aa r X i v : . [ m a t h . N T ] J u l EXPONENTIAL PERIODS AND O-MINIMALITY II
JOHAN COMMELIN AND ANNETTE HUBER
Abstract.
This paper is a sequel to [CHH20]. We complete the com-parison between different definitions of exponential periods, and showthat they all lead to the same notion. In [CHH20], we show that naive exponential periods are absolutely convergent exponential periods. Wealso show that naive exponential periods are up to signs volumes of defin-able sets in the o-minimal structure generated by Q , the real exponentialfunction and sin | [0 , .In this paper, we compare these definitions with cohomological expo-nential periods and periods of exponential Nori motives. In particular,naive exponential periods are the same as periods of exponential Norimotives, which justifies that the definition of naive exponential periodssingles out the correct set of complex numbers to be called exponential periods. Introduction
We strongly advise the reader to read the introduction of the compan-ion paper [CHH20]. Let us now recall the definition of one of the mainprotagonists of that paper.Let k ⊂ C be a subfield such that k is algebraic over k := k ∩ R .See Section 9.1 for more on this condition on k . Recall from [CHH20,Definition 0.2] that a naive exponential period over k is a complex numberof the form Z G e − f ω where G ⊂ C n is an pseudo-oriented (not necessarily compact) closed k -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 } . The definition of generalised naive exponential periods and absolutely con-vergent exponential periods uses the same data, but with weaker conditionson f , ω and G . These definitions are repeated in detail in Definition 9.3.There is an alternative approach of a very different flavour. As far as weunderstand, it is actually the original one: exponential periods appear as theentry of a period matrix in Hodge theory of vector bundles with irregularconnections. We refer to the introduction of [CHH20] for more background.We call the elements in the image of the period pairing H rd n ( X, Y ; Q ) × H n dR ( X, Y, f ) → C Date : July 17, 2020. cohomological exponential periods , see Definition 10.12.Ordinary periods have an even more conceptual interpretation as a C -valued point on the torsor of isomorphisms between the de Rham realisationand the Betti realisation, two fibre functors on the Tannaka category ofmixed (Nori) motives, see [HMS17]. The same picture also applies in thecase of exponential periods. Fres´an and Jossen have developed a fully fledgedtheory of exponential motives in [FJ20].In this sequel to [CHH20] we show that these three approaches yield thesame set of exponential periods. Putting all the pieces of our two paperstogether, we get the following comparison theorem. Theorem (Theorem 13.4) . Let k ⊂ C be a field such that k/k is algebraic.Then the following subsets of C agree:(1) P nv ( k ) , i.e., naive exponential periods over k ;(2) P gnv ( k ) , i.e., generalised naive exponential periods over k ;(3) P abs ( k ) , i.e., absolutely convergent exponential periods over k ;(4) P mot ( k ) , i.e., periods of all effective exponential motives over k ;(5) P coh ( k ) , i.e., the set of periods of all ( X, Y, f, n ) with X a k -variety, Y ⊂ X a subvariety, f ∈ O ( X ) , n ∈ N ;(6) P log ( k ) , i.e., periods of all ( X, Y, f, n ) with ( X, Y ) a log pair, f ∈O ( X ) , n ∈ N ;(7) P SmAff ( k ) , i.e., periods of all ( X • , f • , n ) for ( X • , f • ) ∈ C − (SmAff / A ) , n ∈ N .Moreover, the real and imaginary part of these numbers are up to sign vol-umes of bounded definable sets for the o-minimal structure R sin , exp ,k gener-ated by exp , sin | [0 , and with paramaters in k , see [CHH20, Definition 2.13] . Global structure of the proof.
We recall the following diagram from [CHH20].It explains the global structure of the two papers, and how the different the-orems contribute to the main comparison result.Vol P nv P gnv P abs P log P coh P SmAff P mot[CHH20, Theorem 5.12] [CHH20, Lemma 5.5]Proposition 12.1[CHH20, Corollary 5.20]Proposition 11.1 triv trivProposition 13.3Proposition 13.1 Structure of this paper.
In Section 9 we recall notation and definitionsthat were introduced in [CHH20]. Section 10 is a technical section on thedefinition of cohomological exponential periods. For smooth affine varieties,
XPONENTIAL PERIODS AND O-MINIMALITY II 3 we gave a definition in [CHH20, Section 6]. We now extend this definitionto arbitrary pairs (
X, Y ) of a variety X and a closed subvariety Y ⊂ X .As suggested by the diagram above, Section 11 is devoted to proving P log ( k ) ⊂ P nv ( k ), whereas Section 12 shows the inclusion P gnv ( k ) ⊂ P log ( k ).Finally, in Section 13 we prove the remaining parts, which are all very formal,and glue all the pieces together to obtain the main theorem. Acknowledgements.
We sincerely thank Philipp Habegger with whom wewrote [CHH20], the first part of this series. All our joint discussions and hisnumerous insightful comments have had many direct and indirect influenceson this paper. 9.
Recapitulations
We keep the notation from [CHH20]. We repeat them for the convenienceof the reader.9.1.
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 . The following conditions on k are equivalent: 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.9.2. Categories of varieties.
Let k ⊂ C be a subfield. By variety wemean a quasi-projective reduced separated scheme of finite type over k . By X an we denote the associated analytic space on X ( C ).9.3. Good compactifications.
We say that a pair (
X, Y ) is a log-pair ,if X is smooth variety of pure dimension d , and Y ⊂ X a simple normalcrossings divisor. A good compactification of ( X, Y ) is the choice of an openimmersion X ⊂ ¯ X such that ¯ X is smooth projective, X is dense in ¯ X and¯ Y + X ∞ is a simple normal crossings divisor where ¯ Y is the closure of Y in ¯ X and X ∞ = ¯ X r X . If, in addition, we have a structure morphism f : X → A , we say that ¯ X is a good compactification relative to f if f extends to ¯ f : ¯ X → P . Good compactifications (relative to a structuremorphism) exist by resolution of singularities, see also [CHH20, Section 1.3].9.4. Some semi-algebraic sets.
Let k be as in Section 9.1. Let X be asmooth variety, f ∈ O ( X ), ¯ X a good compactification, X ∞ = ¯ X r X . Wedecompose 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.We denote by B ¯ X ( X ) the oriented real blow-up of ¯ X an in X an ∞ , for detailssee [CHH20, Definition 4.2]. It is a k -semi-algebraic C ∞ -manifold withcorners, see [CHH20, Proposition 4.3].In the case X = A and ¯ X = P , we write ˜ P = B P ( A ). This is amanifold with boundary: the compactification of C ∼ = R by a circle atinfinity, adding one point for each half ray emanating from the origin. JOHAN COMMELIN AND ANNETTE HUBER
For s ∈ C r { } , we write s ∞ for the point of ∂ ˜ P corresponding to thehalf ray s [0 , ∞ ). We say ℜ ( s ∞ ) > ℜ ( s ) >
0, and analogously for ≤ . 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 = { ∞} . See [CHH20, Example 4.1] and [CHH20, Example 8.1] for illustrations.We now return to the situation f : X → A for general smooth X and f .Let ˜ f : B ¯ X ( X ) → ˜ P be the induced map, see [CHH20, Lemma 4.4]. Wealso define B ◦ ¯ X ( X, f ) = B ¯ X ( X ) r { x ∈ ∂ ( B ¯ X ( X )) | π ( x ) ∈ D an0 or ℜ ( ˜ f ( x )) ≤ } ∂B ◦ ¯ X ( X, f ) = B ◦ ¯ X ( X, f ) r X an = B ◦ ¯ X ( X, f ) ∩ ˜ f − ( { s ∞ ∈ ˜ P | ℜ ( s ) > } ) . We are also going to need a variant (see Definition 11.3) B ♯ ¯ X ( X, f ) = B ¯ X ( X ) r { x ∈ ∂ ( B ¯ X ( X )) | π ( x ) ∈ D an0 or ˜ f ( x ) = 1 ∞} ∂B ♯ ¯ X ( X, f ) = B ♯ ¯ X ( X ) r X an . C -homology. In this paper, we denote by ∆ n the simplex { ( x , . . . , x n ) | x i > X i x i < } ⊂ R n . It is open in the ambient space, from which it inherits the standard ori-entation. We denote by ¯∆ n its closure in R n , and define the face maps k i : ¯∆ n − → ¯∆ n as in [War83, (2) p.142].Let X be a C -manifold with corners, see [CHH20, Section 1.5] for moredetails. A C -simplex on X is a C -map σ : ¯∆ n → X of C -manifolds with corners.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 .As we argue in [CHH20, Theorem 1.3], given a C -manifold with corners,the complexes S ∗ ( X ) and S ∗ ( X ) /S ∗ ( ∂X ) compute singular homology of X and of ( X, ∂X ), respectively.9.6.
Semi-algebraic manifolds with corners.
Let k be as in Section 9.1.In [CHH20, Definition 3.1], we introduced the notions of a definable C p -manifolds with corners and definable subsets G ⊂ M with respect to a fixedo-minimal structure. In the present paper we restrict to the case of theo-minimal structure R alg ,k and call them k -semi-algebraic . XPONENTIAL PERIODS AND O-MINIMALITY II 5
Definition 9.1 (See [CHH20, Definition 3.11]) . Let p ≥
1. Let M be a k -semi-algebraic C p -manifold with corners and G a k -semi-algebraic subset.A differential form ω of degree d on G is a continuous section ω : G → Λ d T ∗ M. In order to integrate differential forms, we need a notion of orientability.
Definition 9.2 (See [CHH20, Definition 3.14]) . Fix an integer p ≥
1, let d ≥ M be a k -semi-algebraic C p -manifold withcorners with G ⊂ M a k -semi-algebraic subset of dimension 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 the they agreeon the intersection.(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.By [CHH20, Theorem 3.22], the integral converges absolutely if G is com-pact.9.7. Periods.
Let k be as in Section 9.1. Definition 9.3 (See [CHH20, Definition 0.2], [CHH20, Definition 5.4], [CHH20,Definition 5.17]) . Let k ⊂ C be a subfield, such that k is algebraic over k ∩ R .A complex number α = Z G e − f ω is called(1) naive exponential period over k if G ⊂ C n is a pseudo-oriented closed(not necessarily compact) k -semi-algebraic subset, ω is a rationalalgebraic 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, f ( G ) is contained in a strip S r,s = { z ∈ C | ℜ ( z ) > r, |ℑ ( z ) | < s } ;(2) generalised naive exponential period over k if G ⊂ C n is a pseudo-oriented closed k -semi-algebraic subset, ω is a rational algebraicdifferential form on A nk that is regular on G and f is a rationalfunction 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 ) > } ;(3) absolutely convergent exponential period over k if G ⊂ C n is a pseudo-oriented (not necessarily closed) k -semi-algebraic subset, ω is a ra-tional algebraic differential form on A n , f a rational function on A nk that is regular on G and the closure of f ( G ) in ˜ P is contained in B ◦ . JOHAN COMMELIN AND ANNETTE HUBER
We denote P nv ( k ), P gnv ( k ) and P abs ( k ) the sets of all naive exponential peri-ods over k , all generalised naive periods over k and all absolutely convergentexponential periods over k , respectively.By definition, P nv ( k ) ⊂ P gnv ( k ). By [CHH20, Corollary 5.20], we have P gnv ( k ) = P abs ( k ). For properties and alternative descriptions of these setswe refer for [CHH20, Section 5].10. Exponential periods: the general case
Throughout this section let k ⊂ C be a subfield such that k is algebraicover k = k ∩ R . All varieties are defined over k .We turn to the definition of exponential periods for general ( X, Y ), againfollowing Fres´an and Jossen in [FJ20]. Notation for the smooth affine casewas set-up in [CHH20, Section 6].10.1.
Complexes of varieties.
By SmAff / A we denote the category ofsmooth affine varieties X together with a structure map f : X → A . Notethat we do not require f to be smooth. Let Z [SmAff / A ] be the additivehull of SmAff / A : • the objects are the objects of SmAff / A ; • the morphisms are formal Z -linear combinations of morphisms inSmAff / A , more precisely for connected X we haveHom Z [SmAff / A ] ( X, Y ) = Z [Mor SmAff / A ( X, Y )]; • the disjoint union is the direct sum.We denote by C + (SmAff / A ) the category of bounded below homologicalcomplexes over Z [SmAff / A ].We denote by SmProj / P the category of smooth projective varieties X together with a structure map f : X → P . As in the affine case we define Z [SmProj / P ] and C + ( Z [SmProj / P ]).10.2. Rapid decay homology for complexes.
Recall from [CHH20, Definition 6.6]the description of rapid decay homology for (
X, f ) ∈ SmAff / A . We put S rd ∗ ( X, f ) = S ∗ ( B ◦ ( X, f )) /S ∗ ( ∂B ◦ ( X, f ))where S ∗ ( − ) is as in Section 9.5 the complex of C -simplices. By [CHH20,Theorem 1.3] it computes singular homology.Note that the complex S rd ∗ ( X, f ) depends on the choice of a good com-pactification ¯ X relative to f , but only in a weak way. We want to extendthe construction to complexes of varieties.Let X be a smooth variety, f : X → A . Recall from Section 9.3 thata good compactification of ( X, f ) is a pair ( ¯ X, ¯ f ) where ¯ X is smooth andprojective, ¯ f : ¯ X → P a morphism and X → ¯ X is a dense open immersionsuch that the complement X ∞ is a simple divisor with normal crossing and¯ f extends f . Definition 10.1.
Let X • be a bounded below complex in Z [SmAff / A ]. A good compactification of X • is a bounded below complex ¯ X • in Z [SmProj / P ]together with a morphism of complexes X • → ¯ X • such that for every n themap X n → ¯ X n is a good compactification of ( X n , f ). XPONENTIAL PERIODS AND O-MINIMALITY II 7
Lemma 10.2.
Let X be a smooth variety, f : X → A .(1) The system of good compactifications of ( X, f ) is filtered.(2) Given g : Y → X a morphism of smooth varieties and a good com-pactification of ( X, f ) there is a good compactification ¯ Y of ( Y, f ◦ g ) and morphism ¯ Y → ¯ X over g .Proof. Let X → X and X → X be good compactifications of ( X, f ). Let X ′ be the closure of X in X × P X . Let X → X ′ be a desingularisationmaking the boundary into a divisor with normal crossings. A morphism h : X → X of good compactifications of ( X, f ) is uniquely determined ifit exists because X is dense in X .Let g : Y → X be a morphism of smooth varieties. Let ¯ X be a goodcompactification of X . Choose any compactification Y ′ of Y . Possibly afterreplacing Y ′ by a blow-up, the map g extends to Y ′ . Picking a desingulari-sation ¯ Y of Y ′ finishes the proof of this lemma. (cid:3) Corollary 10.3.
Let ( X • , f • ) be a bounded below (homological) complex in Z [SmAff / A ] . Then the system of good compactifications of ( X • , f • ) is non-empty, filtering and functorial.Proof. We construct ¯ X n by induction on n . For n ≪ n < N .Let X N = S X jN be the decomposition into connected components. Thedifferential d : X N → X N − is of the form d = P mi =1 a i g i for morphisms g i : X j ( i ) N → X N − and a i ∈ Z . Let Y i be a good compactification of X N suchthat g i lifts. Let ¯ X N be a common refinement of Y , . . . , Y m . By construction d lifts to ¯ X N . We need to check that the composition ¯ X N → ¯ X N − → ¯ X N − vanishes. This is a combinatorial identity on the coefficients of the g i . Itcan be checked on the dense open subsets X N → X N − → X N − , where itholds because X • is a complex. This finishes the proof of existence.The same method also produces common refinements of two good com-pactifications and lifts of morphisms of complexes. (cid:3) Recall the functor S rd ∗ computing rapid decay homology. Definition 10.4.
Let ( X • , f • ) be in C + (SmAff / A ). We define S rd ∗ ( X • , f • )as the total complex of the double complex ( S rd m ( X n , f n )) n,m for some choiceof good compactification ( ¯ X • , ¯ f • ) of ( X • , f • ). Remark 10.5.
By Corollary 10.3, this is well-defined up to canonical iso-morphism in the derived category.10.3.
Twisted de Rham cohomology and periods for complexes.
Re-call from [FJ20], see also [CHH20, Section 6.2], that twisted de Rham co-homology of (
X, f ) ∈ SmAff / A is defined as cohomology of the complexΩ ∗ ( X ) with differential Ω p ( X ) → Ω p +1 ( X ) given by dω − df ∧ ω . Definition 10.6.
Let ( X • , f • ) ∈ C + (SmAff / A ). We define H n dR ( X • , f • ) tobe the cohomology of the total complex R Γ dR ( X • , f • ) of the double complexΩ ∗ ( X • , E f • ). JOHAN COMMELIN AND ANNETTE HUBER
Lemma 10.7.
Let ( X • , f • ) ∈ C + (SmAff / A ) . Then the period map of [CHH20, Definition 6.11] extends to a pairing of complexes R Γ dR ( X • , f • ) × S rd ∗ ( X • , f • ) → C , i.e., a morphism of complexes R Γ dR ( X • , f • ) → Hom( S rd ∗ ( X • , f • ) , C ) . Proof.
We apply [CHH20, Lemma 6.12] to each X n , then take total com-plexes. (cid:3) Definition 10.8.
Let ( X • , f • ) ∈ C + (SmAff / A ), n ∈ N . The period pairing for ( X • , f • , n ) is the induced map H n dR ( X • , f • ) × H rd n ( X • , f • ) → C . The elements in the image of this pairing are called the exponential periods of ( X • , f • , n ). We denote the set of these numbers for varying ( X • , f • , n ) by P SmAff ( k ). Remark 10.9.
Fres´an–Jossen interpret these periods as periods for a suit-able category of effective exponential motives. We consider them in Section 13.The usual localisation amounts to inverting π . We do not consider the non-effective case in our paper.10.4. The relative case.
Let X be a variety over k , Y ⊂ X a closed subva-riety and f ∈ O ( X ). We want to define exponential periods for H rd n ( X, Y, f )by reduction to the case C + (SmAff / A ).A simplicial or bisimplicial variety X • → X is called a hypercover of X ,if it is a hypercover for the h-topology. We do not go into details about thistopology, which is introduced and studied in [Voe96]. For our purposes itsuffices to remark that in this case H n ( X an • , Z ) → H n ( X an , Z ) is an isomor-phism. The only examples that we are going to need are open and closedcovers, Section 11.3. We say that a hypercover is smooth and/or affine, re-spectively, if all X n are smooth and/or affine. By resolution of singularities,every hypercover can be refined by a smooth affine hypercover. If g : Y → X is a morphism of varieties, X • → X a smooth affine hypercover, then thereis a smooth affine hypercover Y • → Y and a morphism g • : Y • → X • over g . Lemma 10.10.
Let X be a variety over k , Y ⊂ X a subvariety. Let X • → X be a smooth affine hypercover, Y • → Y a smooth affine hypercover with amorphism Y • → X • of simplicial schemes compatible with the inclusion. Let C ( X, Y ) = Cone( Y • → X • ) be the cone of the associated map of total complexes in C + (SmAff / Z ) . Thenthere is a natural isomorphism H rd n ( X, Y ) ∼ = H rd n ( C ( X, Y )) . Proof.
Fix r ∈ R . We put T r ( X n ) = f − n ( S r ) ⊂ X an n where f n : X n → X → A is the structure map of X n and S r = { z ∈ C |ℜ ( z ) ≥ r } . Bydefinition, X • → X is a universal homological cover, hence the base change T r ( X • ) → T r ( X ) is also a universal homological cover. This implies that the XPONENTIAL PERIODS AND O-MINIMALITY II 9 complex computing homology of X an relative to T r ( X ) is quasi-isomorphicto the total complex of S ∗ ( X an • ) /S ∗ ( T r ( X • )) . By [FJ20, Proposition 3.5.2] (see also [CHH20, Proposition 6.5]) and thefact that S ∗ ( − ) computes singular homology (see [CHH20, Theorem 1.3]),we have for each n and sufficiently large r , a quasi-isomorphism S ∗ ( X an n ) /S ∗ ( T r ( X n )) → S ∗ ( B ¯ X n ( X n )) /S ∗ ( T r ( X n )) ← S rd ∗ ( X n , f n ) . By taking total complexes this gives quasi-isomorphisms of the projectivelimit of the complexes computing rapid decay homology of X and S rd ∗ ( X • , f • ).Note that projective limits are exact in our situation because all homologyspaces are finite dimensional. The same arguments can be applied to Y . Bytaking cones we get the result for relative homology. (cid:3) Given this Lemma, we are led to define:
Definition 10.11 ([FJ20, Definition 7.1.6]) . Let X be a variety over k , f ∈ O ( X ), Y ⊂ X a closed subvariety. Choose C ( X, Y ) ∈ C + (SmAff / A )as in Lemma 10.10.(1) We define H n dR ( X, Y, f ) as cohomology of R Γ dR ( X • , Y • , f ) = R Γ dR ( C ( X, Y )) . (2) We define the period pairing for ( X, Y, f, n ) as the period pairing H rd n ( X, Y, f ) × H n dR ( X, Y, f ) → C for C ( X, Y ).We conclude this section by recalling the definition of a cohomologicalexponential period.
Definition 10.12 (See [CHH20, Definition 6.10]) . Let X be a variety, f ∈O ( X ), Y ⊂ X a closed subvariety, n ∈ N . The elements in the image ofthe period pairing for ( X, Y, f, n ) are called the (cohomological) exponentialperiods 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.
Lemma 10.13.
Let
K/k be an algebraic extension. Then P coh ( K ) = P coh ( k ) . Proof.
The same argument as in the classical case, [HMS17, Corollary 11.3.5],also applies in the exponential case. (cid:3)
Cohomological exponential periods are naive exponentialperiods
The aim of this section is to prove the key comparison in Proposition 11.1.See [CHH20, Proposition 8.4] for the corresponding statement in the specialcase where X is a curve. In that case, the main ideas of the proof are present,but several technicalities are avoided. Proposition 11.1.
Let k ⊂ C be as in Section 9.1. Let ( X, Y ) be a logpair, i.e., X a smooth variety, Y ⊂ X a simple normal crossings divisor.Let f ∈ O ( X ) , and let α be a cohomological exponential period of ( X, Y, f, n ) (see Definition 10.12). Then α is a naive exponential period: P log ( k ) ⊂ P nv ( k ) . Remark 11.2.
This justifies that our fairly restrictive definition of a naiveexponential period was a reasonable choice.The proof is technical and will take the rest of the section.11.1.
Notation.
Throughout, let k be as in Section 9.1, k = k ∩ R .If X is a smooth variety, f ∈ O ( X ), ¯ X a good compactification relativeto f , then we put X ∞ = ¯ X r X . We decompose X ∞ = D ∪ D ∞ where D consists of the horizontal components and D ∞ of the vertical componentsmapping to ∞ in P .As before, we denote by ˜ f : B ¯ X ( X ) → ˜ P the induced map on the orientedreal blow-up of ¯ X an in X an ∞ .Recall from Section 9.4 that B ◦ ¯ X ( X, f ) = B ¯ X ( X ) r { x ∈ ∂ ( B ¯ X ( X )) | π ( x ) ∈ D an0 or ℜ ( ˜ f ( x )) ≤ } ∂B ◦ ¯ X ( X, f ) = B ◦ ¯ X ( X, f ) r X an We introduce a variant.
Definition 11.3.
We put B ♯ ¯ X ( X, f ) = B ¯ X ( X ) r { x ∈ ∂ ( B ¯ X ( X )) | π ( x ) ∈ D an0 or ˜ f ( x ) = 1 ∞} ∂B ♯ ¯ X ( X, f ) = B ♯ ¯ X ( X ) r X an The spaces B ¯ X ( X ) and B ◦ ¯ X ( X, f ) are k -semi-algebraic manifolds withcorners by [CHH20, Proposition 4.3] and B ♯ ¯ X ( X, f ) is a k -semi-algebraicsubset.11.2. A comparison of homology.
The first step in the argument is analternative description of rapid decay homology using B ♯ ( X, f ) rather than B ◦ ( X, f ). Let us motivate why this is needed. We are going to represent ho-mology classes by k -semi-algebraic sets G such that ¯ G ⊂ B ◦ ¯ X ( X, f ). Hence f ( G ) ⊂ B ◦ as in the definition of a generalised naive exponential period. Theproposition will allow us to even choose ¯ G ⊂ B ♯ ¯ X ( X, f ). Hence f ( G ) ⊂ B ♯ and the data defines a naive exponential period. Indeed, the closure of thestrip S r,s = { z ∈ C | ℜ ( z ) > r, |ℑ ( z ) | < s } inside ˜ P is contained in B ♯ . Actually, we can only apply this argument inthe case of smooth X , but see Section 11.3 for the reduction. Proposition 11.4.
Let V be a smooth variety, f ∈ O ( V ) , ¯ V a good com-pactification, n ≥ . Then the natural map H n ( B ♯ ¯ V ( V, f ) , ∂B ♯ ¯ V ( V, f ); Z ) → H n ( B ◦ ¯ V ( V, f ) , ∂B ◦ ¯ V ( V, f ); Z ) is an isomorphism. XPONENTIAL PERIODS AND O-MINIMALITY II 11
Proof.
We are going to show the equivalent statement on cohomology. Thespaces are paracompact Haussdorff and locally contractible, hence we maycompute singular cohomology as sheaf cohomology. We abbreviate B ◦ ( V ) = B ◦ ¯ V ( V, f ) and B ♯ ( V ) = B ♯ ¯ V ( V, f ). Let j ◦ : V an → B ◦ ( V ) and j ♯ : V an → B ♯ ( V ) be the open immersions. Our relative cohomology is computed byapplying R Γ to j ◦ ! Z and j ♯ ! Z , respectively.We compare their higher direct images on a subset of ¯ V an . As in thedefinition of B ◦ ( V ), let ¯ V r V = D ∪ D ∞ such that D ∞ ⊂ ¯ f − ( ∞ ) and f isrational on D . Furthermore let p ◦ : B ◦ ( V ) → ¯ V an r D an0 , and p ♯ : B ♯ ( V ) → ¯ V an r D an0 the projections. We consider the natural map Rp ◦∗ j ◦ ! Z → Rp ♯ ∗ j ♯ ! Z and claim that it is a quasi-isomorphism.We compute its stalks. For x ∈ V an , both sides are simply equal to Z concentrated in degree 0.Let x ∈ D an ∞ r D an0 . The stalk of R i p ♯ ∗ j ♯ ! Z in x is given by the limitof H i ( p ♯ − ( U ) , p ♯ − ( U ) ∩ ∂B ♯ ( V ); Z ) for U running through the system ofneighbourhoods of x . The analogous formula hold for p ◦ . Hence it sufficesto show that H i ( p ♯ − ( U ) , p ♯ − ( U ) ∩ ∂B ♯ ( V ); Z ) → H i ( p ◦− ( U ) , p ◦− ( U ) ∩ ∂B ◦ ( V ); Z )is an isomorphism for all U sufficiently small. This is a local questionon ¯ V . We choose local coordinates z , . . . , z n on ¯ V centered at x such that¯ f ( z , . . . , z n ) = z − d . . . z − d m m , where m is the number of components of D passing through x . Let U ǫ be the polydisc of radius ǫ around the origin. On U ǫ the real oriented blow-up is given by { ( z , . . . , z n , w , . . . , w m ) ∈ B ǫ (0) n × ( S ) m | z i w − i ∈ R ≥ } . We make a change of coordinates by writing z i = r i w i with r i ∈ [0 , ǫ ). Henceover U ǫ the real oriented blow-up is given by( r , . . . , r m , w , . . . , w m , z m +1 , . . . , z n ) ∈ [0 , ǫ ) m × ( S ) m × B ǫ (0) n − m . In it ∂B ♯ ( V ) is the subset of points with r · · · r m = 0, w d . . . w d m m = 1 and ∂B ◦ ( V ) is the subset of points with r · · · r m = 0, ℜ ( w d . . . w d m m ) > p ♯ − ( U ǫ ) and p ◦− ( U ǫ ) and their boundariesseparately. Both p ♯ − ( U ǫ ) and p ◦− ( U ǫ ) are homotopy equivalent to theirintersection with V an , hence they have the same cohomology.We now concentrate on the boundary. In both cases they are fibre bundlesover { ( r , . . . , r m , w , . . . , w m − , z m +1 , . . . , z n ) ∈ [0 , ǫ ) m × ( S ) m − × B ǫ (0) n − m | r · · · r m = 0 } . In the case of ∂B ♯ ( V ), the fibre consists of d m points, the solutions of w d m m =( w d . . . w d m − m − ) − . In the case of ∂B ◦ ( V ), the fibre consist of d m open circlearcs centered around these points. In particular, the inclusion p ♯ − ( U ǫ ) ∩ ∂B ♯ → p ◦− ( U ǫ ) ∩ ∂B ◦ is fibrewise a homotopy equivalence, hence it inducesan isomorphism on cohomology. (cid:3) The goal of this whole section is to express α as a naive exponential period.In order to find the set G as in the definition of a naive exponential period,we are going to choose a k -semi-algebraic triangulation of B ♯ ( V, f ) thatis globally of class C (see [CHH20, Definition 7.1]), and with V as in thesetting of the preceding proposition. Our next goal is therefore to constructa suitable smooth V from the log-pair ( X, Y ).11.3.
Hypercovers.
By definition of cohomological exponential periods, weneed to fix a smooth affine hypercover of our log-pair (
X, Y ). We do thisexplicitly.Let p : S → T be a morphism. Its ˇCech-nerve is the simplicial scheme S • → T with S n = S × T · · · × T S ( n + 1 factors)and the usual face and degeneracy maps. It is a hypercover, if p is a coverfor the h -topology. We need two easy cases.Let X be a smooth variety, U , . . . , U M an affine open cover. We put U = U ∐ · · · ∐ U M → U. Let U • be its ˇCech-nerve. Explicitly, we have U n = a J ∈{ ,...,M } n +1 U J with U ( j ,...,j n ) = n \ i =0 U j i . Singular homology satisfies descent for open covers (the Mayer–Vietorisproperty), hence U • → X is a smooth affine hypercover, the ˇCech-complexdefined by the open cover.For the second special case, let X be a smooth variety, Y ⊂ X a sim-ple normal crossings divisor with irreducible components Y , . . . , Y N . Byassumption they are smooth. We put Y = Y ∐ · · · ∐ Y N → Y. Let Y • be its ˇCech-nerve. Explicitly, we have Y n = a J ∈{ ,...,N } n +1 Y J with Y ( j ,...,j n ) = n \ i =0 Y j i . Singular homology satisfies proper base change, hence Y • → Y is a smoothhypercover, the ˇCech-complex defined by the closed cover.We can combine the two constructions. The bisimplicial scheme Y • ∩ U • → Y is a smooth affine hypercover. In the notation of Lemma 10.10 C ( X, Y ) = Cone( Y • ∩ U • → U • ) . XPONENTIAL PERIODS AND O-MINIMALITY II 13
We write Y − = X , then all terms of C ( X, Y ) are direct sums of objects ofthe form Y n ∩ U m for n ≥ − m ≥ C ( X, Y ). We proceed as follows. Let ¯ X be a goodcompactification of ( X, Y, f ). We choose an open cover U , . . . , U M by affinesubvarieties of X such that ¯ Y + P Mi =1 U i ∞ is still a simple normal crossingsdivisor. This can be achieved by choosing U i as the complement of a generichyperplane U i ∞ in ¯ X . Note that ¯ X is a good compactification of each of the U J . Hence the ˇCech-nerve of the map M a i =1 ¯ X → ¯ X is a good compactification of U • . We denote it ¯ U • . For each I ⊂ { , . . . , N } let ¯ Y I be the closure of Y I in ¯ X . By the transversality assumption it issmooth and a good compactification. Hence¯ Y n = a J ∈{ ,...,N } n +1 ¯ Y J defines a good compactification of Y n and of Y n ∩ U m for all m . The complexCone( ¯ U • ∩ ¯ Y • → ¯ U • ) ∈ C + (SmProj / A )is a good compactification of C ( X, Y ). Corollary 11.5.
Let ( X, Y ) be a log pair, f : X → A . With the notationabove R Γ dR ( X, Y, f ) = R Γ dR ( C ( X, Y )) = Ω ∗ ( C ( X, Y )) and rapid decay homology of ( X, Y, f ) is computed by S rd ∗ ( X, Y, f ) := Cone( S rd ∗ ( ¯ U • ∩ ¯ Y , f • ) → S rd ( ¯ U • , f • )) . Proof.
The statement for de Rham cohomology is simply Definition 10.11.The claim for rapid decay homology is Lemma 10.10 in every degree. (cid:3)
Our next aim is to get a clearer understanding of B ◦ ( − , f ) and B ♯ ( − , f )applied to C ( X, Y ) and its good compactification C ( ¯ X, ¯ Y ).11.4. Real oriented blow-up and closed ˇCech complexes.
Let X besmooth, Y ⊂ X a simple normal crossings divisor, f ∈ O ( X ). Let ¯ X be a good compactification such that Y + X ∞ is a simple normal crossingdivisor and f extends to ¯ X . Let ¯ Y be the closure of Y in ¯ X . Denote by B ¯ X ( Y ), B ◦ ¯ X ( Y, f ) and B ♯ ¯ X ( Y, f ) the closure of Y an in B ¯ X ( X ), B ◦ ¯ X ( X, f ) and B ♯ ¯ X ( X, f ), respectively. As in the last section let Y • → Y and ¯ Y • → ¯ Y bethe ˇCech-complexes for the closed cover of Y and ¯ Y by their irreduciblecomponents.Applying our oriented blow-ups, we get simplicial k -semi-algebraic man-ifolds with corners B ¯ Y • ( Y • ) and B ◦ ¯ Y m ( Y • , f • ) and k -semi-algebraic subsets B ♯ ¯ Y • ( Y • ). Note that B ¯ Y m ( Y m , f m ) = a J ∈{ ,...,N } n +1 B ¯ Y J ( Y J , f J ) ,B ◦ ¯ Y m ( Y m , f m ) = a J ∈{ ,...,N } n +1 B ◦ ¯ Y J ( Y J , f J ) B ♯ ¯ Y m ( Y m , f m ) = a J ∈{ ,...,N } n +1 B ♯ ¯ Y J ( Y J , f J ) . Proposition 11.6.
The simplicial k -semi-algebraic sets B ¯ Y • ( Y • ) → B ¯ X ( Y ) , B ◦ ¯ Y • ( Y • , f • ) → B ◦ ¯ X ( Y, f ) , B ♯ ¯ Y • ( Y • , f • ) → B ♯ ¯ X ( Y, f ) , are the ˇCech-nerves for the corresponding closed covers B ¯ Y ( Y ) → B ¯ X ( Y ) , B ◦ ¯ Y ( Y , f ) → B ◦ ¯ X ( Y, f ) , B ♯ ¯ Y ( Y , f ) → B ♯ ¯ X ( Y, f ) , Proof.
Let Z = Y J for J ⊂ { , . . . , N } m +1 for all m , J . Then ¯ Z is transverseto X ∞ . The description of the real oriented blow-up in local coordinatesimmediately gives B ¯ Z ( Z ) = ¯ Z an × ¯ X B ¯ X ( X ) , B ◦ ¯ Z ( Z, f ) = ¯ Z an × ¯ X an B ◦ ¯ X ( X, f ) ,B ♯ ¯ Z ( Z, f ) = ¯ Z an × ¯ X an B ♯ ¯ X ( X, f ) , In total we have B ¯ Y • ( Y • ) = ¯ Y • × ¯ X B ( X ) ,B ◦ ¯ Y • ( Y • , f • ) = ¯ Y • × ¯ X B ◦ ( X, f ) ,B ♯ ¯ Y • ( Y • , f • ) = ¯ Y • × ¯ X B ♯ ( X, f ) . This gives the claim on ˇCech-nerves. (cid:3)
Corollary 11.7.
Let X be a smooth variety, f ∈ O ( X ) , Y ⊂ X a simplenormal crossings divisor. Choose a good compactification ¯ X of X such that Y + X ∞ is a simple normal crossings divisor. Let B ◦ ¯ X ( Y, f ) and B ♯ ¯ X ( Y, f ) be the closure of Y an in B ◦ ¯ X ( X, f ) and B ♯ ¯ X ( X, f ) , respectively. Then H rd n ( Y, f ) ∼ = H n ( B ◦ ¯ X ( Y, f ) , ∂B ◦ ¯ X ( Y, f ); Q ) ∼ = H n ( B ♯ ¯ X ( Y, f ) , ∂B ♯ ¯ X ( Y, f ); Q ) and H rd n ( X, Y, f ) ∼ = H n ( B ◦ ¯ X ( X, f ) , B ◦ ¯ X ( Y, f ) ∪ ∂B ◦ ¯ X ( X, f ); Q ) ∼ = H n ( B ♯ ¯ X ( X, f ) , B ♯ ¯ X ( Y, f ) ∪ ∂B ♯ ¯ X ( X, f ); Q ) . Proof.
Let Y • → Y be the ˇCech-nerve of the closed cover of Y by the disjointunion of its irreducible components. By Proposition 11.6, the natural map B ◦ ¯ Y • ( Y • , f • ) → B ◦ ¯ X ( Y, f )is a proper hypercover, hence it induces isomorphisms on singular homology. (cid:3)
XPONENTIAL PERIODS AND O-MINIMALITY II 15
Semi-algebraic triangulations of hypercovers.
We use the nota-tion of Section 11.3.Note that the natural map B ¯ X ( U J ) → B ¯ X ( X ) induces an inclusion B ◦ ¯ X ( U J , f ) ⊂ B ◦ ¯ X ( X, f ). Proposition 11.8.
There is a finite dimensional subcomplex S ∆ ∗ ( X, Y, f ) ⊂ S rd ∗ ( X, Y, f ) such that the inclusion is a quasi-isomorphism and every S ∆ n ( X, Y, f ) has afinite basis consisting of k -semi-algebraic C -simplices of the form σ : ¯∆ a → B ♯ ( U b ) a + b = n or σ : ¯∆ a → B ♯ ( U b ∩ Y c ) a + b + c = n − such that σ is a homeomorphism onto its image.Proof. By definition, S rd ∗ ( X, Y ) is the total complex ofCone( S ∗ ( B ◦ ( U • ∩ Y • , f ) , ∂ ) → S ∗ ( B ◦ ( U • , f ) , ∂ )(where ∂ is an abbreviation for ∂B ◦ ( − , f ) as applicable).In order to unify notation, we write Y − = X and also Y I = X for | I | = −
1. We now want to choose compatible k -semi-algebraic triangula-tions in the sense of [CHH20, Section 7]. We first triangulate the base B ¯ X ( X )by applying [CHH20, Proposition 7.4]. We obtain a k -semi-algebraic tri-angulation of the compact k -semi-algebraic manifold with corners B ¯ X ( X )compatible with the finitely many k -semi-algebraic subsets B ♯ ¯ X ( U J ∩ Y I , f )and their boundaries.In the next step, we want to triangulate the (bi)simplicial k -semi-algebraicsets B ♯ ¯ X ( U • ∩ Y • , f ) and B ♯ ¯ X ( U • , f ) and their boundaries such that all struc-ture maps and the maps between them are simplicial. We obtain this simplyby pull-back of the triangulation of the base. By loc. cit. the simplices canbe chosen to be C .We apply [CHH20, Proposition 7.6] to these simplicial complexes andreplace them by the closed core of their barycentric subdivisons. The sub-complexes | cc( βB ♯ ( U a ∩ Y b , f )) | are deformation retracts, hence they have the same homology as ( B ♯ ( U a ∩ Y b , f ) , ∂ ). By Proposition 11.4 their homology also agrees with homology of( B ◦ ( U a ∩ Y b , f ) , ∂ ). The subcomplexes are compact.We now consider the subcomplexes of S ∗ ( B ◦ ( U • ∩ Y • , f ) , ∂ ) and S ∗ ( B ◦ ( U • , f ) , ∂ )that compute the simplicial homology of | cc( βB ♯ ( U • ∩ Y • , f )) | and | cc( βB ♯ ( U • , f )) | relative to their boundaries, respectively. By what we argued above, the in-clusion of subcomplexes into the ambient complexes are quasi-isomorphisms.Let S ∆ ∗ ( X, Y, f ) be the total complex of the cone of the natural map betweenthese subcomplexes. By construction it has a degreewise finite basis of theform given in the proposition. (cid:3)
Proof of Proposition 11.1.
Proof.
Let α = h Ω , Σ i be an exponential period for the log-pair ( X, Y, f ).We want to express it as a naive exponential period.We work with the hypercovers U • and Y • as in Section 11.3. By defini-tion, Σ ∈ H rd n ( X, Y ; Z ). We compute rapid decay homology via the complex S ∆ ∗ ( X, Y, f ) of Proposition 11.8. By definition this means that the cohomo-logy class Σ is represented by a tuple σ bc ∈ S ∆ a ( U b ∩ Y c ) with a + b + c = n − b ≥ c ≥ − ∗ (Cone( U • ∩ Y • → U • )),i.e., a tuple ω bc ∈ Ω a ( U b ∩ Y c ) with a + b + c = n − b ≥ c ≥ − Y − = X ). By definition of the period pairing h Ω , Σ i is obtained by taking a linear combination of the integrals Z σ bc e − f ω bc . Each of the σ bc is a linear combination of k -semi-algebraic strictly-simplicesglobally of class C with values in B ♯ ( U b ∩ Y c ) ⊂ B ◦ ( U b ∩ Y c ).Recall that naive exponential periods form an algebra, hence it suffices toshow that the integrals for the individual simplices define naive exponentialperiods.Let U = U b ∩ Y c ⊂ A N , ω = ω bc ∈ Ω a ( U ). Let T : ¯∆ a → B ♯ ( U, f ) be a k -semi-algebraic C -simplex. Let G = T ( ¯∆ a ) ∩ U an . We equip it with thepseudo-orientation induced from ∆ a . It is a closed k -semi-algebraic subsetof C N because U is affine and the inclusion U an → B ♯ ¯ U ( U, f ) is k -semi-algebraic. Moreover, as U is affine, f | G is the restriction of a polynomial in k [ X , . . . , X N ] to G and ω | G the restriction of an algebraic differential form.We need to check the condition on f ( G ). The closure ¯ G = T ( ¯∆ a ) ⊂ B ♯ ( U, f ) is compact, hence so is its image in B ♯ = C ∪ { ∞} . This impliesthat f ( G ) ⊂ C is contained in a strip S r,s as we want. Compactness of ¯ G also implies that the map ¯ G → ˜ P is proper. The preimage of the circle atinfinity is precisely ¯ G r G , hence f : G → C is also proper.Therefore our α is a linear combination of numbers of the form Z ¯∆ a e − f ◦ T T ∗ ω = Z G e − f ω, which are naive exponential periods. (cid:3) Generalised naive exponential periods are cohomological
Let k ⊂ C be a subfield, k = k ∩ R and assume that k is algebraic over k ,see Section 9.1. Recall from Definition 9.3 the notion of a generalised naiveexponential period. We denote by P gnv ( k ) the set of generalised naive expo-nential periods. Recall from Definition 10.12 the notion of an exponentialperiod of a log pair and the set P log ( k ) of all such numbers.The aim of this section is the proof of the following converse of Proposi-tion 11.1: Proposition 12.1.
Every generalised naive exponential period over k is anexponential period of a log-pair over k : P gnv ( k ) ⊂ P log ( k ) . XPONENTIAL PERIODS AND O-MINIMALITY II 17
More precisely, given • a pseudo-oriented k -semi-algebraic G ⊂ C n of real dimension d , • f a rational function and • ω a rational algebraic d -formas in the definition of a generalised naive exponential period, there are • a smooth affine variety X of dimension d , • a simple normal crossings divisors Y on X , • a function f ∈ O ( X ) induced from the original f , • a homology class [ G ] ∈ H rd d ( X, Y ; Z ) , and • a cohomology class [ ω ] ∈ H d dR ( X, Y, f ) such that h [ ω ] , [ G ] i = Z G e − f ω. Horizontal divisors.
We will need to make the closure of G disjointfrom the components of the divisor that are horizontal relative to f . Westart with a local criterion. Lemma 12.2.
Let k ⊂ R be a real closed field, so that ¯ k = k ( i ) . Let D, E ⊂ A nk be unions of distinct coordinate hyperplanes. In other words, D = (cid:8)Q i ∈ I x i = 0 (cid:9) and E = nQ j ∈ J x j = 0 o , with I, J ⊂ { , . . . , n } and I ∩ J = ∅ . Let G be a semialgebraic subset of A nk ( R ) = R n , such that G isdisjoint from D ( R ) , and such that ¯ G contains the origin. Let ∂ ¯ G = ¯ G r ¯ G int be its boundary in R n . Let U ⊂ A nk ( R ) be an open neighbourhood of theorigin, and assume that G ∩ U is open in R n and ∂ ¯ G ∩ U ⊂ E ( R ) . Then D is empty. ED U ⊂ G Proof.
Without loss of generality, we may assume that U is an open ball.Note that U \ E ( R ) has 2 J connected components. Since G ∩ U is open,and ¯ G contains the origin, we see that G intersects at least one of thesecomponents, say U . Since ∂ ¯ G ∩ U ⊂ E ( R ), we find that U ⊂ G . On theother hand, for every i / ∈ J , it is clear that { x i = 0 } intersects U . Hence D is empty. (cid:3) Setting 12.3.
For the actual proof of Proposition 12.1, we are going to usethe following data: • a real closed field k ⊂ R , hence k ( i ) = ¯ k , • 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 ⊂ Y ( R ) (where ∂G = G r G int inside X ( R )), • a pseudo-orientation on G , • a morphism f : X ¯ k → A k such that f : G → C is proper and suchthat the closure f ( G ) ⊂ ˜ P is contained in B ◦ , • a regular algebraic d -form ω on X ¯ k , • a good compactification ¯ X of X such that f extends to ¯ f : ¯ X ¯ k → P k , • and finally, we denote by D ⊂ ¯ X is the smallest subvariety of ¯ X con-taining all components of ( X ∞ ) ¯ k = ¯ X ¯ k − X ¯ k on which ¯ f is rational. Lemma 12.4.
In this setting, we may choose ¯ X such that, in addition tobeing a good compactification, the closure of G in ¯ X an is disjoint from D an .Proof. Without loss of generality, we may assume that X is connected. If D is empty, we are done. Hence assume that D is not empty. By the propernessassumption on f , we see that ¯ G ∩ D an lies in the preimage of ∞ ∈ P .Let ¯ Y be the Zariski closure of ∂ ¯ G ∪ Y in ¯ X , where ∂ ¯ G = ¯ G r ( ¯ G ) int viewed as subset of ¯ X . It contains the closure of Y in ¯ X , but possibly alsoadditional components mapping to ∞ . By resolution of singularities, we mayfind a modification π : ˜ X → ¯ X such that π − ( X ) → X is an isomorphism,˜ X again smooth and such that ˜ D ∪ E ∪ ˜ Y is a strict normal crossings divisorin ˜ X , where ˜ D and ˜ Y denote the strict transforms of D and Y respectively,and where E denotes the exceptional locus of π . In addition, we may assumethat ˜ D and ˜ Y are disjoint.Let ˜ G denote the strict transform of G under π , i.e., the closure of G ∼ = π − ( G ) in ˜ X an . It is contained in ˜ X ( R ). Since π is proper, the closure of ˜ G in ˜ X ( R ) is contained in π − ( ¯ G ). This means that ∂ ˜ G ⊂ E ∪ ˜ Y .¯ f − ( ∞ ) D∂G ⊂ YG ˜ Y ˜ DE ˜ G We will now show that ˜ D ( R ) is disjoint from the closure of ˜ G in ˜ X ( R ).Suppose that x is contained in their intersection. Since ˜ Y is disjoint from˜ D , we conclude that x ∈ E ( R ). As ˜ Y is closed, there is even an open neigh-bourhood U of x in ˜ X ( R ) such that U is disjoint from ˜ Y ( R ). In particularwe find that G ∩ U is open in ˜ X ( R ), and that ∂ ˜ G ∩ U ⊂ E ( R ). Aftera suitable choice of continuous semialgebraic coordinates, we see that thiscontradicts the conclusion of Lemma 12.2. Therefore the closure of ˜ G isdisjoint from ˜ D ( R ).Since ˜ f = ¯ f ◦ π is not rational on E , we conclude that ˜ X satisfies theconditions of the statement. (cid:3) Proof of Proposition 12.1.
XPONENTIAL PERIODS AND O-MINIMALITY II 19
Proof.
Let α be a generalised naive exponential period. By [CHH20, Lemma 5.6]and Lemma 10.13, we may assume without loss of generality that k ⊂ R andthat k is real closed and hence k ( i ) = ¯ k . Generalised naive exponential peri-ods are absolutely convergent, so we can use the characterisation of [CHH20,Proposition 5.19]. This brings us into Setting 12.3 with α = Z G e − f ω. By Lemma 12.4, we may improve the good compactification ¯ X in such away that the closure of G in ¯ X an is disjoint from the components of X ∞ onwhich ¯ f has a pole. This implies that the closure ¯ G of G in the real orientedblow-up B ¯ X ( X ) is contained in B ◦ ¯ X ( X, f ). Note that ¯ G is compact because B ¯ X ( X ) is. We replace X by X ¯ k from now on.Let Y • and their compactifications be as in Section 11.3. Note that we donot have to pass to an open ˇCech-cover because X is affine. By definition R Γ dR ( X, Y ) d = Ω d ( X ) ⊕ Ω d − ( Y ) ⊕ · · · ⊕ Ω ( Y d − )The tuple ( ω, , . . . ,
0) is a cocycle because dω = 0 and ω | Y = 0, both fordimension reasons. We denote the induced cohomology class by[ ω ] ∈ H n dR ( X, Y, f ) . Recall that G is equipped with a pseudo-orientation. Let G ′ ⊂ G bean oriented semi-algebraic subset with dim( G r G ′ ) < d that representsthe pseudo-orientation. We apply [CHH20, Proposition 7.4] to the semi-algebraic manifold with corners B ◦ ¯ X ( X, f ). Hence we may choose a semi-algebraic triangulation of ¯ G that is globally of class C and that is com-patible with he oriented subset G ′ , and also compatible with the subsets B ◦ ¯ X ( Y J ) ∩ ¯ G , and ∂B ◦ ¯ X ( Y J , f ) ∩ ¯ G for all J Here Y J is the intersection ofirreducible components of Y as in Section 11.3. The top dimensional sim-plices inherit an orientation from G ′ . We use the triangulation of ¯ G to definea cycle ( σ, σ , . . . , σ d − ) in S rd d (Cone( Y • → X )) = S d ( B ◦ ( X, f ) , ∂ ) ⊕ S d − ( B ◦ ( Y , f ) , ∂ ) ⊕ · · · ⊕ S ( B ◦ ( Y d − , f d − , ∂ )where we abbreviate S n − ( B ◦ ( Y i , f i ) , ∂ ) = S n − ( B ◦ ¯ Y i ( Y i , f i )) /S n − ( ∂B ◦ ¯ Y i ( Y i , f i )).In detail: We are given a simplicial complex K and a homeomorphism h : | K | → ¯ G which extends to a C -map on a neighbourhood of | K | . Foreach closed top-dimensional simplex a = [ a , . . . , a d ] ∈ K , we choose a linearisomorphism ¯∆ d → [ a , . . . , a d ]. By composition we obtain a C -map T a : ¯∆ d → [ a , . . . , a d ] h | [ a ,...,ad ] −−−−−−→ B ◦ ¯ X ( X, f ) . It is a homeomorphism onto its image. The image of T a is oriented by theorientation on G ′ . We can arrange for T a to respect this orientation. Theformal linear combination σ = X a ∈ K d T a is a chain on B ◦ ¯ X ( X, f ). Its boundary ˇ ∂σ is a linear combination of ( d − Y i or in ∂B ◦ ¯ X ( X, f ). Let σ ∈ S d − ( Y ) be the chain defined by the simplices in the Y i , ignoringthe ones with image contained in ∂B ◦ ¯ X ( X, f ). By construction, the simplicesappearing in ˇ ∂σ are contained in one of the Y ij , hence they define σ ∈ S d − ( Y ). Recursively, we find all σ a . By construction, ˇ ∂ ( σ, σ , . . . , σ d − ) isa cycle. Let [ G ] ∈ H rd d ( X, Y, f )be its homology class. Because of the special shape of [ ω ], we have h [ ω ] , [ σ ] i = X a ∈ K d Z ¯∆ d T ∗ a ω = X a ∈ K d Z T a ( ¯∆ d ) ω = Z G ω. We have written α as a cohomological period over ¯ k . (cid:3) Conclusion
Fres´an and Jossen develop a fully fledged theory of exponential motivesin [FJ20]. It behaves very much like the theory of ordinary Nori motives.In particular, there is a so-called “basic lemma” for affine pairs (
X, Y, f ).We refer to their book for further details. We denote by P mot ( k ) the set ofperiods of effective exponential motives. Proposition 13.1.
The periods of effective exponential motives are expo-nential periods in the sense of Definition 10.12 for a tuple ( X, Y, f, n ) with X smooth, Y a strict normal crossings divisor and n = dim X . In otherwords, P mot ( k ) ⊂ P log ( k ) . Proof.
By definition, every effective exponential motive is a subquotient ofsome exponential motive of the form H n ( X, Y, f ) for an affine k -variety X , Y ⊂ X a subvariety, f ∈ O ( X ), and X r Y smooth. Hence its periods arealso periods of H n ( X, Y, f ).There is a blow-up π : ˜ X → X such that ˜ X is smooth and ˜ Y = π − ( Y )is a simple normal crossings divisor. By excision for rapid decay homology,we obtain an isomorphism H rd n ( ˜ X, ˜ Y , f ) ∼ = H rd n ( X, Y, f ) . This isomorphism lifts to an isomorphism of motives. Hence they have thesame periods. (cid:3)
Remark 13.2.
By Proposition 12.1 all exponential periods are even realisedas cohomological exponential periods of affine log-pairs. This is not obviousfrom the purely motivic argument given above.
Proposition 13.3.
Periods of complexes of smooth affine varieties are pe-riods of effective exponential Nori motives, i.e., P SmAff ( k ) ⊂ P mot ( k ) . Proof.
The argument is the same as in the case of ordinary Nori motives,see [HMS17, Theorem 11.4.2]. We give a sketch of the proof.By [FJ20, Corollary 3.3.3], we may choose a good filtration F X ⊂ F X ⊂ . . . F n X = X of an affine variety X , i.e., one where in every step the relativehomology is concentrated in a single degree equal to the dimension. By XPONENTIAL PERIODS AND O-MINIMALITY II 21 definition the exponential motives of X are computed as homology of thecomplex of exponential Nori motives . . . H i +1 ( F i +1 X, F i X, f ) → H i ( F i X, F i − X, f ) → . . . . Given a complex X • of affine varieties, we may choose compatible good filtra-tions on all entries of the complex. The exponential motives of X • are definedas homology of the total complex of the double complex H i ( F i X j , F i − X j , f j ).This is compatible with the period computation, hence we have identifiedthe periods of X • with the periods of exponential period motives. (cid:3) Theorem 13.4.
Let k ⊂ C be a field, k = k ∩ R , and assume that k/k isalgebraic. Then the following subsets of C agree:(1) P nv ( k ) , i.e., naive exponential periods over k ;(2) P gnv ( k ) , i.e., generalised naive exponential periods over k ;(3) P abs ( k ) , i.e., absolutely convergent exponential periods over k ;(4) P mot ( k ) , i.e., periods of all effective exponential motives over k ;(5) P coh ( k ) , i.e., the set of periods of all ( X, Y, f, n ) with X a k -variety, Y ⊂ X a subvariety, f ∈ O ( X ) , and n ∈ N ;(6) P log ( k ) , i.e., periods of all tuples ( X, Y, f, n ) with ( X, Y ) a log pair, f ∈ O ( X ) , and n ∈ N ;(7) P SmAff ( k ) , i.e., periods of all tuples ( X • , f • , n ) for ( X • , f • ) ∈ C − (SmAff / A ) and n ∈ N .Moreover, the real and imaginary part of these numbers are up to sign vol-umes of bounded definable sets for the o-minimal structure R sin , exp ,k gener-ated by exp , sin | [0 , and with paramaters in k , see [CHH20, Definition 2.13] .Proof. The statement on volumes of definable sets is [CHH20, Theorem 5.12].The following diagram shows all the inclusions that we have proved be-tween the sets listed above.Vol P nv P gnv P abs P log P coh P SmAff P mot[CHH20, Theorem 5.12] [CHH20, Lemma 5.5]Proposition 12.1[CHH20, Corollary 5.20]Proposition 11.1 triv trivProposition 13.3Proposition 13.1 Therefore we have equality everywhere. (cid:3)
References [CHH20] Johan Commelin, Philipp Habegger, and Annette Huber. Exponential periodsand o-minimality I, 2020. [FJ20] J. Fres´an and P. Jossen. Exponential motives, 2020. version July/20, manuscriptavailable at http://javier.fresan.perso.math.cnrs.fr/expmot.pdf.[HMS17] Annette Huber and Stefan M¨uller-Stach.
Periods and Nori motives , volume 65 of
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of ModernSurveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series.A Series of Modern Surveys in Mathematics] . Springer, Cham, 2017. With con-tributions by Benjamin Friedrich and Jonas von Wangenheim.[Voe96] V. Voevodsky. Homology of schemes.
Selecta Math. (N.S.) , 2(1):111–153, 1996.[War83] F. W. Warner.
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