Exponential periods and o-minimality I
aa r X i v : . [ m a t h . N T ] J u l EXPONENTIAL PERIODS AND O-MINIMALITY I
JOHAN COMMELIN, PHILIPP HABEGGER, AND ANNETTE HUBER
Abstract.
Let α ∈ C be an exponential period. This is the first partof a pair of papers where we show that the real and imaginary partof α are up to signs volumes of sets definable in the o-minimal struc-ture generated by Q , the real exponential function and sin | [0 , . Thisis a weaker analogue of the precise characterisation of ordinary periodsas numbers whose real and imaginary part are up to signs volumes of Q -semi-algebraic sets. Furthermore, we define a notion of naive exponen-tial periods and compare it to the existing notions using cohomologicalmethods. This points to a relation between the theory of periods ando-minimal structures. Introduction
Exponential periods are, roughly speaking, complex numbers of the form(1) Z σ e − f ω where ω is an algebraic differential form, f an algebraic function and σ adomain of integration of algebraic nature. They have a conceptual interpre-tation as entries of the period matrix between twisted de Rham cohomologyand rapid decay homology; more on this later.This paper is the first of two parts on exponential periods and o-minimality.The aim of these papers is to give several definitions that make (1) precise,and to compare these different definitions. Together, the papers prove thefollowing main result. Theorem 0.1 ([CH20, Theorem 13.4]) . Let k ⊂ C be a subfield such that k is algebraic over k = k ∩ R . The following subsets of C agree:(1) naive exponential periods over k ;(2) cohomological exponential periods of triples ( X, Y, f ) where X is asmooth variety over k , Y ⊂ X is a simple normal crossings divisorand f ∈ O ( X ) is a regular function;(3) periods of effective exponential motives over k .Additionally, for every such number its real and imaginary part are up tosigns volumes of compact subsets of R n definable over k in the o-minimalstructure R sin , exp . The most interesting case from the number theoretic point of view is k = Q , or equivalently, Q or Q ∩ R .Let us now explain the notions appearing in this theorem. Date : July 17, 2020.
Naive exponential periods.
We propose the following very explicitdefinition as one way of making (1) precise.
Definition 0.2.
Let k ⊂ C be a subfield such that k is algebraic over k ∩ R .A naive exponential period over k is a complex number of the form Z G e − f ω where G ⊂ C n is a pseudo-oriented (not necessarily compact) closed ( k ∩ R )-semi-algebraic subset, ω is a rational algebraic differential form on A nk thatis regular on G and f is a rational function on A nk such that f is regular andproper on G and, moreover, f ( G ) is contained in a strip S r,s = { z ∈ C | ℜ ( z ) > r, |ℑ ( z ) | < s } . A pseudo-orientation on G is the choice of an orientation on a ( k ∩ R )-semi-algebraic open subset whose complement has positive codimension (andhence measure 0), see Definition 3.14.We check that these integrals converge absolutely. In the case f = 0, werecover the notion of an (ordinary) naive period as introduced by Friedrich in[Fri04], see [HMS17, Definition 12.1.1]. The definition of a naive exponentialperiod is not identical to the definition given by Kontsevich–Zagier in [KZ01, § On o-minimality.
In his “Esquisse d’un Programme”, Grothendieckset forth the need for, and the principles of, some form of “tame” topology.O-minimality provides a good theory of “tame” subsets of R n , avoidingCantor sets, fractals, the graph of a space-filling curve and sin(1 /x ). Inrecent years, o-minimality has seen spectacular applications in algebraicgeometry, most notably as an important tool in the proof of the Andr´e–Oort conjecture for A g by Tsimerman following work of many people, seethe survey [KUY18].The ‘o’ in “o-minimality” stands for “order”. The concept was first intro-duced by in work of Van den Dries in [vdD84] and Pillay–Steinhorn [PS84] atabout the same time that Grothendieck was writing his “Esquisse d’un Pro-gramme”. We recall the definition and some basic properties of o-minimalstructures in Section 2.By work of Wilkie [Wil96], Van den Dries and Miller [vdDM94], the struc-ture of subsets of R n defined using + , · , < , the elements of k ∩ R , the realexponential function exp, and the restriction of the analytic function sin tothe bounded interval [0 ,
1] is an example of an o-minimal structure. Wedenote it by R sin , exp ,k . Theorem 0.3 (See Theorem 5.12) . Let α be a naive exponential periodover k . Then its real and imaginary part are up to signs volumes of compactsubsets of R n definable in R sin , exp ,k . This generalises a result for ordinary periods: their real and imaginarypart are volumes of compact semi-algebraic sets, see [HMS17, Proposition 12.1.6]together with [VS15]. There is a significant difference though: in the caseof ordinary periods, we also have the converse implication. The volume ofa compact Q -semi-algebraic set is by definition a naive period. This is no XPONENTIAL PERIODS AND O-MINIMALITY I 3 longer clear or even expected in the exponential setting. The definable sub-sets appearing in the theorem are of a special shape. For example, we do notneed to iterate the functions exp and sin | [0 , . The number e e is definablein the o-minimal structure and hence also appears as a volume. We do notexpect it to be an exponential period. Question 0.4.
Is there a natural way to charaterise definable sets whosevolumes are naive exponential periods?0.3.
Exponential periods and cohomology.
The origins of the theoryof exponential periods lie in a version of Hodge theory for vector bundleswith irregular connections. To our knowledge such a theory was first con-sidered by Deligne, see [DMR07, p. 17]. A systematic study of the periodisomorphism was started by Bloch and Esnault in [BE00], and fully devel-oped by Hien [Hie07]. He establishes a period isomorphism between deRham cohomology of the connection and a suitable homology theory. Thespecial and central case of exponential connections is treated by Hien andRoucairol [HR08]. If X is a smooth variety over a field k ⊂ C , f ∈ O ( X ) aregular function, they consider the twisted de Rham complex Ω ∗ f with differ-ential ω dω − df ∧ ω . Its hypercohomology is twisted de Rham cohomology .They define rapid decay homology of X an (see Section 6.1) taking the roleof singular cohomology in the classical case and a period pairing H rd n ( X, Q ) × H n dR ( X, f ) → C inducing a perfect pairing after extending scalars to C . The numbers in theimage of the pairing are the exponential periods. Their study in their ownright was proposed by Kontsevich and Zagier in [KZ01].As in the classical case, the theory can be extended to singular varietiesand also relative cohomology. A full-fledged theory of exponential motivesis being developed by Fres´an–Jossen in [FJ20]. Their books also containsa very accessible account of the constructions and the proof of the periodisomorphism. They also give many examples of interesting numbers thatappear as exponential periods.We prove: Theorem 0.5 (Propositions 12.1 and 11.1 of [CH20]) . A complex number α is a naive exponential period over k if and only if there is a smooth variety X over k , a simple normal crossings divisor Y , and f ∈ O ( X ) such that α is in the image of the period pairing H rd n ( X, Y, Q ) × H n dR ( X, Y, f ) → C . Again this generalises the result for ordinary periods, see [HMS17, Theo-rem 12.2.1]. Actually, the theorem also holds for general X and Y or evenall periods of effective exponential Nori motives, see [CH20, Theorem 13.4].The general proof is quite technical. In the present paper, we include thearguments in the curve case, see Section 8. It is more accessible, yet alreadycontains the main ideas.0.4. Method of proof.
The strategy is very similar to the case of ordinaryperiods. Algebraic varieties admit triangulations by semi-algebraic simplices.
JOHAN COMMELIN, PHILIPP HABEGGER, AND ANNETTE HUBER
This allows us to represent homology classes by semi-algebraic sets. In thesimplest case the period pairing on cohomology has the shape( σ, ω ) Z σ e − f ω, suggesting the relation to naive periods. Conversely, the Zariski closure ofa semi-algebraic set G is an algebraic variety X , and the Zariski closure ofits boundary is a closed subvariety Y ⊂ X .The main new tool compared to the classical case is the real oriented blow-up of a smooth analytic variety at some divisor. In the simplest case of P and the divisor ∞ , it is the compactification of C by a circle at infinity. Thepoints correspond to the directions of half rays. Its use is of long standingin the theory of irregular connections. Hien and Roucairol and also theexposition of Fres´an–Jossen use it to establish the period isomorphism inthe exponential case. Indeed, rapid decay homology of X can be computedas the homology of a certain partial compactification B ◦ ( X an , f ) of X an relative to its boundary, see Proposition 6.5. For details on B ◦ ( X an , f ) seeDefinition 6.3 and Section 8.2. It is still semi-algebraic, more precisely, asemi-algebraic manifold with corners.However, this is not yet enough to bound the imaginary part of f ( G ),something that is crucial in showing that R G e − f ω is the volume of a de-finable set in the o-minimal structure R exp , sin ,k . Recall that the complexexponential is not definable, only the real exponential and sin (or cos) re-stricted to bounded intervals. We introduce a smaller semi-algebraic subset B ♯ ( X, f ) of B ◦ ( X, f ). The actual key step in the proof of our main theoremis the comparison between the homology of B ♯ ( X, f ) and B ◦ ( X, f ) in [CH20,Proposition 11.4]. In the simplest case, they agree because a half-circle iscontractible to a single point.There are two reasons for the considerable length of the present paper andits companion: on the one hand, we aim for readers without a backgroundin o-minimality and/or in the classical theory of periods and have chosen toreproduce definitions from the literature and to give detailed arguments andreferences. We have also added a section on the case of curves that is notneeded for the proof of the main theorems, but should be more accessibleand still uses all of the main ideas.On the other hand, we ran into many technical problems. The first twoare addressed in the present paper, the last in [CH20]. • For example, we do not know if the real oriented blow-up of a smoothvariety can be embedded into R n preserving both the semi-algebraicand differentiable structure. Instead we introduce the notion of asemi-algebraic (or more general: definable) manifold, at the price ofhaving to extend results well-known for semi-algebraic subsets of R n to the manifold setting. • The standard triangulation results in semi-algebraic geometry or forsets definable in an o-minimal structure only give facewise differen-tiability of the simplices. This is not strong enough for a straightfor-ward application of the Theorem of Stokes—something that we needfor a well-defined period pairing depending only on homology classes.Our way out is by a result of Ohmoto–Shiota [OS17] who prove
XPONENTIAL PERIODS AND O-MINIMALITY I 5 the existence of C -parametrisations with applications to periods inmind. We can then use a subtle version of Stokes’s theorem provedby Whitney in [Whi57] for “regular” differentials on C -manifolds. • Finally, the period isomorphism has a simple description only inthe case of a smooth affine variety. The general case is handled byhypercovers. This involves some checking of strict compatibilitiesbetween our real oriented blow-ups and their subspaces and a checkthat the abstract period pairing is still realised by integration.0.5.
Structure of the papers.
The following diagram explains the globalstructure of the two papers, and how the different theorems contribute tothe main comparison result.Vol P nv P gnv P abs P log P coh P SmAff P motTheorem 5.12 Lemma 5.5 [CH20, Proposition 12.1]Corollary 5.20[CH20, Proposition 11.1] triv triv[CH20, Proposition 13.3][CH20, Proposition 13.1] Section 8 proves part of the central triangle in the case of curves. In thisspecial case the main ideas of the proof are present, but several delicateproblems are avoided. P nv P logProposition 8.3 Proposition 8.4 Outlook.
Our comparison results point to a deeper relation betweenperiods and o-minimal theory. While the case of ordinary periods—withtheir incarnations as entries of periods matrices or as volumes of semi-algebraic sets—might be seen as a coincidence, this second instance suggeststhat this is not the case. Bakker, Brunebarbe, Klingler and Tsimerman havebeen pursuing a project of making a systematic use of tame geometry inHodge theory and apply it successfully to questions related to the Hodgeconjecture. A central tool was their GAGA theory merging complex spaceswith o-minimal geometry. We hope that the period isomorphism can also beextended to the o-minimal setting, providing a new point of view on periodnumbers.
JOHAN COMMELIN, PHILIPP HABEGGER, AND ANNETTE HUBER
Acknowledgements.
Many thanks to Amador Martin-Pizarro for teach-ing two of us (Commelin and Huber) not only the formalism but also theintuition of o-minimal theory. The review of o-minimal theory in Section 2owes a lot to his talks. We also thank Fabrizio Barroero and Reid Barton fordiscussions on o-minimality and Lou van den Dries for carefully explainingaspects of C p -cell decomposition and triangulation.We thank Marco Hien, Ulf Persson and Claus Scheiderer for answeringquestions on the real oriented blow-up. Stefan Kebekus shared his insightson blow-ups in algebraic geometry. Finally, we appreciated the help of Na-dine Große with the theory of integration.We thank Amador Martin-Pizarro and Javier Fr´esan for their commentson our first draft. 1. Notation
Fields of definition. If z is a complex number, we write ℜ ( z ) and ℑ ( z ) for its real and imaginary part. Let k ⊂ C be a subfield. We denoteby k the intersection k ∩ R , by ¯ k the algebraic closure of k in C , and by˜ k the real closure of k in R . Note that k is not automatically algebraicover k . (For example, let a, b ∈ R such that trdeg Q ( Q ( a, b )) = 2, andconsider k = Q ( a + bi ). Then k = Q .) The following conditions on k areequivalent: k ⊂ k is alg. ⇐⇒ k ⊂ ¯ k is alg. ⇐⇒ ˜ k ⊂ ¯ k is alg. ⇐⇒ [¯ k : ˜ k ] = 2 . If k satisfies these conditions, so does every intermediate extension k ⊂ L ⊂ C with k ⊂ L algebraic.1.2. Categories of varieties.
Let k ⊂ C be a subfield. By variety we meana quasi-projective reduced separated scheme of finite type over k . By X an we denote the associated analytic space on X ( C ).1.3. Good compactifications.
We say that a (
X, Y ) is a log-pair if X issmooth of pure dimension d , and Y a simple normal crossings divisor. A goodcompactification of ( X, Y ) is the choice of an open immersion X ⊂ ¯ X suchthat ¯ X is smooth projective, X is dense in ¯ X and ¯ Y + X ∞ is a simple normalcrossings divisor where ¯ Y is the closure of Y in ¯ X and X ∞ = ¯ X r X . If, inaddition, we have a structure morphism f : X → A , we say that ¯ X is a goodcompactification relative to f if f extends to ¯ f : ¯ X → P . Let f : X → A be a morphism. Consider the graph of f in X × A and take its Zariskiclosure ¯ X ′′ inside ¯ X ′ × P , where ¯ X ′ is a projective variety containing X asa Zariski open and dense subset. We may consider X as a Zariski open anddense subset of ¯ X ′′ . The projection ¯ X ′′ → P is a morphism that extends f . By applying Hironaka’s Theorem we see that a good compactificationrelative to f exists.1.4. Some semi-algebraic sets.
Let k be as in Section 1.1. Let X be asmooth variety, ¯ X a good compactification, X ∞ = ¯ X r X . We denote by B ¯ X ( X ) the oriented real blow-up of ¯ X an in X an ∞ , for details see Definition 4.2.It is a k -semi-algebraic C ∞ -manifold with corners, see Proposition 4.3.In the case X = A , ¯ X = P , we write ˜ P = B P ( A ). This is a manifoldwith boundary: the compactification of C ∼ = R by a circle at infinity, one XPONENTIAL PERIODS AND O-MINIMALITY I 7 for each half ray. For s ∈ C r { } , we write s ∞ for the point of ∂ ˜ P corresponding to the half ray s [0 , ∞ ). We say ℜ ( s ∞ ) > ℜ ( s ) >
0. Weput B ◦ = ˜ P r { s ∞ ∈ ∂ ˜ P | ℜ ( s ∞ ) ≤ } = C ∪ { s ∞ | ℜ ( s ) > } ,∂B ◦ = B ◦ r C = { s ∞ | ℜ ( s ) > } ,B ♯ = ˜ P r { s ∞ ∈ ∂ ˜ P | s ∞ 6 = 1 ∞} = C ∪ { ∞} ,∂B ♯ = B ♯ r C = { ∞} . If G ⊂ R n is k -semi-algebraic, we will also denote by ∂G the complement G r G int where G int is the interior of G inside X ( R ) where X is the Zariski-closure of G in A nk . If G is of dimension d , then ∂G is of dimension at most d − G is closed.1.5. C -homology. In this paper, we denote by ∆ n the standard simplexas normalised in [War83]:∆ n = ( ( x , . . . , x n ) | x i > X i x i < ) ⊂ R n . It is open in the ambient space. We denote by ¯∆ n its closure in R n . Wefix the standard orientation. We define the face maps k i : ¯∆ n − → ¯∆ n as in[War83, (2) p.142]. Moreover, for any topological space X and subspace Y ,we let H n ( X, Y ; R ) denote n -th singular homology with coefficients in thering R .A manifold with corners is a second countable Hausdorff topological spacefor which every point has a neighborhood that is homeomorphic to an opensubset of R n × R m ≥ . Say p ≥
1. We will assume that each manifold withcorners is equipped with a set of charts which need not be maximal; later onthis set will be finite. A map defined on a subset A of R n with values in R m is called C p if it extends to a C p map on an open neighborhood of A in R n with values in R m . A C p -manifold with corners is a manifold with cornerssuch that all transition maps betweens charts are C p . A map between two C p -manifolds with corners is called C p if it is C p on all charts. Definition 1.1. A C -simplex on X is a continuous map σ : ¯∆ n → X such that for any chart φ : U → V ⊂ R n × R m ≥ with U open in X thecomposition φ ◦ σ | σ − ( U ) : σ − ( U ) → V extends to a C -map on an openneighbourhood of σ − ( U ) in R n with target R n + m .Let S n ( X ) be the space of formal Q -linear combinations of C -simplicesof dimension n . For A ⊂ X closed, we denote S n ( A ) ⊂ S n ( X ) the subspacespanned by simplices with image in A .The restriction of σ to a face is again C , hence the usual boundaryoperator ∂ turns S ∗ ( X ) into a complex. The barycentric subdivision of a C -simplex is again C . JOHAN COMMELIN, PHILIPP HABEGGER, AND ANNETTE HUBER
Remark 1.2. If ω is an n -form of class C , then σ ∗ ω = g d t ∧ · · · ∧ d t n fora C -function g on ¯∆ n . Hence the Lebesgue integral converges (absolutely). Theorem 1.3.
Let X be a C -manifold with corners. Then the complex S ∗ ( X ) of C -chains computes singular homology of X and S ∗ ( X ) /S ∗ ( ∂X ) computes singular homology of X relative to its boundary ∂X .Proof. It is equivalent to prove the result in cohomology instead. The argu-ment for the C ∞ -case and smooth manifolds is given in [War83, Section 5.31].It works without changes in the C -case, even for a manifold with corners.The boundary ∂X is not a C -manifold itself, but only a closed subset ina C -manifold with corners. The constructions of loc. cit. still apply. E.g.the partition of unity needed on p. 193 is constructed on X , not on ∂X . Onp. 194/196, U is not an open ball (the manifold case) or the intersection ofan open ball with R n ≥ × R n (the case of a manifold with corners) but theboundary of the latter. If σ is a simplex with values in the boundary, thenso does ˜ h p ( σ ) of Equation (21). (cid:3) Theorem 1.4 ([Whi57, Chapter III, §§ . Let X be a C -manifold withcorners. Let ω be an n -form of class C on X and let σ : ¯∆ n +1 → X be a C -simplex. Then Z σ dω = Z ∂σ ω. Proof.
We first recall the notion of regular differential form in Euclideanspace introduced by Cartan [Whi57, Section 16]. (Warning: this notionis unrelated to the usual concept of regularity in algebraic geometry.) Acontinuous r -form ω on an open subset U ⊂ R n is regular if there exists acontinuous and necessarily unique ( r + 1)-form ω ′ (then called dω ) such thatfor every oriented linear ( r + 1)-simplex ∆ ⊂ U we have Z ∆ ω ′ = Z ∂ ∆ ω. Let ω and σ be as in the hypothesis. After passing to a barycentricsubdivision we may assume that σ takes values in the domain of a chart U → V ⊂ R l × R m ≥ , with U ⊂ X open. Working in this chart, σ extends toa C -map on a open neighbourhood Ω ⊂ R n +2 of ¯∆ n +1 with target R l + m .We may identify ω with a C -form on an open subset R ⊂ R l + m containingthe image of Ω.Note that Whitney’s notion of smooth means C in modern terms, seeloc. cit. p. 15. Thus ω is regular on R and ω ′ is the usual dω by Stokes’sTheorem for linear simplices, see [Whi57, Theorem 14A].By [Whi57, Theorem 16B], the pull-back of a regular form under a C -map is again regular and commutes with d . So σ ∗ ω is regular on Ω with dσ ∗ ω = σ ∗ dω . By definition of regularity, we have R ¯∆ n +1 σ ∗ dω = R ∂ ¯∆ n +1 σ ∗ ω when considering ¯∆ n +1 as a linear simplex in R n +2 with the usual orientation.So the formula of the Theorem holds. (cid:3) O-minimal structures
For the purposes of our paper it is helpful to think of o-minimal geome-try as a generalisation of semi-algebraic geometry. The canonical reference
XPONENTIAL PERIODS AND O-MINIMALITY I 9 for o-minimality is [vdD98]. Within the encyclopedia of mathematics, o-minimality is firmly rooted in the field of mathematical logic and more par-ticularly model theory. In this section we briefly survey the essentials in afashion that is geared towards geometers with no background in model the-ory. The reader is warned in advance that some of the definitions presentedbelow are severe mutations of more general concepts in model theory.
Definition 2.1 ( § . A structure on a non-empty set R is asequence S = ( S m ) m ∈ Z ≥ such that for each m ≥ S m is a boolean subalgebra of the power set P ( R m ): that is, ∅ ∈S m , and S m is closed under complements and binary unions andintersections;(2) if A ∈ S m , then R × A and A × R belong to S m +1 ;(3) { ( x , . . . , x m ) ∈ R m | x = x m } ∈ S m ;(4) if A ∈ S m +1 , then π ( A ) ∈ S m , where π : R m +1 → R m is the projec-tion onto the first m coordinates.We are actually only going to need the case R = R , but in this section wewill present the definitions in the general setting. Definition 2.2.
Let k ⊂ R be a subfield. An example of a structure that isrelevant to the topic of this paper is the structure of k -semi-algebraic sets over R consisting of those subsets of R m that are of the form (cid:8) x ∈ R m | f ( x ) = . . . = f k ( x ) = 0 and g ( x ) > , . . . , g l ( x ) > (cid:9) for some polynomials f i , g j ∈ k [ X , . . . , X m ].It is a non-trivial fact that the collection of semi-algebraic sets satisfiesthe final condition in Definition 2.1. This result is known as the Tarski–Seidenberg theorem. The structure does not change when we replace k byan algebraic subextension in R , hence we may assume k to be real closed.A structure can often be “generated” by a smaller collection of sets.This leads to the following concept (one that is more faithful to the model-theoretic point of view). We follow the terminology of [vdD98]. Definition 2.3 (Def 5.2 of [vdD98]) . A model theoretic structure R =( R, ( S i ) i ∈ I , ( f j ) j ∈ J ) consists of a set R , called its underlying set , relations S i ⊂ R m ( i ) ( i ∈ I, m ( i ) ∈ N ), and functions f j : R n ( j ) → R ( j ∈ J, n ( j ) ∈ N ). If n ( j ) = 0, we call f j a constant and identify it with its unique value.If R = ( R, ( S i ) i ∈ I , ( f j ) j ∈ J ) is a model theoretic structure, and C ⊂ R asubset, then we denote the model theoretic structure ( R, ( S i ) i ∈ I , ( f j ) j ∈ J , ( c ) c ∈ C )by R C . The elements of C are called parameters . Definition 2.4 ( § . (1) Let R = ( R, ( S i ) i ∈ I , ( f j ) j ∈ J ) bea model theoretic structure. We denote with Def( R ) the smalleststructure on R that contains the S i , for i ∈ I , and the graphs of thefunctions f j (for j ∈ J ).(2) A subset A ⊂ R m is called definable in R if A ∈ Def( R ) m . Afunction f : R m → R n is definable in R if its graph Γ( f ) = { ( x, y ) | y = f ( x ) } ⊂ R m × R n = R m + n is definable in R . A point x ∈ R m isdefinable in R if the singleton { x } ⊂ R m is definable in R . (3) Let C ⊂ R be a subset. A subset/function/point is definable in R with parameters from C or definable over C in R or C -definable in R if it is definable in R C .The following Proposition serves two purposes: it makes the relation ofthe previous definitions with logic apparent, and it is a useful result forshowing that certain sets are definable. Proposition 2.5. If R = ( R, ( S i ) i ∈ I , ( f j ) j ∈ J ) is a model theoretic structure,and C ⊂ R a subset, then a subset A ⊂ R m is definable in R with parametersfrom C if and only if there exists a formula φ [ x , . . . , x m , y , . . . , y n ] in “thefirst-order language of R ” and elements c , . . . , c n ∈ C such that A = (cid:8) ( a , . . . , a m ) ∈ R m | φ [ a , . . . , a m , c , . . . , c n ] (cid:9) . Proof.
See [vdD98, Exercise 1, Chapter 1.5]. (cid:3)
Example 2.6. (1) From now on, we will denote by R alg the model the-oretic structure ( R , <, , , + , · ) and (consistent with Definition 2.4)for every subfield k ⊂ R we denote by R alg ,k the model theoreticstructure obtained from R alg by adding elements in k as constants.This is justified by the fact that the structure Def( R alg ,k ) consistsprecisely of the k -semi-algebraic sets introduced in Definition 2.2. In-deed, they are defined by first-order formulas in the language of R alg with parameters from k .(2) Let A ⊂ R m be a k -semi-algebraic set. Using Proposition 2.5 itbecomes straightforward to show that the topological closure ¯ A ⊂ R m is semi-algebraic. Indeed¯ A = (cid:8) x ∈ R m | ∀ ε ∈ R , ∃ y ∈ A, ε > → | x − y | < ε (cid:9) , which is clearly a first-order formula. Remark 2.7.
For our purposes it is essential to keep track of parameters.For example, π is R -definable in R alg but not Q -definable in R alg . Whendealing with definable sets we usually explicitly mention the scope of ourparameters. Definition 2.8.
We say that a model theoretic structure R expands ( R , <, , , + , · )if its underlying set is R , and if it contains the relation < , the constants 0 , , · with their usual interpretations.Now we are finally ready for the central notion. Definition 2.9 (See § § . A model theoretic structure R expanding ( R , <, , , + , · ) is o-minimal if the R -definable subsets of R areexactly the finite unions of points and (possibly unbounded) open intervalsin R . Remark 2.10.
Note that in this definition, Van den Dries considers R -definable subsets of R in R . In particular, it is not required that everyinterval is definable in R without introducing additional parameters. Example 2.11.
Since the R -semi-algebraic subsets of the real line are ex-actly finite unions of points and (possibly unbounded) open intervals, we seethat R alg is an o-minimal structure. XPONENTIAL PERIODS AND O-MINIMALITY I 11
Note that N and Z are not definable subsets in any o-minimal structure,because of the finiteness condition in the definition. In particular, the func-tions sin : R → R and exp : C → C (after identifying C with R ) cannot bedefinable in any o-minimal structure. Definition 2.12.
The model theoretic structure ( R , <, , , + , · , exp) will bedenoted by R exp . Here exp : R → R is the real exponential function (andnot the complex one, this is important!). Definition 2.13.
Let F an be the collection of restricted analytic functions,that is, functions f : R m → R that are zero outside [0 , m and such that f | [0 , m can be extended to a real analytic function on an open neighbourhoodof [0 , m .We denote by R an the model theoretic structure ( R , <, , , + , · , F an ) andby R an , exp the model theoretic structure ( R , <, , , + , · , F an , exp). Finally,we denote by R sin , exp the model theoretic structure ( R , <, , , + , · , sin | [0 , , exp).For every subfield k ⊂ R , we denote R sin , exp ,k the model theoretic structurewhere we adjoin all elements in k as parameters.This is one of the protagonists in this paper. Remark 2.14.
The model theoretic structure R sin , exp will be of most in-terest to us. Note that if the interval I ⊂ R is definable with parametersin C , then the functions sin | I and cos | I are definable in R sin , exp with param-eters in C . Indeed, one may use the identity sin ( θ ) + cos ( θ ) = 1 to definecos( θ ) for θ ∈ [0 , θ ) can be arbitrarily extended usingcos( − θ ) = cos( θ ) and cos(2 θ ) = 2 cos ( θ ) −
1. This allows one to define π :it is twice the smallest positive zero of cos. Finally, one can define sin onarbitrary bounded definable intervals by translating cos by π/ Theorem 2.15.
The model theoretic structures R exp , R an , R an , exp , and R sin , exp are o-minimal.Proof. For R an , the result was proven by Van den Dries in [vdD86]. Wilkieproved that R exp is o-minimal in [Wil96]. Building on Wilkie’s result (thatwas already announced in 1991), Van den Dries and Miller [vdDM94] showedthat R an , exp is o-minimal. Finally, R sin , exp is o-minimal because its definablesets are definable in the o-minimal structure R an , exp and it expands R alg . (cid:3) Remark 2.16.
A fundamental fact about o-minimal structures is that eachdefinable set is a finite disjoint union of basic building blocks called cells . Ifthe set is defined over a subfield k ⊂ R , then so are the cells. This followsfrom the Cell Decomposition Theorem [vdD98, Theorem 2.11, Chapter 3],see also [vdD98, Chapter 3, Section 2.19, Exc. 4]. Using this theorem onecan introduce a good notion of dimension of definable sets that behavesas one expects intuitively. For example, if X is a nonempty definable set,then dim( ¯ X r X ) < dim( X ). See [vdD98, Chapter 4] for details and otherproperties of the dimension. Remark 2.17.
For the reader well versed in o-minimality we remark thatfor the remainder of this text, our o-minimal structures will always expand( R , <, , , + , · ). In particular, a definable subset of R n is connected if andonly if it is definably connected . Moreover, the word compact retains itsmeaning from point set topology. Definable manifolds
Fix an arbitrary o-minimal structure S expanding ( R , <, , , + , · ). anda subfield k ⊂ R . In the remainder of this section, all definable sets areunderstood to be definable in S with parameters from k unless otherwisespecified. Definition 3.1.
Let 0 ≤ p ≤ ∞ .(1) A definable C p -manifold with corners M is a C p -manifold with cor-ners together with the choice of a finite atlas ( φ i : U i → V i ⊂ R n i × R m i ≥ ) i ∈ I such that the V i are open in R n i × R m i ≥ and defin-able and the transition maps φ ij = φ j ◦ φ − i are definable and ofclass C p on their domain. Its boundary ∂M is the union of thepreimages of the boundaries of R n i × R m i ≥ ⊂ R n i + m i under the φ i .(2) A subset G ⊂ M is called definable if φ i ( G ∩ U i ) is definable in R n i for all i . We say G is an affine definable set if M = R n , i.e., if it isa definable set in the sense of Definition 2.4 and Proposition 2.5.(3) A subset N of a definable C p -manifold with corners M is called a submanifold if there is a C p -manifold M and a C p -immersion M → N that is a homeomorphism where N carries the subspace topology.I.e. our submanifolds are embedded and have no corners.(4) Let ( M, φ i ) , ( N, ψ j ) be definable C p -manifolds with corners. A mapof definable C p -manifolds with corners is called a definable C p mapif all ψ j ◦ f ◦ φ − i are definable and C p on their domain. Remark 3.2.
The definition of definable manifold includes the choice ofa finite atlas. The finiteness condition is important, as, for example, wedo not want manifolds with infinitely many connected components. So wecannot work with a maximal atlas. However, we could work with an equiv-alence class of finite atlases. Alternatively, one may rephrase the definitionin the language of locally ringed sites, using the Grothendieck topologyof definable open subsets and finite covers. The definition of a definablemanifold is inspired by and related to the semialgebraic spaces of Delfsand Knebusch [DK81] and the complex analytic definable spaces of Bakker–Brunebarbe–Tsimerman [BBT18]. See Chapter 10 § Remark 3.3.
Robson (see [Rob83]) showed that all semi-algebraic spaces(the C -case of the above definition) are actually affine. However, it is notclear to us if this extends to the C p -setting. The above notion is generalenough for our needs. Example 3.4.
Let ¯∆ ⊂ R n be the closed simplex spanned by v , . . . , v m ∈ k n . Then ¯∆ is a definable C p -manifold with corners for all p ≥
0. As thisexample shows, the boundary of a manifold with corners does not have aninduced structure of C p -manifold for p = 0. We are particularly interestedin the case p = 1 because every affine definable set G has a triangulationsuch that the maps ¯∆ → G are maps of definable C -manifolds in the abovesense. See [OS17] and [CP18], and also Proposition 7.4, where we quote thisresult. XPONENTIAL PERIODS AND O-MINIMALITY I 13
Another well-known example are cells. We refer to Chapter 3 of [vdD98]for the definition and basic properties of C -cells. Chapter 7.3 [vdD98]introduces C p -cells and proves the decomposition theorem for p = 1, thegeneral case is similar. Example 3.5.
Let C ⊂ R n be a definable C p -cell of dimension d . It iseasy to see that there is a of coordinates { x i , . . . , x i d } on R n inducing adefinable homeomorphism φ = ( x i , . . . , x i d ) : C → φ ( C ) ⊂ R d . We give C the structure of an affine definable C p -manifold using the chart φ . Then theinclusion C → R n is a definable C p -map of definable C p -manifolds. In otherwords, cells are definable C p -submanifolds of R n . Definition 3.6.
Fix an integer p ≥
1, let d ≥ M bea definable C p -manifold with corners with G ⊂ M a definable subset. Wedefine Reg d ( G ) to be the set of x ∈ G that admit an open neighbourhood U in M such that G ∩ U is a submanifold of M of dimension d . Remark 3.7.
The set Reg d ( G ) is open in G , it is empty if dim G < d .If dim( G ) = d , it is the maximal subset of G that is a submanifold of M having connected components of dimension d . If G and H are disjointdefinable subsets of M , then in general there is no inclusion between thetwo sets Reg d ( G ∪ H ) and Reg d ( G ) ∪ Reg d ( H ).The following lemma adapts to our situation the fact that the p -regularpoints of given dimension of a definable set constitute a definable set. Lemma 3.8.
Let
M, G, and
Reg d ( G ) be as in definition 3.6. Then Reg d ( G ) is a definable subset of M and dim G r Reg d ( G ) < d if dim G = d .Proof. Assuming the first claim we begin by proving the last claim by con-tradiction. Suppose H = G r Reg d ( G ) has dimension dim G = d . Thereis a chart of M on which V ∩ H becomes a definable set of dimension d .So we may assume H ⊂ G ⊂ R n × R m ≥ . We fix a C p -cell decompositionof G partitioning H and G r H . One cell in H must have top dimensiondim H = dim G and this cell has a point not contained in the closure of anyother cell. This point lies in Reg d ( G ), which is a contradiction.To show that Reg d ( G ) is definable it suffices to work in a single chart.So without loss of generality G is a definable subset of R n × R m ≥ of dimen-sion d . We use the classical theory of differential manifolds to characterizesubmanifolds locally as graphs of functions. I.e., Reg d ( G ) is the set of pointsof G that have an open neighbourhood in M in which G is the graph of a C p map defined on an open subset of a projection of R n + m to d differentcoordinates. The argument laid out in [vdDM96, B.9] applies directly to ourslightly more general situation, and implies the definability of Reg d ( G ). (cid:3) Lemma 3.9.
Let G ⊂ R n be a definable subset of dimension d . Let π : R n → R d denote the projection to the first d coordinates. There are pairwise dis-joint definable open subsets G , G , . . . , G N of Reg d ( G ) with dim G r ( G ∪· · · ∪ G N ) < d such that all fibres of π | G have positive dimension and suchthat π | G i : G i → π ( G i ) is a chart for all i ∈ { , . . . , N } .Proof. Without loss of generality G = Reg d ( G ). Let G ′ be the set of pointsof G that are isolated in their fibre of π | G . It is definable, see Corollary 1.6, Chapter 4 [vdD98]. Each fibre of π | G ′ is discrete and thus finite with uni-formly bounded cardinality, see Corollary 3.7, Chapter 3 [vdD98]. Let N belargest cardinality of a fibre.By definable choice, Proposition 1.2(i) Chapter 6 [vdD98], applied to thegraph of π | G ′ there is a definable section ψ : π ( G ′ ) → G ′ of π | G ′ , i.e. π ◦ ψ isthe identity. The image ψ ( π ( G ′ )) is a definable set. It lies in G ′ but possiblymissing some branches. The set of missing points G ′ = G ′ r ψ ( π ( G ′ )) is alsodefinable. Now π | G ′ certainly still has finite fibres, but the maximal fibrecount dropped to N −
1. We repeat this step and find a section ψ : π ( G ′ ) → G ′ and again the fibre count of π on G ′ = G ′ r ψ ( π ( G ′ )) drops by one.After N steps, all fibres are exhausted. We obtain definable maps ψ , . . . , ψ N defined on subsets of π ( G ′ ) whose images cover G ′ and are pairwise disjoint.But the ψ i may fail to be continuous. By the Cell Decomposition Theorem,[vdD98, Chapter 3, Theorem 2.11] applied to the domain of each ψ i , we get,after adjusting N and renaming, finitely many continuous definable maps ψ i : C i → G ′ on cells C i ⊂ R d with S i ψ i ( C i ) = G ′ and with π ◦ ψ i theidentity for all 1 ≤ i ≤ N . Observe that the ψ i ( C i ) remain pairwise distinct.Suppose dim C i = d , then C i is open in R d . As G is a manifold, invarianceof domain implies that ψ i ( C i ) is open in G and ψ i : C i → ψ i ( C i ) is ahomeomorphism. Thus π | ψ i ( C i ) : ψ i ( C i ) → C i is a chart. We can safelyignore cells C i with dim C i < d ; the union H = S i :dim C i 1. Fix a cell decomposition of G r G ′ and let G bethe union of all d -dimensional cells; then G is open, and possibly empty, inthe submanifold G . We add the remaining cells to H . We retain dim H < d and the lemma follows from G = G ∪ S i :dim C i = d ψ ( C i ) ∪ H . (cid:3) Lemma 3.10. Let p ≥ and ( M, φ i ) , ( N, ψ j ) be definable C p -manifoldswith corners. The the bundles T M , T ∗ M and their exterior powers havea natural structure of a definable C p − -manifold with corners. Moreover, adefinable C p -map f : M → N induces definable C p − -maps df : T M → T N and d ∗ f : T ∗ N → T ∗ M .Proof. We only have to verify definability. This holds because the deriv-ative of a definable differentiable function is definable. Indeed, in the 1-dimensional case the graph Γ( f ′ ) of the derivative is given by the formula (cid:26) ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ∀ ε > , ∃ δ > , ∀ x ′ , | x ′ − x | < δ → (cid:12)(cid:12)(cid:12)(cid:12) f ( x ′ ) − f ( x ) x ′ − x − y (cid:12)(cid:12)(cid:12)(cid:12) < ε (cid:27) . (cid:3) Many properties of affine definable sets extend immediately to the non-affine case. This is in particular the case for the notion of dimension andthe stratification by submanifolds. We want to use these facts in order tointegrate differential forms. Definition 3.11. Let p ≥ 1. Let ( M, φ i ) be a definable C p -manifold withcorners and G ⊂ M a definable subset. A differential form ω of degree d on G is a continuous section ω : G → Λ d T ∗ M. It is called definable , if it is definable as a map in the sense of Definition 3.1 (4). XPONENTIAL PERIODS AND O-MINIMALITY I 15 In the affine case, we can give an explicit description: Let x , . . . , x n bethe standard coordinates on R n . For I = { i , . . . , i d } ⊂ { , . . . , n } a subsetwith i < i < · · · < i d we write as usuald x I = d x i ∧ · · · ∧ d x i d . A differential form on G can be written uniquely as ω = X I a I d x I with a I : G → R continuous. It is definable if and only if the a I are definable. Remark 3.12. Note that we do not put differentiability conditions or re-quire that ω extends to a neighbourhood of G . Lemma 3.13. Let p ≥ , and f : M → N be a definable C p -map of defin-able manifolds with corners. Let G ⊂ M and H ⊂ N be definable subsetswith f ( G ) ⊂ H . Then the pull-back of a differential form on H defines adifferential form on G . If ω is definable, so is f ∗ ω | G .Proof. By definition, f ∗ ω | G : G → Λ d T ∗ M is the composition G → H → Λ d T ∗ N → Λ d T ∗ M of continuous maps. In particular, it is definable if ω is definable. (cid:3) As usual, we can only expect a well-defined integration theory for differ-ential forms on oriented domains. Definition 3.14. Fix an integer p ≥ 1, let d ≥ M be a definable C p -manifold with corners with G ⊂ M a definable subset ofdimension d .(1) A pseudo-orientation on G is the choice of an equivalence class of adefinable open subset U ⊂ Reg d ( G ) such that dim( G r U ) < d andan orientation on U . Two such pairs are equivalent if they agree onthe intersection. We thereby obtain an equivalence relation.(2) Given a pseudo-orientation on G with U as in (1) and a differentialform ω of degree d on G , we define Z G ω := Z U ω if the integral on the right converges absolutely. Remark 3.15. The same definition also allows us to integrate a d -form ω over a G of dimension smaller than d : in this case Reg d ( G ) = ∅ and theintegral is set to 0. Such integrals occur in our formulas and are to be readin this way. Lemma 3.16. Let p ≥ . Let G be a definable subset of a definable C p -manifold with corners M .(1) The integral is well-defined, i.e., independent of the choice of repre-sentative for the pseudo-orientation.(2) By restriction a pseudo-orientation on G also induces the choice ofa pseudo-orientation on every definable subset G ′ ⊂ G with dim G =dim G ′ . (3) The choice of a pseudo-orientation on G induces a choice of a pseudo-orientation on every definable superset G ⊂ G ′′ such that dim( G ′′ r G ) < d , in particular on ¯ G .(4) Let π : G ′ → G be a definable modification, i.e., there is an opendefinable subset U ⊂ Reg d ( G ) with dim( G r U ) < d such that π | U ′ : U ′ = π − ( U ) → U is an isomorphism of definable C -manifoldsand dim( G ′ r U ′ ) < d . Then the choice of a pseudo-orientation on G induces a pseudo-orientation on G ′ .Proof. If U , U ⊂ Reg d ( G ) are definable open such that dim( G \ U i ) < d ,then the same is true for U ∩ U . Hence it suffices to consider the case U ⊂ U . By assumption the orientation on U restricts to U . We have Z U ω = Z U ω because U r U has measure 0. The left hand side converges absolutely ifand only if the right hand side does.We fix a pseudo-orientation on G , i.e., an orientation on some U ⊂ Reg d ( G ) such that dim( G r U ) < d .Let G ′ ⊂ G , U ′ = U ∩ Reg d ( G ′ ). The orientation on U restricts to anorientation on U ′ . We have dim( G ′ r U ′ ) < d , hence this data defines thepseudo-orientation on G ′ .Let G ⊂ G ′′ , U ′′ = U ∩ Reg d ( G ′′ ). The orientation on U restricts to anorientation on U ′′ . As dim( G ′′ r G ) < d , we also have dim( G ′′ r U ′′ ) < d ,hence again this data defines a pseudo-orientation on G ′′ .The case of a modification combines the two operations. (cid:3) Corollary 3.17. Let G, H ⊂ M be definable subsets of dimension at most d of a definable manifold with corners, equipped with a pseudo-orientationon G ∪ H . Let ω be a definable differential form of degree d on G ∪ H . Thenwith the restricted pseudo-orientations Z G ∪ H ω = Z G ω + Z H ω − Z G ∩ H ω and the left hand side is finite if and only if all terms on the right are.Proof. We may assume dim G = dim H = d . We can decompose G ∪ H intothe disjoint subsets G ∩ H, G r H, H r G . Hence it suffices to check theformula in the case where the two sets are disjoint.We start with an orientation on a definable open subset U ⊂ Reg d ( G ∪ H ) with dim( G ∪ H ) r U < d . The pseudo-orientations on G and H arerepresented by the restricted orientations on V = U ∩ Reg d ( G ) and W = U ∩ Reg d ( H ), respectively. Then V ∪ W represents our pseudo-orientation on G ∪ H . By definition and by the standard computation rules for integrationon manifolds, we find Z G ∪ H ω = Z V ∪ W ω = Z V ω + Z W ω = Z G ω + Z H ω. (cid:3) Remark 3.18. (1) As in the case of ordinary orientations, the valueof the integral depends on the choice of pseudo-orientation. Notethat even a simple definable set like an interval U , admits infinitely XPONENTIAL PERIODS AND O-MINIMALITY I 17 many different pseudo-orientations. If U has n connected compo-nents, there are 2 n possible orientations and we can cut up U asmuch as we like.(2) For each G the choice U = Reg d ( G ) is canonical if it is possibleto choose an orientation on this set. However, the behaviour ofReg d ( G ) under standard topological operations is complicated. It isnot true that the choice of an orientation on Reg d ( G ) also inducesan orientation on Reg d ( ¯ G ) (take G = R r { } ). Neither is it truethat Reg d ( G ′ ) ⊂ Reg d ( G ) if G ′ ⊂ G (take the x -axis in the union ofthe coordinate axes in R ). Our more flexible notion sidesteps theseproblems.(3) Note also that every non-empty definable set G admits a pseudo-orientation because open cells are orientable and G admits a celldecomposition.(4) The restriction operation described in the proof of Lemma 3.16(2) iswell-defined in the following sense. Two representatives of a pseudo-orientation on G restrict to representatives of the same pseudo-orientationon G ′ . Moreover, the same holds true for the extension operationdescribed in the proof of part (3). Finally, extending a pseudo-orientation from G to G ′′ and then restricting it back to G recoversthe original pseudo-orientation. So the extension in part (3) of thelemma is unique. Remark 3.19. If G ⊂ R n is a definable open with the standard orientationand ω = d x ∧ · · · ∧ d x n , then R G ω = vol( G ). This number is always finiteif G is bounded.We will see that the example of the volume form is really the general case,but before that we need to establish a technical lemma. Lemma 3.20. Let ( M, φ i ) be a definable manifold with corners, x ∈ M .Then there is a definable open neighourhood U x ⊂ M with compact closureand such that ¯ U x ⊂ U i for some i .Proof. We fix i such that x ∈ U i . Recall that V i = φ i ( U i ) is open in H := R n × R m ≥ . Hence there is a definable 0 < r < ∞ such that the open ball H ∩ B r ( φ i ( x )) is contained in V i . Let a ∈ H be definable with distance atmost r/ φ i ( x ). Put V x = B r/ ( a ) ∩ H . Then ¯ V x ⊂ B r ( φ i ( x )) ∩ H is acompact set contained in V i . We put U x = φ − i ( V x ). (cid:3) Lemma 3.21. A finite Z -linear combination of volumes of definable boundedopen subsets of R d is up to sign the volume of a definable bounded open subsetof R d .Proof. All contributions with a positive coefficient can be combined into asingle one by taking the disjoint union of translates of the definable sets. Inthe same way all contributions with a negative coefficient can be combinedinto a single one. So it suffices to prove that the difference of the voluminaof two definable bounded open subsets of R d is up to sign the volume of adefinable bounded open subset of R d . The argument of Viu-Sos, see [VS15,Section 4] in the semi-algebraic setting works identically in the definablecase and provides what we want. (cid:3) Recall that we work with k -definable sets in a fixed o-minimal structure S expanding ( R , <, , , + , · ). Theorem 3.22. Let p ≥ , and ( M, φ i ) be a definable C p -manifold withcorners, G ⊂ M a pseudo-oriented compact definable subset of dimension d .Let ω be a differential form of degree d on G as in Definition 3.11. Then Z G ω converges absolutely. If ω is definable, then the value is up to a sign thevolume of a definable bounded open subset of R d +1 .Proof. We are going to rewrite our integral as a finite Z -linear combinationof other integrals. Eventually these summand will be absolutely convergent,proving absolute convergence of the original integral. In the definable case,every summand will be written as a difference between volumes of boundeddefinable open subsets of R d +1 . By Lemma 3.21 this will imply that theoriginal volume is up to a sign the volume of a single definable boundedopen subset of R d +1 and hence finish the proof of the theorem.We begin by showing how to reduce to the case M = R n . By Lemma 3.20each point x ∈ G has a definable open neighbourhood U x in M such that¯ U x is compact and contained in one of the finitely many charts of M . Byhypothesis G is compact, so it is covered by finitely many such neighbor-hoods, let us call them U , . . . , U a . The ¯ U i and their multiple intersectionsinherit a pseudo-orientation from G . By the inclusion-exclusion principle,Corollary 3.17, we have Z G ω = a X i =1 Z ¯ U i ω − X i 1. All G i inhert a pseudo-orientation from G and all π | G i with i ≥ G i by a finite union of open subsets, againup-to a subset of dimension d − 1, and assume that all G , . . . , G N carry anorientation in the classical sense and that π | G i : G i → π ( G i ) is orientationpreserving. Thus Z G ω = N X i =0 Z G i ω by Corollary 3.17 if all integrals on the right converge absolutely. Thus itsuffices again to treat a single R G i ω . XPONENTIAL PERIODS AND O-MINIMALITY I 19 We begin with the easy case i = 0. By assumption, all fibres of π | G have positive dimension, hence π | ∗ G = 0 on differential forms of degree d .Thus the restriction of ω = a d x ∧ · · · ∧ d x d to G vanishes. Hence R G ω converges absolutely with value 0, the volume of ∅ .Now we treat G i with i ≥ 1. Then π | G i : G i → π ( G i ) ⊂ R d is a chartand thus has an inverse ψ i : π ( G i ) → G i . Note that ψ i is of C p -class . Let y , . . . , y d denote the coordinates of R d . The integral Z π ( G i ) a ◦ ψ i d y ∧ · · · ∧ d y d converges absolutely as a is continuous on the compact G and thus in par-ticular bounded on G i . Finally, ψ ∗ i ( d x ∧ · · · ∧ d x d ) = d y ∧ · · · ∧ d y d as π ◦ ψ i is the identity. Thus Z G i a d x ∧· · ·∧ d x d = Z π ( G i ) ψ ∗ i ( a d x ∧· · ·∧ d x d ) = Z π ( G i ) a ◦ ψ i d y ∧· · ·∧ d y d converges absolutely.Suppose that ω is definable, then a is definable. It remains to show that R ψ i ( G i ) a ◦ ψ d y ∧ · · · ∧ d y d is the volume of a definable bounded open subsetof R d +1 . This integral equals Z C + a ◦ ψ i d y ∧ · · · ∧ d y d − Z C − | a ◦ ψ i | d y ∧ · · · ∧ d y d with C ± = { y ∈ ψ i ( G i ) | ± a ( ψ i ( y )) > } both definable bounded and openin R d . Hence it equals vol( U + ) − vol( U − ) with U ± = { ( y, z ) ∈ C ± × R :0 < z < | a ( ψ i ( y )) |} . Note that U ± are both definable bounded and open in R d +1 . This difference is the volume of a definable bounded open subset of R d +1 by Lemma 3.21. (cid:3) Remark 3.23. Let us explain why we cannot replace R d +1 by R d in thetheorem above. Consider the half-circle G = { ( x, y ) ∈ R × (0 , ∞ ) | x + y =1 } . It is relatively compact, semi-algebraic and definable without parameters.Then Reg ( G ) = G for all p ≥ 1. Now R G y d y = ± R − p − y d y = ± π/ π is transcendental, R G y d y cannot be the volume of Q -semi-algebraicsubset of R . Remark 3.24. The natural way of computing the integral is to pull thedifferential form back to a chart (via the inverse of the chart map) andevaluate there. However, this pull-back involves a Jacobian matrix. Itsentries are not bounded in general, hence convergence is not automatic.Here is an explicit example: Let M = R , G = { ( y , y ) | y ∈ [0 , } , ω = a d x + b d x for continuous a, b on G . We have Reg ( G ) = { ( y , y ) | y ∈ (0 , } . It is a submanifold. We can use the projections π and π to thefirst or second coordinate as a chart. In both cases the image in R is theopen interval I = (0 , ψ : I → G of π is t ( t, √ t ). ItsJacobian matrix is (cid:18) , √ t (cid:19) . The second entry is unbounded on I . We have ψ ∗ ω = ( a ◦ φ ) d t + ( b ◦ φ ) 12 √ t d t. The coefficient function is unbounded. (Note that a ◦ φ and b ◦ φ arebounded because a and b are. Note also that differentiability of a and b does not come into play. It suffices that they are continuous.) The solutionis to treat the summands a d x and b d x separately and use the projection π for the first summand and π for the second summand. We then interpret a d x = π ∗ (( a ◦ φ ) d t ) , b d x = π ∗ (( b ◦ φ ) d t )and the convergence issue disappears. Remark 3.25. A similar convergence argument for integrals can also befound in [HKT15]. Alternatively, convergence also follows from the existenceof triangulations that are strictly of class C , shown in [OS17] and [CP18].These references treat explicitly the case of C ∞ -forms, but actually thisassumption is not needed.4. Oriented real blowup The oriented real blowup is a natural construction in the context of semi-algebraic geometry. Nevertheless, it seems that little is written about itfrom this point of view. The construction is discussed in § I.3 of [Maj84], § R alg ) with suitable parameters. For a general discussion we refer to theaforementioned sources.Let X be a topological space, let π : L → X be a complex (topological)line bundle on X , and let s : X → L be a section. Let L ∗ be the complementof the zero section. We put B ∗ L,s = { l ∈ L ∗ | s ( π ( l )) ∈ R ≥ l } . If s ( x ) = 0, then B ∗ L,s contains L x r { } , otherwise, it contains the uniqueopen half-ray generated by s ( x ). In particular, B ∗ L,s is stable under thefibrewise action of R > .Following [Gil], we call the quotient the simple oriented real blowup :Blo L,s ( X ) = B ∗ L,s / R > . The simple oriented real blowup comes equiped with a natural projectionmap π : Blo L,s ( X ) → X that is an isomorphism outside the zero locus of s .If X is a complex analytic space, and D ⊂ X an effective Cartier divisor,and s the tautological section of O ( D ), then we will write B D , and Blo D ( X )for B O ( D ) ,s and Blo O ( D ) ,s ( X ) respectively. Example 4.1. The blowup ˜ P := Blo ∞ ( P C ) is a compactification of C bya circle at infinity. The details of the following picture will be explained aswe describe the general situation in local coordinates. XPONENTIAL PERIODS AND O-MINIMALITY I 21 r ∞ i ∞ θ ∞ U ε,R ¯ S r For every z ∈ S = { z ∈ C | | z | = 1 } there is a point z ∞ on the boundary:it is the point of intersection of the boundary and the half-ray z · R ≥ . Asystem of open neighbourhoods around z ∞ is given by the sets U ε,R = { w ∈ C | | w | > R and | arg( w ) − arg( z ) | < ε }∪ { w ∞ | | arg( w ) − arg( z ) | < ε } for small ǫ and positive real R .The closure of the set S r = { z ∈ C | ℜ ( z ) ≥ r } is given by the union of S r and the half-circle { z ∞ | ℜ ( z ) ≥ } .Suppose that L , . . . , L n are line bundles on X with respective sections s , . . . , s n , and put L = L ⊗ · · · ⊗ L n with section s ⊗ · · · ⊗ s n . We maythen form the fibre productBlo L ,s ( X ) × X · · · × X Blo L n ,s n ( X )which naturally maps to Blo L,s ( X ). Definition 4.2. Let X be a smooth analytic space, and let D ⊂ X be asimple normal crossings divisor. Denote the (smooth) irreducible compo-nents of D by D , . . . , D n . The oriented real blowup of X in D , denoted byOBl D ( X ) is the fibre productBlo D ( X ) × X · · · × X Blo D n ( X ) . Note that OBl D ( X ) comes with a natural projection map to X .One topological intuition for OBl D ( X ) is the complement of a tubularneighbourhood of D in X . We now make this picture precise by a descriptionin local coordinates.Consider a domain U in C n and D = D ∪ . . . ∪ D m the union of thefirst m coordinate hyperplanes (intersected with U ). In that case we havethe following explicit description of OBl D ( U )(2) { ( z , . . . , z n , w , . . . , w m ) ∈ C n × ( S ) m | ( z , . . . , z n ) ∈ U, z i w − i ∈ R ≥ for 1 ≤ i ≤ m } and π is the projection ( z , . . . , z n , w , . . . , w m ) ( z , . . . , z n ). In particular,it is a C ∞ -manifold with corners. Local coordinates are defined byOBl D ( U ) → R m ≥ × ( S ) m × C n − m (3) ( z , . . . , z n , w , . . . , w m ) (cid:18) z w , . . . , z m w m , w , . . . , w m , z m +1 , . . . , z n (cid:19) . In particular, this gives the blow-up the the structure of a manifold withcorners. As a consequence, we obtain the following result. Proposition 4.3. Let k ⊂ C be a field which is algebraic over k = k ∩ R .Let X be a smooth algebraic variety over k and let D ⊂ X be a simplenormal crossings divisor. Then the oriented real blowup OBl D ( X an ) cannaturally be endowed with a structure of k -semi-algebraic C ∞ -manifold withcorners (see Definition 3.1) in such a way that the natural projection map π : OBl D ( X an ) → X an is morphism of k -semi-algebraic C ∞ -manifolds withcorners.Proof. Without loss of generality k = ˜ k is real closed and k = ¯ k alge-braically closed. Let ( ¯ X, ¯ D ) be a good compactification of the log pair( X, D ). It suffices to prove the proposition for ( ¯ X, ¯ D ) because OBl D ( X an )is the preimage of X an in OBl ¯ D ( ¯ X ). In other words, without loss of gener-ality X an is compact.Without loss of generality X is connected. By definition, for every point x ∈ X , there is a Zariski-open neighbourhood U x and an ´etale map p : U x → A d (with d = dim X ) such that p ( x ) = 0 and D ∩ U x = p − ( { z · · · z m = 0 } ).By the semi-algebraic implicit function theorem, the map p an is invertible onan open ball B x around 0 in C d . Let V x = p − ( B x ) ⊂ X an . The coordinatefunctions z , . . . , z m are both holomorphic and k -semi-algebraic. Hence thepreimage π − ( V x ) ⊂ OBl D ( X an )has the shape described after Definition 4.2. The map (3) defines a chart.More precisely, we also need to cover S ⊂ R by finitely many semi-algebraiccharts. As X an is compact, finitely many of the V x suffice to cover X an . Thetransition maps are C ∞ and k -semi-algebraic because the transition mapsbetween the p ( V x ) are holomorphic and k -semi-algebraic. (cid:3) Lemma 4.4. The construction of the oriented real blowup is functorial: Let X and X be analytic spaces, and let D i ⊂ X i be a simple normal crossingsdivisor. Let f : X → X be a morphism, such that f − ( D ) ⊂ D . Thenthere is a natural morphism ˜ f such that the following diagram commutes: OBl D ( X ) OBl D ( X ) X X ff If f is a morphism of smooth algebraic varities, then ˜ f is a C ∞ -morphismof semi-algebraic manifolds with corners.Proof. Compute in local coordinates. (cid:3) XPONENTIAL PERIODS AND O-MINIMALITY I 23 Remark 4.5. In the future, it will often be the case that we start with avariety X that is not complete, and consider the oriented real blow-up ofthe boundary divisor X ∞ of a completion ¯ X of X . In such a situation, wewill write B ¯ X ( X ) instead of OBl X ∞ ( ¯ X ). Remark 4.6. It is not clear to us whether OBl D ( X ) is affine as semi-algebraic C -manifold with corners. In other words, does there exist a semi-algebraic C -embedding of OBl D ( X ) into R n ? Compare with Remark 3.3.5. Naive exponential periods Let k ⊂ C , k = k ∩ R and assume that k is algebraic over k , see thediscussion in Section 1.1. Recall from Definition 0.2 the notion of a naiveexponential period. We denote P nv ( k ) the set of naive exponential periods.Let ˜ P denote the real oriented blow-up of P at the point at infinity, seeExample 4.1.5.1. Examples of integrals. We first consider some instructive examples. Example 5.1. Let G = [1 , ∞ ) ⊂ C , f = z , ω = d z . Consider Z G e − f ω = Z ∞ e − t d t = Z − e − s s d s. It does not converge. Indeed, the image f ( G ) = (0 , 1] is not closed, hence f : G → C is not proper. The properness condition in the definition of anaive exponential period was added to exclude cases like this. Example 5.2. Once again let G = [1 , ∞ ) ⊂ C , f = z , but ω = z . Asin the previous example, the data does not satisfy the definition of a naiveexponential period because f : G → C is not proper. However, this time Z G e − f ω = Z ∞ e − t t d t = Z e − s d s converges. It can be understood as a naive exponential period with G ′ =[0 , f ′ = z , ω ′ = d z . Example 5.3. Let s ∈ S with ℜ ( s ) > 0. Consider the half ray G s = { rs | r ≥ } , f = z , ω = d z . If s = 1, this data does not satisfy the definitionof a naive exponential period because f ( G s ) = G s does not have boundedimaginary part. Nevertheless, Z G s e − f d z = Z ∞ e − rs s d r = − e − rs (cid:12)(cid:12)(cid:12) ∞ = 1converges and is obviously an exponential period. Note that it is independentof s . Actually, G s defines a class in H rd1 ( A , { } ; Z ), see Section 6.1 below,because its closure in ˜ P is contained in B ◦ = B ◦ P ( A , id). The homologyclass is independent of s (fill in the triangle between G and G s , the thirdedge is in ∂B ◦ ). The period integral only depends on the homology class,hence the independence follows from the abstract theory as well. We do notallow G s in our definition of a naive exponential period, but the same numbercan be obtained as a naive exponential period for G . This is a generalfeature, see [CH20, Proposition 11.4]. In Definition 5.4 we will introducethe notion of a generalised naive exponential period which allows all G s . General properties.Definition 5.4. A generalised naive exponential period over k is a complexnumber of the form Z G e − f ω where G ⊂ C n is a pseudo-oriented closed k -semi-algebraic subset, ω is arational algebraic differential form on A nk that is regular on G and f is arational function on A nk such that f is regular and proper on G and, moreover,the closure of f ( G ) in ˜ P is contained in B ◦ = C ∪ { s ∞ | s ∈ S , ℜ ( s ) > } .We denote P gnv ( k ) the set of generalised naive exponential periods.We are going to show in Corollary 5.11 that these generalised naive expo-nential periods converge absolutely. For the rest of this section we assumeabsolute convergence. Lemma 5.5. Naive exponential periods are generalised naive exponentialperiods.Proof. The condition f ( G ) ⊂ S r,s implies f ( G ) ⊂ B ◦ . (cid:3) Lemma 5.6. The sets P nv ( k ) and P gnv ( k ) are ¯ k -algebras. Moreover, P nv ( k ) = P nv (¯ k ) and P gnv ( k ) = P gnv (¯ k ) .Proof. The arguments are the same for both notions. We formulate it fornaive exponential periods.For the first statement we use the same argument as for f = 0, see [HMS17,Proposition 12.1.5]:We give the argument for the second. Let L/k be a finite subextensionof ¯ k/k . Since k is algebraic over k , the extension L/L with L = L ∩ R isalso algebraic, for every finite extension L/k . Hence, P nv (¯ k ) = S L/k P nv ( L )where L runs through all finite subextensions of ¯ k/k . Thus it suffices toshow that P nv ( k ) = P nv ( L ) for L/k finite.We view A nL → Spec( L ) → Spec( k ) as an affine k -variety contained in A n +1 k . We call it ˜ A . Then ˜ A × k C = [ σ : L → C A n C where σ runs through all embeddings of L into C fixing k . If R G e − f ω isa naive exponential period over L , then f and ω are defined over k whenviewed on ˜ A ⊂ A n +1 . The extension L /k is algebraic, hence every L -semialgebraic set is also k -semialgebraic. (cid:3) In particular, we can move between k , ¯ k , ¯ k ∩ R and k = k ∩ R with-out changing the set of naive exponential or generalised exponential naiveperiods. Remark 5.7. The assumption G ⊂ C n = ( A n ) an is surprising when com-paring to the literature on ordinary periods. Most period references workwith semi-algebraic G ⊂ R n . The two points of view are not equivalenteven though of course C n ∼ = R n as semi-algebraic manifolds. We work with f ∈ k ( z , . . . , z n ) and ω ∈ Ω dk ( z ,...,z n ) /k . Simply replacing C by R in the XPONENTIAL PERIODS AND O-MINIMALITY I 25 definition would eliminate all non-real periods (at least if we assume k ⊂ R as we may by the above). In the case of ordinary periods, a complex num-ber is a period if and only if its real and imaginary part can be written as R G ω with G ⊂ R n and ω ∈ Ω dk ( z ,...,z n ) /k . We cannot show the same simplecharacterisation in the exponential case and it is very likely false. Lemma 5.8. Let k = k ⊂ R . The following are equivalent for α ∈ C :(1) The number α is a naive exponential period over k .(2) It can be written as α = Z G e − f ω with G ⊂ R n a pseudo-oriented closed k -semi-algebraic subset ofdimension d , f ∈ k ( i )( z , . . . , z n ) regular on G such that f | G : G → C is proper with image contained in S r,s and ω ∈ Ω dk ( i )( z ,...,z n ) /k ( i ) isregular on G .(3) Its real and imaginary part can be written as ℜ ( α ) = Z G (cid:16) cos( f )e − f ω + sin( f )e − f ω (cid:17) ℑ ( α ) = Z G (cid:16) − sin( f )e − f ω + cos( f )e − f ω (cid:17) with G ⊂ R n a pseudo-oriented closed k -semi-algebraic subset of di-mension d , f , f ∈ k ( z , . . . , z n ) regular on G such that f | G , f | G : G → R are proper, f ( G ) is bounded from below, f ( G ) is bounded, and ω , ω ∈ Ω dk ( z ,...,z n ) /k regular on G .Moreover, f , f in (3) are the real and imaginary parts of f in (2), respec-tively, and similarly for ω , ω . Finally, α is a generalised naive exponentialperiod if and only if it can be written as in (2) with f ( G ) ⊂ B ◦ .Proof. Let G, f, ω as in the definition of a naive exponential period. Bydefinition G ⊂ C n with coordinates z , . . . , z n . By sending a complex num-ber to its real and imaginary part we view G as a real subset G ′ of C n with coordinates x , y , x , y , . . . , x n , y n . Let Σ : C n → C n be given by( x , y , . . . , x n , y n ) ( x + iy , . . . , x n + iy n ). By definition Σ( G ′ ) = G , com-patible with the pseudo-orientation. Put f ′ = Σ ∗ ( f ) and ω ′ = Σ ∗ ( ω ). Thenby the transformation rule Z G ′ e − f ′ ω ′ = Z G e − f ω. Note that f ′ and ω ′ are defined over k ( i ). This shows that (1) implies (2).Conversely, a number of the form in (2) is by definition a naive exponentialperiod over k ( i ). By Lemma 5.6 it is also a naive exponential period over k , so (2) implies (1). Let G, f, ω as in (1). We put f = f + if and ω = ω + iω and compute e − f ω . The conditions on f and ω are equivalent tothe conditions on f , f and ω , ω . So properties (2) and (3) are equivalent.The final claim follows as the equivalence proof of (1) and (2). (cid:3) Convergence and definability. The conditions on our domain ofintegration can be reformulated. Lemma 5.9. Let f : A n → P be a rational function over k and let G ⊂ C n be closed a semi-algebraic set such that f is regular and proper on G . Let ω be a rational differential form on A n over k . Let X ⊂ P n be the complementof the polar loci of f and ω , ¯ X a good compactification of X such that f extends to ¯ f : ¯ X → P . Let ¯ G be the closure of G in the real oriented blow-up B ¯ X ( X ) of ¯ X at the divisor at infinity, see Remark 4.5, and G ∞ = ¯ G r G .(The case G ∞ = ∅ is allowed.)Then f extends to a semi-algebraic C ∞ -map ˜ f : B ¯ X ( X ) → ˜ P of compactsemi-algebraic C ∞ -manifolds with corners with boundary mapping G ∞ to ∂ ˜ P . Moreover,(1) (Naive exponential periods) f ( G ) ⊂ S r,s for some r, s if and only if ˜ f ( G ∞ ) ⊂ { ∞} .(2) (Generalised naive exponential periods) f ( G ) ⊂ B ◦ if and only if ˜ f ( G ∞ ) ⊂ B ◦ .Proof. By definition of X , we have ¯ f − ( ∞ ) ⊂ ¯ X r X . By Lemma 4.4 weget an induced C ∞ -morphism of semi-algebraic manifolds with corners ˜ f .Let ( g i ) i ≥ be a sequence in G converging to g ∈ ¯ G . Assume g ∈ G ∞ . Wehave g ∈ ∂B ¯ X ( X ) because G ⊂ X an is closed. In particular, the image of g in ¯ X an is in the complement of X an .We claim that ˜ f ( g ) / ∈ C . Assume ˜ f ( g ) ∈ C ⊂ ˜ P . Note that lim ˜ f ( g i ) =˜ f ( g ) by continuity. As f is proper, f ( G ) ⊂ C is closed. All ˜ f ( g i ) are in f ( G ), hence so is ˜ f ( g ). Let D ⊂ C be a closed disk around ˜ f ( g ). It iscompact, hence so is its preimage E := ˜ f | − G ( D ) ⊂ G . There is N ≥ f ( g i ) ∈ D for all i ≥ N . Hence their preimages g i are in E . As E iscompact, the limit point g is in E , in particular in G . This is a contradiction.We have shown that ˜ f ( G ∞ ) ⊂ ∂ ˜ P .Note that f ( G ) = ˜ f ( ¯ G ). Hence (2) is obvious. For (1) note that ¯ S r,s ∩ ∂ ˜ P = { ∞} . Hence f ( G ) ⊂ S r,s implies ˜ f ( G ∞ ) ⊂ { ∞} . Conversely,consider a small open neighbourhood U of 1 ∞ in ˜ P . It intersects C insidesome strip of the form S r,s . As ¯ G is compact, so is G ′ = ¯ G r ˜ f − ( U ). Theimage f ( G ′ ) is compact, so bounded in C . By enlarging r and s , we ensurethat both f ( G ′ ) and f ( G ) ∩ U are contained in the same strip. (cid:3) Lemma 5.10. Let f and G be as in the definition of a generalised naiveexponential period. Let ¯ G be the compactification of G as in Lemma 5.9 and G ∞ = ¯ G r G . Let c be a rational function on A n which is regular on G . Theextension of e − f c by on G ∞ yields a continuous function on ¯ G .Proof. Let ( g i ) i ≥ be a sequence in G converging to g ∈ G ∞ . Then | e − f ( g i ) | = e −ℜ ( f ( g i )) → f ( g i ) tends to ˜ f ( g ) ∈ ∂B ◦ . The function c has at worst a pole in g ,but the exponential factors decays faster than | c ( g i ) | grows. In totallim i →∞ | e − f ( g i ) c ( g i ) | = 0 . (cid:3) XPONENTIAL PERIODS AND O-MINIMALITY I 27 Corollary 5.11. Assume that G, f, ω define a generalised naive exponentialperiod. Then Z G e − f ω converges absolutely.Proof. We apply Theorem 3.22 to ¯ G ⊂ B ¯ X ( X ) as in Lemma 5.9. It iscompact. By Lemma 5.10, the C ∞ -form e − f ω on G extends to a continuousform on ¯ G . This is enough. (cid:3) Theorem 5.12. If a number α ∈ C is a naive exponential period over k ,then its real and imaginary part are up to signs volumes of compact subsets S ⊂ R n defined in the o-minimal structure R sin , exp = ( R , <, , , + , · , sin | [0 , , exp) with parameters from k .Proof. By Lemma 5.6, we may assume k = k . We use the characterisationof naive exponential periods given in parts (2) and (3) of Lemma 5.8. Thus α = R G e − f ω with ℜ ( α ) = Z G (cid:16) cos( f )e − f ω + sin( f )e − f ω (cid:17) , ℑ ( α ) = Z G (cid:16) − sin( f )e − f ω + cos( f )e − f ω (cid:17) where G ⊂ R n is closed and k -semi-algebraic of dimension d carrying apseudo-orientation, f , f ∈ k ( z , . . . , z n ) are regular and proper on G , f ( G )is bounded from below, f ( G ) is bounded, and ω , ω ∈ Ω dk ( z ,...,z n ) /k .We want to apply Theorem 3.22. Again we apply it to the compact k -semialgebraic C ∞ -manifold with corners B ¯ X ( X ) of Lemma 5.9 and theclosure ¯ G of G B ¯ X ( X ). It is compact and a semi-algebraic subset of B ¯ X ( X ),hence definable in R sin , exp . The forms ℜ (e − f ω ) and ℑ (e − f ω ) are definableon G ∞ because they vanish identically. Hence it remains to verify the defin-ability on the affine G itself. The forms ω and ω are algebraic, in particulardefinable. By assumption f is bounded, hence using Remark 2.14 the func-tion sin( f ) is definable in our o-minimal structure. The same is true forcos( f ) because cos( f ) = sin( f + π/ π is definable in the o-minimalstructure R sin , exp . (cid:3) Remark 5.13. The above argument does not work for generalised naiveexponential periods. It is essential that the imaginary part of f is boundedon G . However, we are going to show (see [CH20, Theorem 13.4]) that everygeneralised naive exponential period is actually a naive exponential period,hence the consequence still applies. Remark 5.14. In contrast to the case of ordinary periods, we do not expectthat all volumes of definable sets in this o-minimal structure are naive expo-nential periods. The above argument only produces very special definablesets: there is no need of nesting exp or sin | [0 , . The Euler number e is defin-able (as exp(1)), hence also e e (as exp(e)). The number e is known to be anexponential period (e.g., R (e s + 1) d s ). However, we do not see an obviousway to write e e as an exponential period. It would be very interesting togive a characterisation of the sets that do occur. The definition of Kontsevich and Zagier. In § exponential period inthe sense of Kontsevich–Zagier is “an absolutely convergent integral of theproduct of an algebraic function with the exponent of an algebraic function,over a real semi-algebraic set, where all polynomials entering the definitionhave algebraic coefficients”. We take this to mean numbers of the form Z G e − f ω where G ⊂ R n is semi-algebraic over ˜ Q = Q ∩ R , f ∈ Q ( z , . . . , z n ), and ω a rational algebraic differential form defined over Q such that the integralconverges absolutely. It is not clear to us if they want dim( G ) = n . Inthis case, there is a prefered orientation from the orientation of R n , in thegeneral case we have to orient G .We have shown that naive and generalised naive exponential periods over Q are absolutely convergent. In particular, a generalised naive exponentialperiod over Q is an exponential period in the sense of Kontsevich–Zagier.What about the converse? Example 5.15. Let G = [1 , ∞ ) ⊂ R , f = iz , ω = z d z . Then Z G e − f ω = Z ∞ t e − it d t = Z ∞ t cos( − t ) d t + i Z ∞ t sin( − t ) d t converges absolutely because sin and cos are bounded by 1. However, thedata does not define a generalised naive exponential period. The interval G is not a cycle for rapid decay homology of ( A , { } ). We do not havelim t →∞ ℜ ( f ( t )) → ∞ on G .Hence: Conjecture 5.16. There are exponential periods in the sense of Kontsevich–Zagier which are not (generalised) naive exponential periods. This is in the spirit of the period conjecture: if a number is not obviouslya period, than it is not. As the example demonstrates, the condition onabsolute convergence only implies that f ( G ) ⊂ ˜ P is contained in C ∪{ s ∞| s ∈ S , ℜ ( s ) ≥ } . The above example uses the boundary point i ∞ . For such f ,the absolute convergence of the integral depends on the choice of ω .We propose the following modification: Definition 5.17. An absolutely convergent exponential period over k is acomplex number obtained as the value of an absolutely convergent integralof the form(4) Z G e − f ω where G ⊂ C n is a pseudo-oriented (not necessarily closed) k -semi-algebraicsubset, ω is a rational algebraic differential form on A nk that is regular on G , f a rational function on A nk regular on G and the closure of f ( G ) in ˜ P iscontained in B ◦ .We denote P abs ( k ) the set of all absolutely convergent exponential periodsover k . XPONENTIAL PERIODS AND O-MINIMALITY I 29 Remark 5.18. The regularity condition for f and ω on G is harmless. Wemay replace G by the open subset G ′ of points in which f and ω are finite.The value of the integral only changes if dim( G − G ′ ) = dim( G ), i.e., if thereis an open U ⊂ G on which f or ω are infinite. The integral R U e − f ω doesnot make sense in this case, so we definitely want to exclude it. Note thatthe condition on f ( G ) excludes Example 5.15 where we have f ( G ) = [ i, i ∞ ]and i ∞ / ∈ B ◦ .We are going to show that every absolutely convergent exponential periodis a generalised naive exponential period. Also for later use, let us be moreprecise. Proposition 5.19. Let α be an absolutely convergent exponential periodover k ⊂ R with domain of integration as in (4) of dimension d . Then thereare: • a smooth affine variety X over k of dimension d , • a simple normal crossings divisor Y ⊂ X , • a closed k -semi-algebraic subset G ⊂ X ( R ) of dimension d such that ∂G = G r G int is contained in Y , • a pseudo-orientation on G , • a morphism f : X k ( i ) → A k ( i ) such that f | G : G → C is proper andsuch that the closure f ( G ) ⊂ ˜ P is contained in B ◦ , • a regular algebraic d -form ω on X k ( i ) ,such that α = Z G e − f ω. Proof. We start with a presentation α = Z G e − f ω with G of dimension d as in the definition of an absolutely convergent ex-ponential period and modify the data without changing the value. In par-ticular, G is equipped with a pseudo-orientation. With the same trick as inLemma 5.8, we may assume that G ⊂ R n = A nk ( R ) is k -semi-algebraic with f, ω algebraic over k ( i ).Let X ⊂ P nk be the Zariski-closure of G . It is an algebraic variety definedover k of dimension d , see the characterisation of dimension in [BCR98,Definition 2.8.1]. Moreover, dim X ( R ) = d as a real algebraic set. Byassumption, f is a rational map on X ,k ( i ) . After replacing X by a blow-upcentered in the smallest subvariety of X defined over k containing the locusof indeterminancy of f , it extends to a morphism f : X ,k ( i ) → P k ( i ) . Byconstruction, G ⊂ X ( R ). Let G := ¯ G ⊂ X an0 be the closure. It is containedin X ( R ) and compact because X an0 is. It inherits a pseudo-orientation from G . Let Y ⊂ X be the union of X , sing and the Zariski closure of ∂G , wherethe boundary is taken inside X ( R ). It has dimension less than d .As the next step, let π : X → X be a resolution of singularities suchthat the preimage Y of Y is a divisor with normal crossings. The map π is an isomorphism outside Y . As Y ⊂ X has codimension at least 1,the intersection G ∩ Y ( R ) has real codimension at least 1 in G . Let G 10 JOHAN COMMELIN, PHILIPP HABEGGER, AND ANNETTE HUBER be the “strict transform” of G in X an1 , i.e., the closure of the preimage of U = G r ( G ∩ Y ( R )). By construction ∂G ⊂ G r π − ( U ) ⊂ Y . Let ω = π ∗ ω . By Lemma 3.16 (4), the set G inherits a pseudo-orientation.Moreover, Z G e − f ◦ π π ∗ ω = Z G e − f ω, where the left hand side converges absolutely because the right hand sidedoes.We claim that after further blow-ups, we can reach X → X preservingthe properties of X , Y , and G such that, in addition, points of G in thepolar locus of ω are contained in the polar locus of f .We first prove the claim. Let X , ∞ be the polar locus of f and X ,ω thepolar locus of ω , i.e., the smallest closed subvarieties over k such that theirbase change to k ( i ) contains the poles of f and ω , respectively. Note that G is disjoint from X , ∞ because f is regular on G and G is contained inthe real points of X .Let x ∈ G be a point such that f is regular, but ω has a pole. Let U be a small compact neighbourhood of x in G in which f is regular. Byassumption, Z U e − f ω converges absolutely. As f is regular on U , the factor e − f and its inverseare bounded. Hence the absolute convergence of the integral is equivalentto absolute convergence of the integral Z U ′ ω . This case already shows up in the case of ordinary periods, see the proof of[HMS17, Lemma 12.2.4]. The argument is due to Belkale and Brosnan in[BB03]. After a blow-up X → X we find holomorphic coordinates suchthat the pull-back ω of ω has the shapeunit × n Y j =1 z e j j d z ∧ · · · ∧ d z n with e j ∈ Z . Absolute convergence is only possible if e j ≥ j , i.e., if ω is regular on U . This finishes the proof of the claim.Let X be the complement of the polar loci of f and ω , Y = X ∩ Y , f and ω the restrictions of f and ω to X , and G = X an ∩ G . The map f : G → ( P ) an is proper, and hence so is f : G → C . The data satisfiesall properties stated in the proposition, with the exception that X is onlyquasi-projective rather than affine. We have X ⊂ P Nk for some N . Let H bethe hypersurface defined by the equation X + · · · + X N = 0. Then P Nk r H isaffine. Note that G ∩ H an = ∅ because G ⊂ P N ( R ) and H ( R ) = ∅ . Hence wemay replace X by X r ( X ∩ H ), making it quasi-affine. Now X is of the form X ′ r V ( s , . . . , s m ) for finitely many s i ∈ O ( X ′ ). Let H ′ = V ( s + · · · + s m ).Note that X ( R ) = ( X r H ′ )( R ). Hence we may replace X by its open subset X ′ r H ′ , making it affine. (cid:3) XPONENTIAL PERIODS AND O-MINIMALITY I 31 Corollary 5.20. The set of absolutely convergent exponential period equalsthe set of generalised naive exponential periods: P gnv ( k ) = P abs ( k ) . Proof. By Corollary 5.11, every generalised naive exponential period is anabsolutely convergent exponential period.Let α be an absolutely convergent exponential period. By the same argu-ment as for naive exponential periods (see Lemma 5.6), we may replace k by k ∩ R . We apply Proposition 5.19. Let X ′ ⊂ X be a dense open affinesubvariety, G ′ = G ∩ X ′ an . As G ⊂ X ( R ) is of full dimension, we havedim( G r G ′ ) < dim( G ), hence the integral does not change when restrictingto the open subset G ′ of G . We replace X , G by X ′ , G ′ . Now X ⊂ A n . Themorphism f : X k ( i ) → A k ( i ) extends to a rational morphism A nk ( i ) → A k ( i ) .The differential form ω on X extends to a rational differential form on A nk ( i ) .This data satisfies the assumptions of the definition of a generalised naiveexponential period. (cid:3) Remark 5.21. It is not clear to us if it is equivalent to restrict to G ⊂ R n of dimension n in the definition of an absolutely convergent exponential pe-riod. We tend to expect that it fails to be true. The analogous statement forordinary periods holds true because they turn out to be volumes of boundedsemi-algebraic sets (see [HMS17, Section 12.2], also [VS15]). We have re-placed this by our Theorem 3.22. Close inspection of the proof only showsthat every naive exponential period (and hence by [CH20, Theorem 13.4]also all absolutely convergent exponential periods) can be written as a Z -linear combinations of numbers of the form Z G e − f d x ∧ · · · ∧ d x n for G ⊂ R n of dimension n , f : G → C continuous with semi-algebraic realand imaginary part. Remark 5.22. We pick up again on Example 5.15. As explained previously,the integral R ∞ e − it d tt converges absolutely, but does not obviously define ageneralised naive period. We concentrate on the real part. Integration byparts gives Z ∞ cos( t ) t d t = cos(1) − Z ∞ sin( t ) t d t = cos(1) − π Z sin( t ) t d t because of the classical identity R ∞ t ) t d t = π . Note that the function sin( t ) t is entire, so there are no convergence issues with the last integral. Thenumbers cos(1) and π/ R sin , exp of Definition 2.13. The same is true for the function sin( t ) t . Hence we havewritten our number as the volume of a set that is definable in R sin , exp . Note,however, that the formula does not give a presentation as an absolutely convergent exponential period. We have Z sin( t ) t d t = Z ℑ (cid:18) e it t (cid:19) d t, but the real part does not converge for the choice G = (0 , f = iz , and ω = d zz . 6. Review of cohomological exponential periods Throughout this section let k ⊂ C be a subfield. All varieties are definedover k .We give the definition of exponential periods following [FJ20] concentrat-ing on the smooth affine case at the moment.6.1. Rapid decay homology. [FJ20, 1.1.1] Given a real number r , let S r = { z ∈ C | ℜ ( z ) ≥ r } . Definition 6.1. Let X be a complex algebraic variety, Y ⊂ X a subvariety, f ∈ O ( X ). The rapid decay homology of ( X, Y, f ) is defined as H rd n ( X, Y, f ) = lim r →∞ H n ( X an , Y an ∪ f − ( S r ); Q ) . For r ′ ≥ r , there is a projection map on relative homology, so this reallyis a projective limit. A direct limit construction using singular cohomologyyields rapid cohomology H n rd ( X, Y, f ). It is dual to rapid decay homology.By [FJ20, 3.1.2], these limits stabilise, so it suffices to work with a single,big enough r . Indeed: Theorem 6.2 (Verdier [Ver76, Corollaire 5.1]) . There is a finite set Σ ⊂ C such that f | f − ( Cr Σ) : f − ( C r Σ) → C r Σ is a fibre bundle. As S r is contractible, this implies that all f − ( S r ) with r sufficiently largeare homotopy equivalent to a fibre of f .There is an alternative description of H rd n ( X, f ) which is better suitedto the computation of periods. It is originally due to Hien and Roucairol,see [HR08]. We follow the presentation of Fres´an and Jossen in [FJ20, Sec-tion 3.5].We fix a smooth variety X and f ∈ O ( X ). Let ¯ X be a good com-pactification, i.e., such that ¯ X is smooth projective, X ∞ = ¯ X r X isa divisor with normal crossing and f extends to ¯ f : ¯ X → P . We de-compose X ∞ = D ∪ D ∞ into simple normal crossings divisors such that¯ f ( D ∞ ) = {∞} and ¯ f : D → P is dominant on all components, i.e., intovertical and horizontal components. Definition 6.3. We denote by π : B ¯ X ( X ) → ¯ X an the real oriented blow-upOBl X ∞ ( ¯ X ), see Definition 4.2. Let ˜ f : B ¯ X ( X ) → ˜ P be the induced map,see Lemma 4.4. We also define B ◦ ¯ X ( X, f ) = B ¯ X ( X ) r (cid:16) π − ( D an0 ) ∪ ˜ f − ( { s ∞ ∈ ˜ P |ℜ ( s ) ≤ } ) (cid:17) ,∂B ◦ ¯ X ( X, f ) = B ◦ ¯ X ( X, f ) r X an = B ◦ ¯ X ( X, f ) ∩ ˜ f − ( { s ∞ ∈ ˜ P |ℜ ( s ) > } ) . XPONENTIAL PERIODS AND O-MINIMALITY I 33 We are going to omit the subscript ¯ X as long as it does not cause confu-sion.At this point we only consider them as topological spaces. In fact B ◦ ¯ X ( X, f )is a semi-algebraic C ∞ -manifold with corners. Remark 6.4. Our definition of B ◦ ( X, f ) does not agree with B ◦ , FJ = X an ∪ ˜ f − ( { s ∞ ∈ ˜ P |ℜ ( s ) > } )as defined by Fres´an–Jossen [FJ20, Section 3.5] and the earlier rapid decayliterature. The two definitions differ if D and D ∞ intersect. They agree inthe curve case where D ∩ D ∞ = ∅ is automatic. If the intersection is non-empty, then B ◦ , FJ is not a manifold with corners whereas B ◦ ( X, f ) alwaysis. The issue is also addressed in [MH17, Section 2]. Proposition 6.5 ([FJ20, Proposition 3.5.2]) . Let X be a smooth varietyover k . For sufficiently large r , the inclusion induces natural isomorphisms H n ( X an , f − ( S r ); Q ) ∼ = H n ( B ( X ) , ˜ f − ( S r ); Q ) ∼ = H n ( B ◦ ( X, f ) , ∂B ◦ ( X, f ); Q ) . In particular, H rd n ( X, f ) ∼ = H n ( B ◦ ( X, f ) , ∂B ◦ ( X, f ); Q ) . Proof. Their proof is correct with the modified notion of B ◦ ( X, f ). (cid:3) Recall that S ∗ ( M ) denotes the complex of Q -linear combinations of C -simplices for a C p -manifold with corners M . Recall also Definition 1.1. Itcomputes singular cohomology by Theorem 1.3. Definition 6.6. Let X be a smooth variety, f ∈ O ( X ). Choose a goodcompactification ¯ X . We put S rd ∗ ( X, f ) = S ∗ ( B ◦ ¯ X ( X, f )) /S ∗ ( ∂B ◦ ¯ X ( X, f )) . Remark 6.7. Fres´an and Jossen work with piecewise C ∞ -simplices instead,see [FJ20, Section 7.2.4]. We opt for the slightly more complicated notionof C -simplices as opposed to C ∞ -simplices because they are well-suited forworking with our semi-algebraic sets.6.2. Twisted de Rham cohomology: the smooth case. Let X/k bea smooth variety, f ∈ O ( X ). We define a vector bundle with connection E f = ( O X , d f ) with d f (1) = − df . The de Rham complex DR( E f ) has thesame entries as the standard de Rham complex for X , but with differentialΩ p → Ω p +1 given by dω − df ∧ ω . Definition 6.8. Let ( X, f ) be as above. We define algebraic de Rhamcohmology H ∗ dR ( X, Y, f ) of ( X, f ) as hypercohomology of DR( E f ).If X is affine, this is nothing but cohomology of the complex R Γ dR ( X ) := [ O ( X ) d f −→ Ω ( X ) d f −→ . . . ] . The definition needs to be extended to the relative cohomology of singularvarieties. We first consider a special case. Let X be smooth and Y ⊂ X asimple divisor with normal crossings. Let Y • → Y be the ˇCech-nerve of thecover of Y by the disjoint union of its irreducible components, see Section 1.It is a smooth proper hypercover. In particular, H n ( Y an • , Z ) = H n ( Y an , Z ). Definition 6.9. Let X be a smooth variety, Y ⊂ X a divisor with simplenormal crossings. We define algebraic de Rham cohomology H ∗ dR ( X, Y, f ) of( X, Y, f ) as hypercohomology of Cone (cid:0) π ∗ DR( E f | Y • ) → E f (cid:1) [ − The period isomorphism. Hien and Roucairol established the exis-tence of a canonical isomorphism H n rd ( X, f ) ⊗ Q C → H n dR ( X, f ) ⊗ k C for smooth affine varieties X see [HR08, Theorem 2.7]. It is also explainedand extended to the relative case for any variety X and subvariety Y byFres´an and Jossen, see [FJ20, Theorem 7.6.1]. We refer to it as the periodisomorphism . It induces a period pairing (5) h− , −i : H n dR ( X, Y, f ) × H rd n ( X, Y, f ) → C . Definition 6.10. Let X be a variety, f ∈ O ( X ), Y ⊂ X a closed subvariety, n ∈ N . The elements in the image of the period pairing (5) are called the (cohomological) exponential periods of ( X, Y, f, n ).We denote P coh ( k ) the set of cohomological exponential periods for vary-ing ( X, Y, f, n ) over k . We denote P log ( k ) the subset of cohomological expo-nential periods for varying ( X, Y, f, n ) such that ( X, Y ) is a log-pair.The construction of the period map is non-trivial. Fortunately, we onlyneed its explicit description in a special case. Definition 6.11. Let X be smooth affine. We define a pairingΩ n ( X ) × S rd n ( X an , f ) → C by mapping ( ω, σ ) to Z σ e − f ω an . Lemma 6.12. The pairing is well-defined and induces a morphism of com-plexes Ω ∗ ( X, f ) → Hom( S rd ∗ ( X an , f ) , C ) . On cohomology it induces the pairing (5).Proof. Let ω ∈ Ω n ( X ), σ an n -dimensional C -simplex in S rd n ( X an , f ). Thesmooth form ω an on X an defines a smooth form e − f ω an on B ◦ ( X, f ). (Notethat e − f ω an vanishes to any order on ∂B ◦ ( X, f ), so it can be extended by0 to a neighbourhood of the boundary). Hence the integral is well-defined.The compatibility with the boundary map translates as Z σ e − f d f ω an = Z ∂σ e − f ω an which holds by Stokes’s formula (see Theorem 1.4) because d f ω = dω − df ∧ ω .The construction is the one of [FJ20, Chapter 7.2.7], only with our S ∗ ( X )instead of their complex, see Remark 6.7. (cid:3) By taking double complexes, this extends to general X and Y . We willdiscuss this in detail in [CH20, Section 10]. At this point, we handle thesimplest case. XPONENTIAL PERIODS AND O-MINIMALITY I 35 Example 6.13. Let X be a smooth affine variety, Y ⊂ X a smooth closedsubvariety, f ∈ O ( X ). Then relative twisted de Rham cohomology is com-puted by the complex R Γ dR ( X, Y, f ) = Cone (Ω ∗ ( X ) → Ω ∗ ( Y )) [ − (cid:2) Ω ( X ) → Ω ( X ) ⊕ Ω ( Y ) → Ω ( X ) ⊕ Ω ( Y ) → . . . (cid:3) with differential induced by d f and restriction. Its rapid decay homology iscomputed by the complex S rd ∗ ( X, Y, f ) = Cone (cid:16) S rd ∗ ( Y, f ) → S rd ∗ ( X, f ) (cid:17) . Explicitly: let ¯ X be a good compactification of X such that f extends to amorphism on ¯ X with target P k and such that the closure ¯ Y of Y in ¯ X is agood compactification as well. ThenCone( S rd ∗ ( Y, f ) → S rd ∗ ( X, f )) =[ S rd0 ( X, f ) ← S rd1 ( X, f ) ⊕ S rd0 ( Y, f ) ← S rd2 ( X, f ) ⊕ S rd1 ( Y, f ) ← . . . ] . Let σ be a cycle in S rd n ( X, Y ), i.e., a chain σ X on X such that ∂σ X = σ Y issupported on Y . In the second incarnation, we identify it with ( σ X , − σ Y ).Let ω be cocycle in R Γ dR ( X, Y, f ), i.e., a pair of differential forms ( ω X , ω Y ) ∈ Ω n ( X ) ⊕ Ω n − ( Y ) such that dω X = 0, dω Y = ω X | Y . Their period is h [ ω ] , [ σ ] i = Z σ X ω X − Z σ Y ω Y . Triangulations We fix a real closed field ˜ k ⊂ R and work with semi-algebraic sets of R N defined over ˜ k . We expect that everything holds in general for o-minimalstructures, but we do not need this for our application. We use the set-up of[vdD98, Chapter 8] for complexes. It is not completely standard, but veryconvenient for us.Let n ∈ N . Let a , . . . , a n ∈ ˜ k N be affine independent. The open n -simplex defined by these vectors is the set σ = ( a , . . . , a n )= ( n X i =0 λ i a i ∈ R N : for all i we have λ i > λ + · · · + λ n = 1 ) . We fix the orientation given by dλ ∧ · · · ∧ dλ n . The closure of σ is denotedby [ a , . . . , a n ] and obtained by relaxing to λ i ≥ a , . . . , a n ] a closed k -simplex . The points a , . . . , a n are uniquelydetermined by [ a , . . . , a n ] and thus by σ . As usual, a face of σ is a sim-plex spanned by a non-empty subset of { a , . . . , a n } . Then [ a , . . . , a n ] is adisjoint union of faces of σ . We write τ < σ if τ is a face of σ and τ = σ .A finite set K of simplices in R N is called a complex if for all σ , σ ∈ K the intersection σ ∩ σ is either empty or the closure of common face τ of σ and σ . Van den Dries’s definition does not ask for τ to lie in K . So the polyhedron spanned by K | K | = [ σ ∈ K σ may not be a closed subset of R N . We call K a closed complex if | K | isclosed or equivalently, if for all σ ∈ K and all faces τ of σ , we have τ ∈ K .Note that S σ ∈ K σ is a disjoint union, this is an advantage of working with“open” simplices. We write K for the complex obtained by taking all facesof all simplices in K . Note that K is ˜ k -semi-algebraic. Definition 7.1. Let M be a ˜ k -semi-algebraic C -manifold with corners, A ⊂ M be a ˜ k -semi-algebraic subset. A semi-algebraic triangulation of A is a pair ( h, K ) where K is a complex and where h : | K | → A is a ˜ k -semi-algebraic homeomorphism. We say that it is globally of class C , if h extendsto a C -map on an open neighbourhood of | K | .Let B ⊂ A be a ˜ k -semi-algebraic subset. We say that ( h, K ) is compatiblewith B if Φ( B ) is a union of members of K . Remark 7.2. Note that there are weaker definitions of C -triangulationsin the literature, see for example Remark 9.2.3(a) [BCR98] or [Shi97, Chap-ter II]. However, Ohmoto-Shiota have shown the existence of semi-algebraictriangulations globally of class C for locally closed semi-algebraic subsetsof R N , see [OS17]. Czapla-Paw lucki [CP18] show even stronger regularityproperties (that we do not need) in the o-minimimal setting.7.1. Existence of triangulations. Our aim is to triangulate semi-algebraicmanifolds with corners, see Definition 3.1. Lemma 7.3. Let X a compact ˜ k -semi-algebraic C -manifold with cornerswith atlas ( φ i : U i → V i | i = 1 , . . . , N ) . Then there are ˜ k -semi-algebraicfunctions of class C f , . . . , f m : X → [0 , such that for every j there is i • such that the support of f j is contained in U i , • there is an open subset W j ⊂ U i on which f j is identically ,and, moreover, the W j are a cover of X .Proof. For each P ∈ X we fix a chart U i containing P . Each V i is an opensubset of some R n i × R m i ≥ . There is an open ball B in R n i + m i centeredat φ i ( P ) of radius r > φ i ( P ) ∈ B ∩ R n i × R m i ≥ ⊂ V i . Let f ′ P : R n i + m i → [0 , 1] be a ˜ k -semi algebraic C -function that is identically 1on the open ball of radius r/ φ i ( P ) and with support containedcompletely in B . We denote by W P the preimage in U i of the said ball ofradius r/ f P the composition f ′ P ◦ φ i extended by zero on X r V i . As X is compact, there are finitely many P , . . . , P m such that W P ∪· · ·∪ W P m = X . The lemma follows with W j = W P j and f j = f P j . (cid:3) Proposition 7.4. Let X be a compact ˜ k -semi-algebraic C -manifold withcorners, A , . . . , A M semi-algebraic subsets of X . Then there is a ˜ k -semi-algebraic triangulation of X compatible with A , . . . , A M that is globally ofclass C . XPONENTIAL PERIODS AND O-MINIMALITY I 37 Proof. As a first step, we ignore the regularity issue and consider X as a com-pact ˜ k -semi-algebraic space. By [Rob83, Theorem 1] it is affine. By [BCR98,Theorem 9.2.1] or [vdD98, Theorem 8.2.9] it admits a ˜ k -semi-algebraic tri-angulation ( h, K ) of X compatible with A , . . . , A M . Using [OS17], we aregoing to construct a ˜ k -semi-algebraic homeomorphism Φ : | K | → | K | whichrespects all simplices and such that h ◦ Φ is C . This new triangulation hasthe required properties.In detail: Let φ , . . . , φ N and f , . . . , f m be as in Lemma 7.3. The map f j φ i : X → R d is well-defined and ˜ k -semi-algebraic. We apply the “panelbeating” of [OS17, Corollary 3.3] to the maps g j = ( f j φ i ) ◦ h : | K | → X → R d . Note that they formulate the results in the semi-algebraic setting, but theypoint out that the proof is written in a way that it also applies in othersettings such as ours.This gives us a ˜ k -semi-algebraic homeomorphism Φ : | K | → | K | respect-ing all simplices such that all g j ◦ Φ are C . We claim that h ◦ Φ is C . As W , . . . , W m cover X , it suffices to check the claim after restricting to thepreimage of some W j . By definition, a map is C if its composition with φ i is. This is the case because f j φ i = φ i on W j and g j is C . (cid:3) Remark 7.5. We briefly sketch what the C -triangulation result of Ohmoto–Shiota [OS17] boils down to in the 1-dimensional setting. This suffices forthe context considered in Section 8. Say γ : [0 , → R N is a continuous˜ k -semi-algebraic map, then there exist 0 = t < · · · < t m = 1 in ˜ k such thatall γ | ( t i ,t i +1 ) are C . Thus γ is represented in homology by a chain of pathsthat are C on (0 , γ is such a path. For ℓ ∈ Z suffientlylarge the right-sided derivative of t γ ( t ℓ ) at t = 0 exists and vanishes; thisfollows from asymptotic behavoir of semi-algebraic functions see [vdDM96,4.12]. Extending t γ ( t ℓ ) to the left with value γ (0) yields a C -functionon ( −∞ , t = 1 by reparametrizing with1 − (1 − t ) ℓ and extending to the right with value γ (1).7.2. A deformation retract. We are going to show that, up to deforma-tion, a complex K can be identified with a closed complex. The argumentsare similar to the ones in [vdD98, Chapter 8 (3.5)]. Compare also Friedrich’s[HMS17, Proposition 2.6.8] and its proof.If σ is a simplex in R N , then b ( σ ) denotes its barycenter. Let K ⊂ R N be a complex. We denote by β ( K ) its barycentric subdivision as defined in[vdD98, Chapter 8 (1.8)]. Note that | K | = | β ( K ) | .We define the closed core of a complex K ascc( K ) = { σ ∈ K | K contains all faces of σ } . Then cc( K ) is a subcomplex of K . It is a closed complex by definition.But it can be empty: consider a complex consisting of a single simplex ofpositive dimension. This problem is remedied by passing to the barycentricsubdivision. More precisely, if K is non-empty, then cc( β ( K )) is non-empty.Indeed, the barycenter b ( σ ) of σ ∈ K defines a face ( b ( σ )) of β ( K ); it mustlie in cc( β ( K )). Finally, note that if L is a subcomplex of K , then β ( L ) ⊂ β ( K ) andcc( L ) ⊂ cc( K ), so we have cc( β ( L )) ⊂ cc( β ( K )). Proposition 7.6. Let K be a complex. There exists a ˜ k -semi-algebraicretraction r : | K | = | β ( K ) | → | cc( β ( K )) | with the following properties.(i) For each x ∈ | K | the half open line segment [ x, r ( x )) is contained inthe simplex of β ( K ) containing x .(ii) The map H ( x, t ) = (1 − t ) x + tr ( x ) is a ˜ k -semi-algebraic strong deformation retraction H : | K | × [0 , →| K | onto | cc( β ( K )) | . We use a variation of the arguments found in § 3, Chapter 8 [vdD98]. Proof. Let b = b ( σ ) be a vertex of β ( K ). As in loc. cit. we define a continuoussemi-algebraic function λ σ : | K | = | β ( K ) | → [0 , | τ | if b is not a vertex of τ ∈ β ( K ) and equals the barycen-tric coordinate with respect to b if it is.Let us define furthermore Λ( x ) = X σ ∈ K λ σ ( x ) . We claim that Λ( x ) > x ∈ | K | . Indeed, x is contained in asimplex ( b ( σ ) , . . . , b ( σ n )) of β ( K ); here σ < · · · < σ n are open simplices of K and σ n ∈ K . In particular, λ σ n ( x ) > 0. Thus the contribution comingfrom σ n to the sum Λ( x ) is strictly positive. As all other contributions arenon-negative we find Λ( x ) > 0, as desired.We are ready to define r ( x ) for x ∈ | K | as r ( x ) = P σ ∈ K λ σ ( x ) b ( σ )Λ( x ) . Thus r : | K | → R m is ˜ k -semi-algebraic and continuous.Let us verify that r ( | K | ) ⊂ | cc( β ( K )) | . Say x ∈ | K | and let σ , . . . , σ n beas before. Say σ ∈ K . We recall that λ σ ( x ) > σ is among { σ , . . . , σ n } . Let σ i < · · · < σ i k = σ n be those among the σ , . . . , σ n thatlie in K . So r ( x ) = P kj =0 α j b ( σ i j ) with coefficients α j ∈ [0 , 1] such that P kj =0 α j = 1. Observe that α j > x ∈ ( b ( σ ) , . . . , b ( σ n )). Thus r ( x ) ∈ ( b ( σ i ) , . . . , b ( σ i k )). Finally, b ( σ i j ) ∈ σ i j ∈ K for all j . Therefore, β ( K ) contains all faces of the simplex ( b ( σ i ) , . . . , b ( σ i k )) which must thusbe an element of cc( β ( K )). We conclude r ( x ) ∈ | cc( β ( K )) | . So the targetof r is cc( β ( K )), as claimed.Moreover, ( b ( σ i ) , . . . , b ( σ i k )) is a face of ( b ( σ ) , . . . , b ( σ n )) ∈ β ( K ), henceby convexity the ray [ x, r ( x )) is in the simplex of β ( K ) containing x .We now verify that r is a retraction. We still assume x ∈ ( b ( σ ) , . . . , b ( σ n ))as above. Note that x = P σ ∈ K λ σ ( x ) b ( σ ) and P σ ∈ K λ σ ( x ) = 1. If λ σ ( x ) > XPONENTIAL PERIODS AND O-MINIMALITY I 39 σ ∈ K , then σ is among σ , . . . , σ n . Hence n X i =0 λ σ i ( x ) b ( σ i ) = x and n X i =0 λ σ i ( x ) = 1 . Now suppose x ∈ | cc( β ( K )) | . By definition, β ( K ) contains all faces of( b ( σ ) , . . . , b ( σ n )). In particular, b ( σ i ) ∈ | K | and hence σ i ∈ K for all i . SoΛ( x ) = 1 and r ( x ) = x . In particular, r is a retraction.Keeping the notation above for x ∈ ( b ( σ ) , . . . , b ( σ n )) ∈ β ( K ), we find forall t ∈ (0 , 1) that(1 − t ) x + tr ( x ) = 1Λ( x ) n X i =0 ((1 − t )Λ( x ) + tw σ i ( x )) λ σ i ( x ) b ( σ i ) , here w σ i is constant 1 if σ i ∈ K and constant 0 else wise. Each factor inthe sum on the right is strictly positive, which implies (1 − t ) x + tr ( x ) ∈ ( b ( σ ) , . . . , b ( σ n )). As we have seen before, r ( x ) = x for x ∈ | cc( β ( K )) | .Altogether, this proves claim (ii). (cid:3) The case of curves Let k ⊂ C be a subfield which is algebraic over k = k ∩ R . For sim-plicity, we assume that k is algebraically closed. In this section we willshow that naive exponential periods of the form R G e − f ω where G is 1-dimensional are the same as cohomological exponential periods of smoothmarked curves. This comparison is a special case of the general result in[CH20, Theorem 13.4], but we include it to illustrate the key ideas of thegeneral proof, while avoiding several technical problems.This section is organised as follows: first we give some elementary exam-ples of cohomological exponential periods and explain why they are naiveexponential periods. This is followed by an intermezzo in which we describethe oriented real blow-up of a marked curve, because it features several timesin the remainder of the section. Finally, we prove the inclusions announcedabove.8.1. Examples of cohomological exponential periods. In Section 5.1we saw explicit examples of naive exponential periods. We will now look atsome examples of cohomological exponential periods, before considering thecase for general curves. Example 8.1. We start with the simplest non-trivial case: X = A , Y = { } , f = id. Then H rd1 ( A , { } , id) = H ( B ◦ ( A , id) , { } ∪ ∂B ◦ ( A , id); Q ).Both B ◦ = B ◦ ( A , id) and its boundary are contractible, hence H rd1 ( A , { } , id)is of dimension 1. The generator is the path from 0 to a point on ∂B ◦ , i.e.,one of the G s of Example 5.3. We use G = [0 , ∞ ) because it is in thesubspace B ♯ as defined in Section 1.4. B ◦ : C ∞ i ∞ B ♯ : C ∞ The boundary in singular homology maps it to the class of the point 0 withmultiplicity − k [ z ] P ( dP − P d z,P (0)) −−−−−−−−−−−−→ k [ z ] d z ⊕ k. As in Example 6.13, the periods of ( Q d z, a ) are computed as Z G e − z Q d z − a. The general theory tells us that H ( A , { } , id) also has dimension 1. Itis easy to see that (d z, 0) is a not in the image of the differential: Indeed P dP − P d z is injective, and the preimage of d z under this injectionis the constant polynomial − 1, which does not have constant coefficient 0.Hence (d z, 0) generates our cohomology. The periods of ( A , { } , id , 1) areprecisely the elements k as Z G e − z d z = 1 . Unsurprisingly, these elements are naive exponential periods as explainedin Example 5.3. We now turn to X = A , Y = { } and f = z n . In thiscase the boundary of B ◦ ( A , f ) has n components, hence H rd1 ( A , { } , f )is of dimension n . As generators for homology we can use the n differentpreimages of [0 , ∞ ) under z z n . They are of the form G s m for m =0 , . . . , n − s = e πi/n . The boundary map in singular homology mapseach of them to the point 0 with multiplicity − k [ z ] P ( dP − nz n − P d z,P (0)) −−−−−−−−−−−−−−−−→ k [ z ] d z ⊕ k. All elements in H ( A , { } , f ) are represented by pairs ( QdZ, a ). Theirperiods are computed as Z G s e − z n Q d z − a. These are naive exponential periods. Remark 8.2. The preceding example provides an explicit instance of [CH20,Proposition 11.4] which is an important ingredient in the final comparisontheorem: rapid decay homology is not only computed by B ◦ ( A , f ), but alsoby B ♯ ( A , f ) = C ∪ ˜ f − (1 ∞ ) so we can choose intervals with end points E in { } ∪ ˜ f − (1 ∞ ).8.2. The oriented real blow-up of a marked curve. Let ¯ C be a smoothprojective complex curve, or in other words, a compact Riemann surface. Let¯ f : ¯ C → P be a non-constant meromorphic function. Let Q , . . . , Q n ∈ ¯ C denote the poles of ¯ f , let P , . . . , P m be some points on ¯ C distinct fromthe Q i , and denote by C ⊂ ¯ C the complement of { P , . . . , P m , Q . . . Q n } .Denote by f : C → A the restriction of ¯ f to C .We now consider the real oriented blow-up B ( C ) = B ¯ C ( C ) and the map˜ f : B ( C ) → ˜ P induced by f . It adds a circle to ¯ C an in each of the points P i and Q j . The algebraic map f : C → A induces a semi-algebraic map XPONENTIAL PERIODS AND O-MINIMALITY I 41 of manifolds with boundary ˜ f : B ( C ) → ˜ P . The circles around the P i aremapped to f ( P i ) ∈ C . The circles around the Q i are mapped to the circle atinfinity of ˜ P . As in Definition 6.3 let B ◦ ( C, f ) ⊂ B ( C ) be the open subsetof points either in C an or mapping to ℜ ( s ∞ ) > P .So it removes the circles around the P i ’s and some circle segments from thecircles around the the Q j ’s.The following figure illustrates the case ¯ C = P . Q Q Q Q P P A -dimensional comparison. We now show that generalised naiveexponential periods are cohomological exponential periods. Proposition 8.3. Let α = R G e − f ω be a generalised naive exponential periodover k as in Definition 5.4. Assume that dim( G ) = 1 . Then α is a cohomo-logical exponential period for a tuple ( C, Y, f, , where C is a smooth curvedefined over k , Y ⊂ C is finite set of points, and f : C → A k is a regularfunction. This is a special case of [CH20, Proposition 12.1]. Proof. By Corollary 5.11, every generalised naive exponential period is abso-lutely convergent. Hence we may apply Proposition 5.19 to obtain a smoothaffine curve C over k , a finite set of points Y ⊂ C ( k ), a pseudo-oriented1-dimensional k -semi-algebraic subset G of C ( R ) with endpoints in Y , afunction f : C k → A k that is proper on G and such that f ( G ) ⊂ B ◦ , and aregular 1-form ω on C k , such that α = R G e − f ω . By abuse of notation wereplace C and Y by C k and Y k from now on.Certainly, the form ω defines a class [ ω ] ∈ H ( C, Y, f ).The semi-algebraic set Reg ( G ) is semi-algebraically homeomorphic to afinite union of open intervals and circles. We may consider connected compo-nents separately. Thus, without loss of generality, Reg ( G ) is homeomorphicto an open interval and G its closure in C an . The semi-algebraic set G ishomeomorphic to either a circle, or to an interval with 0, 1 or 2 end pointsin C an . By assumption, we are given an orientation on the complementof finitely many points of G . We may consider these intervals separately,enlarging Y if necessary.Let ¯ C be a smooth compactification of C , and ¯ G the closure of G in B ( C ).It is compact because B ( C ) is. Lemma 5.9 implies ¯ G ⊂ B ◦ ( C, g ). By construction, the boundary of ¯ G is contained in Y ∪ ∂B ◦ ¯ C ( C, f ). Itdefines a class [ G ] ∈ H rd1 ( C, Y, f ) = H ( B ◦ ( C, g ) , Y ∪ ∂B ◦ ( C, f ); Q ). Finally,as in Example 6.13, the period pairing of these classes is computed as h [ ω ] , [ G ] i = Z ¯ G e − f ω = α. This proves the result: α is indeed a cohomological exponential period. (cid:3) Converse direction. We now want to express cohomological expo-nential periods as naive exponential periods. This means that we startwith a marked curve Y ⊂ C , and cohomology classes γ ∈ H rd1 ( C, Y ) and ω ∈ H ( C, Y, f ). We want to show that the period pairing h ω, γ i is a naiveexponential period. Let us sketch the ingredients of the proof:( i ) The first step is the observation that rapid decay homology H rd1 ( C, Y )is computed as the ordinary homology of the space B ◦ ( C, f ).( ii ) We then note that B ◦ ( C, f ) is homotopic to a certain subset B ♯ ( C, f ).We will give an ad hoc definition of this subset here, for the generaldefinition see [CH20, Definition 11.3].This step is crucial, because in the next step it will allow us toobtain semi-algebraic sets G whose image is contained in a suitablestrip: f ( G ) ⊂ S r,s . See also Remark 8.2.( iii ) Finally, we use semi-algebraic triangulation results and the delicateProposition 7.6 to realise γ as a linear combination of homologyclasses of semi-algebraic sets. This will allow us to realise h ω, γ i as naive exponential period. Proposition 8.4 ([CH20, Proposition 11.1]) . Let C ⊂ A n be a smooth affinecurve over k , f ∈ O ( C ) , and Y ⊂ C a proper closed subvariety. Then everycohomological exponential period of ( C, Y, f, is a naive exponential period.Proof. By definition, f ∈ k [ C ]. We also write f for a polynomial in k [ z , . . . , z n ]representing it. As C is affine, the twisted de Rham cohomology H ( C, Y, f )is a quotient of Ω ( C ) ⊕ L y ∈ Y k hence every element is represented by a tuple( ω, a y ). We also write ω for the element of Ω ( A n ) representing ω ∈ Ω ( C ). Step 1. Let ¯ C be a smooth compactification of C and let Z = ¯ C r C bethe points at infinity. By Proposition 6.5, H rd1 ( C, Y ; Z ) = H ( B ◦ ( C, f ) , Y an ∪ ∂B ◦ ( C, f ); Z ) . We decompose Z = Z f ∪ Z ∞ such that f is regular in the points of Z f andhas a pole in the points of Z ∞ . Let d z ≥ f at z ∈ Z .The oriented real blow-up of ¯ C in Z replaces each point z ∈ Z an by a circle S z . It is compact. The boundary is a disjoint union of circles. The map˜ f : B ( C ) → ˜ P maps these circles either to C (the case z ∈ Z f ) or to thecircle at infinity (the case z ∈ Z ∞ ). In the latter case, the map on the circleis a d z to 1 cover.By definition the subset B ◦ ( C, f ) is the union of the preimage of C an andthose points P in the circles S z above z ∈ Z an ∞ that are in the preimage ofthe half circle { w ∞ | ℜ ( w ) > } . Hence the boundary of B ◦ ( C, f ) consistsof d z many circle segments for every z ∈ Z an ∞ . XPONENTIAL PERIODS AND O-MINIMALITY I 43 Step 2. Now consider the smaller subset B ♯ ( C, f ) defined as the union ofthe preimage of C an and the points P in the circles S z above z ∈ Z an ∞ that arein the preimage of 1 ∞ . Hence the boundary ∂B ♯ ( C, f ) of B ♯ ( C, f ) consistsof d z many disjoint points for every z ∈ Z an ∞ . In particular the boundaries of B ◦ ( C, f ) and B ♯ ( C, f ) are homotopy equivalent. Both B ◦ ( C, f ) and B ♯ ( C, f )are homotopy equivalent to C an . Thus H rd1 ( C, Y ; Z ) = H ( B ♯ ( C, f ) , Y an ∪ ∂B ♯ ( C, f ); Z ) . Note that B ♯ ( C, f ) is not a manifold with corners, hence we are not ableto interpret the right hand side in the sense of C -homology as defined inSection 1.5. However, it is a topological space so ordinary singular homologyis perfectly well-defined and this is how we interpret the right-hand side. Step 3. The space B ◦ ( C, f ) is a k -semi-algebraic C ∞ -manifold withboundary. By Proposition 7.4, it has a k -semi-algebraic triangulation com-patible with B ♯ ( C, f ), Y and ∂B ♯ ( C, f ) that is globally of class C . Inparticular, the points in Y an ∪ ∂B ♯ ( C, f ) are vertices. By Proposition 7.6,the closed core of its barycentric subdivision is a strong deformation retrac-tion of B ♯ ( C, f ). We denote the closed core by A . Hence H ( B ♯ ( C, f ) , Y an ∪ ∂B ♯ ( C, f ); Z ) = H ( A, Y an ∪ ∂B ♯ ( C, f ); Z ) . The subcomplex A is compact, hence simplicial and singular homology of A agree. Therefore every homology class is represented by a linear combina-tion of closed semi-algebraic 1-simplices in A . The triangulation is C , hencethe closed 1-simplices in the triangulation of C define elements of S ( C, f ).In all, each homology class in H rd1 ( C, Y ; Z ) is represented by linear combina-tion of C -paths in B ♯ ( C, f ) with boundary in Y an ∪ ∂B ♯ ( C, f ). The periodintegral is defined by integrating e − f ω on these paths, see Definition 6.11and Lemma 6.12.Let γ : [0 , → A be one these simplices. We put G = γ ([0 , ∩ C an . Weneed to check that it satisfies the conditions needed for naive exponentialperiods. The closure ¯ G = γ ([0 , G by at most two points, theend points. The image ˜ f ( ¯ G ) in ˜ P is compact and contained in B ♯ ( C, f ),hence f ( G ) is contained in a suitable strip S r,s for r, s > 0. The map˜ f : ¯ G → ˜ P is proper because ¯ G is compact. By definition, the preimage˜ f − (1 ∞ ) does not contain any points of G . Hence f : G → C is also proper.We conclude that R G e − f ω is a naive exponential period.Our cohomological period was a linear combination of such. The same ar-guments as in the case of ordinary periods (see [HMS17, Proposition 12.1.5])show that a linear combination of naive exponential periods is a naive expo-nential period. (cid:3) References [BB03] P. Belkale and P. Brosnan. 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