categories. J. Reine Angew. Math., 626:39–74, 2009.
[PW06] J. Pila and A. J. Wilkie. The rational points of a definable set. Duke Math. J., 133(3):591–
616, 2006.
[Wen14] Chris Wendl. Lectures on holomorphic curves in symplectic and contact geometry. 2014.