Buchstaber, Ochanine, Krichever, and Witten genera

Buchstaber , Ochanine, Kriche v er , and W itten g enera Mikhail Kornev Abstract W e introduce a new class of f ormal group laws whose modulus square c onstruction yields Buchstaber’s f amily of polynomials. This class is related to , but does not coincide with, the f amily of formal group la ws associated with the Krichev er genus. W e compute the v alues of the corresponding Hirzebruch genus on theta divisors and c omplex proje ctive sp aces, describe its relation to the Ochanine, Krichev er, and Witten genera, and sho w how this c onstruction gives ex amples not arising from Hirzebruch’s elliptic genera of lev el n . 1 In troduction In [ Buc90 ], a f amily of polynomials B a ( z ; x, y ) = ( x + y + z − a 2 xyz ) 2 − 4 ( 1 + a 3 xyz ) ( xy + yz + xz + a 1 xyz ) , (1) where a = ( a 1 , a 2 , a 3 ) , was introduced. It was shown that each polynomial denes a two- value d f ormal group G a , univ ersal in the class of tw o-value d formal groups obtained b y the modulus square c onstruction (see [ BN71 ]) from the class of f ormal groups F ( u, v ) = u 2 λ 1 ( v ) − v 2 λ 1 ( u ) uλ 2 ( v ) − vλ 2 ( u ) , (2) where λ k ( u ) ∈ A [ [ u ] ] , k = 1 , 2 ; the ring A has a rather c omplic ated structure (see [ BU15 ]), but A ⊗ Q  Q [ b 1 , b 2 , b 3 , b 4 ] , b 1 = α, b 2 = ℘ ( z ) , b 3 = ℘ ′ ( z ) , b 4 = g 2 , with deg b i = − 2 i , i = 1 , . . . , 4 , and ℘ ( z ) = ℘ ( z ; g 2 , g 3 ) is the W eierstrass function satisfying the equation ( ℘ ′ ( z ) ) 2 = 4 ℘ ( z ) 3 − g 2 ℘ ( z ) − g 3 . The class of groups ( 2 ) c ontains the f ormal groups corresponding to the T odd genus, the signature, the Ochanine (also elliptic) genus [ Och87 ], and the Krichev er genus. In [ Buc90 ], it 1 was prov ed that the class F ( u, v ) is a realization of the class of Krichever formal groups (see [ Kri90 ]) with exponential f Kr ( x ) = e αx Φ ( x, z ) , where Φ ( x, z ) = Φ ( x, z ; g 2 , g 3 ) = σ ( z − x ) σ ( z ) σ ( x ) e ζ ( z ) x (3) is the Baker–Akhiezer function, and σ ( z ) = σ ( z ; g 2 , g 3 ) , ζ ( z ) = d dz log σ ( z ) are W eierstrass functions (see, f or ex ample, [ Lan87 ], [ WW96 ]). In the present p aper, a new class of formal groups F Bc ( u, v ) =       u  Q ( v ) − v  Q ( u ) u 2 − v 2 ! 2 + a 3 u 2 v 2       − 1 / 2 , (4) where Q ( t ) = 1 − a 1 t 2 + a 2 t 4 − a 3 t 6 is introduc ed, whose modulus square is giv en by the family of po lynomials ( 1 ) . The f ormal group la w F Bc ( u, v ) belongs to the f amily of f ormal group la ws ( 2 ) if and only if a 3 = 0 . In the case of a 3 = 0 , we get the Ochanine genus. Theref ore, for a 3 ≠ 0 , the c orresponding genus is distinct from the elliptic genera of level n introduc ed b y Hirzebruch (see [ HBJ92 , Appendix III, Se ction 1]). The p aper is organized as f ollows. In Section 2 , w e recall the basics of the theory of two-v alued formal groups. Section 3 is dev oted to the central result of our paper, Theorem 2 . Corollary 2.1 says that the formal group ( 4 ) denes a Hirzebruch genus called the Buc hsta ber genus . As applic ations, w e compute its values on the smooth theta divisors and comp lex projectiv e spac es. It turns out that the values Bc ( Θ n ) belong to the ring Z [ a 1 , a 2 , a 3 ] , see Proposition 4 . In Propositions 5 and 6 , we c ompare the Buchstaber, Ochanine, Krichev er and Witten genera. This paper develops ideas from the note “F ormal Groups and Buchstaber Genus,” to appear in Uspekhi M atematicheskikh Nauk. The author is grateful to V ictor M. Buchstaber for useful discussions of this w ork. 2 The M odulus Square Construction In [ BN71 ], a modulus square construction and the corresponding two-v alued f ormal group in the comp lex c obordism ring Ω U w ere introduce d. The general theory of tw o-value d f ormal groups was develope d in [ Buc75 ]. F or the denition, examp les, and properties of n -value d groups, see the surv ey [ Buc06 ], or rec ent w orks [ BK25b , BK25a ]. Let us re call the modu lus square c onstructi on . Let ζ 1 = pr ∗ 1 ( ζ ) and ζ 2 = pr ∗ 2 ( ζ ) be quaternionic line bundles, where ζ is the universal quaternionic line bundle ov er H P ∞ , and pr j : H P ∞ × H P ∞ → H P ∞ are two projections onto 2 the Cartesian f actors. The embedding S 1 → Sp ( 1 ) of the multip licativ e circle group S 1 into the group of unit quaternions Sp ( 1 ) induces a map ι : C P ∞ → H P ∞ (5) betw een their classifying spac es. The pullback of the bundle ζ j ( j = 1 , 2 ) under the map ι dec omposes as a sum η j ⊕ η j , where η j is the univ ersal comp lex line bundle. W e hav e the f ollo wing pullback diagram: ( η 1 ⊕ η 1 ) ⊗ C ( η 2 ⊕ η 2 ) / /   ζ 1 ⊗ C ζ 2   C P ∞ × C P ∞ ι × ι / / H P ∞ × H P ∞ There is an isomorphism of c omple x v ector bundles: ( η 1 ⊕ η 1 ) ⊗ C ( η 2 ⊕ η 2 )  ( η 1 η 2 ⊕ η 1 η 2 ) ⊕ ( η 1 η 2 ⊕ η 1 η 2 ) . Note that the bundles ξ 1 : = η 1 η 2 ⊕ η 1 η 2 and ξ 2 : = η 1 η 2 ⊕ η 1 η 2 admit quaternionic structures. Let Pic ( M ) be the Picard group of isomorphism classes [ ξ ] of comp lex linear bundles ov er a comp lex manif old M . Consider an in v olution σ : [ ξ ] ↦→ [ ξ ] . By abuse of notation, we will write ξ instead of [ ξ ] . The points of the c orresponding orbit sp ace X : = Pic ( M ) / σ are identied with unordere d pairs [ ξ, ξ ] . W e get the 2-valued c oset group (see [ Buc06 , Section 6] f or the denition) with multiplic ation [ ξ, ξ ] ∗ [ η, η ] = [ [ ξη, ξ η ] , [ ξ η, ξ η ] ] and neutral element [ 1 C , 1 C ] , where 1 C denotes a class of a trivial c omplex line bundle. Note that ζ 1 ⊗ C ζ 2 cannot admit a quaternionic structure (an ex ercise in representation theory). But the bundle ζ 1 ⊗ C ζ 2 ⊗ C ζ 3 ov er H P ∞ × H P ∞ × H P ∞ does admit one. A natural isomorphism ( ζ 1 ⊗ C ζ 2 ) ⊗ C ζ 3  ζ 1 ⊗ C ( ζ 2 ⊗ C ζ 3 ) is equiv alent to the associativity of the two-v alued operation ∗ . This tw o-valued group is a ge ometric realization of a more general construction: Proposition 1. Let G be an abe lia n group and σ : g ↦→ − g a n i nvoluti on. Denote by X t he orbit space G / σ wit h points [ g, − g ] . Then X ca rries a n invo lutive com mutative two-valu ed coset group stru cture wit h oper ati on [ g, − g ] ∗ [ h, − h ] = [ [ g + h, − g − h ] , [ g − h, − g + h ] ] wit h neutr al e lement [ e, e ] , a nd e is a unit. 3 Proof. This fo llows dire ctly from the coset c onstruction, see [ Buc06 , Section 6]. Denition 1. The two-v alued group c onsidered in Proposition 1 is called the modu lus square constru cti on for an abelian group G . The modulus square c onstruction has the f ollo wing innitesimal analogue. Let F ( u, v ) be a commutativ e f ormal group ov er an algebra A with logarithm g ( u ) and exponential f ( u ) (they are dened o v er A ⊗ Q ). Let x = uu , y = vv . Then, b y denition, a two-va lued forma l group G F is giv en by the la w x ∗ y = [ F ( u, v ) F ( u, v ) , F ( u, v ) F ( u, v ) ] , (6) neutral element x = 0 and inv erse in v ( x ) = x . Dene the f ollo wing formal series: z 1 = F ( u, v ) F ( u, v ) , z 2 = F ( u, v ) F ( u, v ) Ψ 1 = z 1 + z 2 , Ψ 2 = z 1 z 2 . (7) One can check that these series belong to the ring A [ [ x, y ] ] , i. e. Ψ 1 = Ψ 1 ( x, y ) and Ψ 2 = Ψ 2 ( x, y ) , see [ BN71 , Lemma 2.21]. Henc e, we get the f ollo wing Proposition 2. The two-valu ed formal group G F is determined by t he roots z 1 , z 2 of a quadr ati c po lynomial over t he ring A [ [ x, y ] ] ( see ( 7 ) for the notati on ) : z 2 − Ψ 1 ( x, y ) z + Ψ 2 ( x, y ) = 0 , x ∗ y = [ z 1 , z 2 ] . (8) A general denition of a multi-value d formal group w as giv en in [ Buc75 , Section 1]. Denition 2. In the notation of ( 7 ), the series B ( x ) = g ( u ) g ( u ) = − g ( u ) 2 (9) is calle d a loga rithm of the tw o-valued f ormal group G F with the multiplic ation ( 6 ). The series B ( x ) in [ BN71 ] was c alled the logarithm f or the f ollowing reason: Proposition 3. In t he notations of ( 7 ) and ( 9 ) : z 1 , 2 = B − 1   B ( x ) ±  B  y   2 ! . 4 Proof. W e hav e B − 1 ( (  B ( x ) ±  B ( y ) ) 2 ) = B − 1 ( − ( g ( u ) ± g ( v ) ) 2 ) . It is enough to check that B ( z 1 , 2 ) = − ( g ( u ) ± g ( v ) ) 2 . Consider the case of z 1 (the case of z 2 is c ompletely analogous). W e hav e B ( z 1 ) = B ( V V ) = − g ( V ) 2 = − ( gg − 1 ( g ( u ) + g ( v ) ) ) 2 = − ( g ( u ) + g ( v ) ) 2 , where V = g − 1 ( g ( u ) + g ( v ) ) . This is ex actly what was re quired. Rec all the topologic al applic ations of the group G F dev eloped in [ BN71 ]. Consider the f ormal group F ( u, v ) = F U ( u, v ) of geometric c omple x cobordisms ov er the ring Ω U = U ∗ ( pt ) with the logarithm g ( u ) = g U ( u ) and the exponential f ( u ) = f U ( u ) (see [ Nov67 , Appendix 1]). Introduc e classes z 1 , z 2 ∈ U 4 ( C P ∞ × C P ∞ ) as Chern classes in U -theory: z 1 : = c U 2 ( η 1 η 2 ⊕ η 1 η 2 ) = c U 1 ( η 1 η 2 ) c U 1 ( η 1 η 2 ) = F U ( u, v ) F U ( u, v ) , z 2 : = c U 2 ( η 1 η 2 ⊕ η 1 η 2 ) = c U 1 ( η 1 η 2 ) c U 1 ( η 1 η 2 ) = F U ( u, v ) F U ( u, v ) , where u = c U 1 ( η 1 ) , v = c U 1 ( η 2 ) , u = c U 1 ( η 1 ) , v = c U 1 ( η 2 ) . Let x = ι ∗ p Sp 1 ( ζ 1 ) = uu and y = ι ∗ p Sp 1 ( ζ 2 ) = vv ∈ Sp 4 ( C P ∞ ) , where p Sp 1 ( ζ j ) ∈ Sp 4 ( H P ∞ ) denotes the Borel class in Sp -theory , and ι is the map ( 5 ) . Dene series: Ψ 1 : = p Sp 1 ( ι ∗ ( ζ 1 ⊗ C ζ 2 ) ) = z 1 + z 2 , Ψ 2 : = p Sp 2 ( ι ∗ ( ζ 1 ⊗ C ζ 2 ) ) = z 1 z 2 . Let z be a generator of H 4 ( H P ∞ , Z ) such that ι ∗ z = − t 2 , where t is a generator of H 2 ( C P ∞ , Z ) . Let ζ be a univ ersal quaternionic bundle ov er H P ∞ , and η be a univ ersal com- ple x bundle ov er C P ∞ . As we already know , x = ι ∗ c U 2 ( ζ ) = c U 1 ( η ) c U 1 ( η ) = uu . By abuse of notation, sometimes we will drop the map ι ∗ . Then [ BN71 , page 93], ch U ( g ( u ) ) = t and ch U ( B ( x ) ) = − t 2 = z , where ch U ( x ) is a Chern–Dold character ch U : U ∗ ( H P ∞ ) → H ∗ ( H P ∞ , Ω U ⊗ Q )  Ω U ⊗ Q [ z ] , which was introduc ed in [ Buc70 ]. Thus, the inv erse pow er series B − 1 ( z ) calle d exponential c oincides with the Chern character: B − 1 ( x ) = ch U ( z ) ∈ H ∗ ( H P ∞ , Ω U ⊗ Q ) . All the 2-v alued f ormal groups of the f orm ( 8 ) w ere classied b y Buchstaber in [ Buc75 ]. Rec all one of the main results of that paper: 5 Theorem 1 [ Buc75 , The orem 6.4] . Let x ∗ y = { z | z 2 − Ψ 1 ( x, y ) z + Ψ 2 ( x, y ) = 0 } (10) be a n arbitr a ry two-valu ed formal group G ( R ) in formal power seri es over an arbitr a ry Q - a lgebr a R . Let B ( x ) be its loga rithm. Then B ( x ) satisfi es the dierentia l equation 1 2 φ 1 ( x ) B ′ ( x ) + 1 8 φ 2 ( x ) B ′ ′ ( x ) = 1 wit h the i nitial c ondition B ( 0 ) = 0 , where φ 1 ( x ) = 𝜕 Ψ 1 ( x, y ) 𝜕 y     y = 0 , φ 2 ( x ) = 𝜕 σ ( x, y ) 𝜕 y     y = 0 , σ ( x, y ) = Ψ 2 1 − 4 Ψ 2 . If G ( R ) is of t he first type, t hat is, Ψ 2 ( x, y ) ≡ ( x − y ) 2 mod deg 3 , t hen the seri es B ( x ) defines a strong isomorp hism of this two-va lued formal grou p with t he e lementary two-va lued forma l group defi ned by the po lynomia l z 2 − 2 ( x + y ) z + ( x − y ) 2 . Moreover, for t he loga rithm of G ( R ) we have: B ( x ) = ©   « √ x ∫ 0 d t  φ ( t 2 ) ª ® ® ¬ 2 , φ 2 ( x ) = 8 x ∫ 0 φ 1 ( t ) d t, (11) where φ ( t ) = φ 2 ( t ) / ( 16 t ) and φ ( 0 ) = 1 . 3 The Buchstaber Genus A c entral result of this paper is the f ollo wing Theorem 2. (i) Let u : = I ( x ) = √ x ∫ 0 d t  1 + a 1 t 2 + a 2 t 4 + a 3 t 6 , g 2 = 4 a 2 1 3 − a 2 ! , g 3 = 4 a 1 a 2 3 − 2 a 3 1 27 − a 3 ! . (12) 6 Then x ( u ) = 1 ℘ ( u ; g 2 , g 3 ) − a 1 / 3 . ( If g 3 2 − 27 g 2 3 = 0 , t hen the fu nction ℘ ( u ; g 2 , g 3 ) = − d 2 d u 2 log σ ( u ; g 2 , g 3 ) corresponds to a degener ation of t he W eierstr ass σ -functi on ) . (ii) The loga rithm a nd exponential of t he formal two-va lued grou p G a a re given by the fo llow- ing forma l seri es over the a lgebr a Q [ a 1 , a 2 , a 3 ] : B ( x ) = I 2 ( x ) = ©   « √ x ∫ 0 d t  1 + a 1 t 2 + a 2 t 4 + a 3 t 6 ª ® ® ¬ 2 , B − 1 ( x ) = 1 ℘ ( √ x ; g 2 , g 3 ) − a 1 / 3 . (iii) Let Q ( t ) : = 1 − a 1 t 2 + a 2 t 4 − a 3 t 6 . Then t he law F Bc ( u, v ) , its exponentia l f Bc ( u ) , a nd its log arit hm g Bc ( u ) have t he form: F Bc ( u, v ) =       u  Q ( v ) − v  Q ( u ) u 2 − v 2 ! 2 + a 3 u 2 v 2       − 1 / 2 , f Bc ( u ) = 1  ℘ ( u ; g 2 , − g 3 ) + a 1 3 , g Bc ( u ) = u ∫ 0 d t  1 − a 1 t 2 + a 2 t 4 − a 3 t 6 . (iv) The forma l group F Bc ( u, v ) is universa l in t he class of a ll si ng le-valu ed formal grou ps with t he conditi on u = − u , for whic h the modu lus square constru ction yie lds the two-valu ed forma l group G a wit h the two-va lued law ( 1 ) . (v) The intersecti on of the c lasses F ( u, v ) a nd F Bc ( u, v ) coi ncides wit h the Ocha nine genus. This is c har acterized by t he conditi ons λ 1 ( u ) ≡ 1 for F ( u, v ) a nd a 3 = 0 for F Bc ( u, v ) . Proof. (i) is obtaine d by a series of substitutions u = t 2 → u = 1 v → v = w − a 1 3 in the integral I ( x ) . 7 (ii) f ollo ws from the rst part of Theorem 6.4 in [ Buc75 ] and from (i) . More conc retely , rewrite the tw o-valued la w ( 1 ) in the f orm ( 10 ), where Ψ 1 ( x, y ) = 2 x + 2 y + 4 a 1 xy + 2 a 2 x 2 y + 2 a 2 xy 2 + 4 a 3 x 2 y 2 1 − 2 a 2 xy − 4 a 3 x 2 y − 4 a 3 xy 2 + a 2 2 x 2 y 2 − 4 a 1 a 3 x 2 y 2 , Ψ 2 ( x, y ) = x 2 − 2 xy + y 2 1 − 2 a 2 xy − 4 a 3 x 2 y − 4 a 3 xy 2 + a 2 2 x 2 y 2 − 4 a 1 a 3 x 2 y 2 . Direct c omputations giv e φ 1 = 2 ( 1 + 2 a 1 x + 3 a 2 x 2 + 4 a 3 x 3 ) , φ 2 ( x ) = 16 x ( 1 + a 1 x + a 2 x 2 + a 3 x 3 ) . The desired f ormula f or B ( x ) fo llows from f ormula ( 11 ). (iii) Ac cording to the modulus square c onstruction (see Section 2 ), B ( x ) = g ( u ) g ( u ) = − g ( u ) 2 , where x = uu and g ( u ) is the logarithm of some f ormal group F ( u, v ) . Assume that u = − u , or, equivalently , g ( − u ) = − g ( u ) . Then u = √ − x . From this and part (i) it f ollo ws that g ( u ) = u ∫ 0 Q ( t ) − 1 / 2 d t, where Q ( t ) : = 1 − a 1 t 2 + a 2 t 4 − a 3 t 6 . Applying p art (i) to the case u : = I ( w 2 ) giv es the exponential w ( u ) = f Bc ( u ) . Now let us derive the la w F Bc ( u, v ) . Let ˜ ℘ ( z ) : = ℘ ( z ; g 2 , − g 3 ) , u = f Bc ( z ) , v = f Bc ( w ) . Then ˜ ℘ ( z ) = 1 u 2 − a 1 3 , ˜ ℘ ′ ( z ) = − 2  Q ( u ) u 3 . Substituting these f ormulas into the addition law ˜ ℘ ( z + w ) = − ˜ ℘ ( z ) − ˜ ℘ ( w ) + 1 4  ˜ ℘ ′ ( z ) − ˜ ℘ ′ ( w ) ˜ ℘ ( z ) − ˜ ℘ ( w )  2 and carrying out straightf orward c omputations, we obtain the la w F Bc ( u, v ) . (iv) fo llows from the previous points. (v) Assume that F Bc ( u, v ) = F ( u, v ) f or some series λ 1 , λ 2 . Since f Bc is an odd function, w e must require that f Kr ( u ) = e αu Φ ( u, z ) 8 be odd. Henc e V : = σ ( z − u ) σ ( z + u ) e 2 ( ζ ( z ) − α ) u ≡ 1 . From 𝜕 log V 𝜕 u     u = 0 = 0 and 𝜕 3 log V 𝜕 u 3     u = 0 = 0 w e obtain α = 0 and ℘ ′ ( z ) = 0 . It is known (se e, f or examp le, [ Kri80 ]) that Φ ( u, z ) Φ ( − u, z ) = ℘ ( z ) − ℘ ( u ) . Henc e, from the oddness condition, w e obtain: f 2 Kr ( u ) = 1 Φ 2 ( u, z ) = 1 ℘ ( z ) − ℘ ( u ) . Let ℘ ′ ( u ) 2 = 4 ( ℘ ( u ) − e 1 ) ( ℘ ( u ) − e 2 ) ( ℘ ( u ) − e 3 ) . Then from f ′ Kr ( u ) 2 =  1 + ( e 1 − e 2 ) f Kr ( u ) 2   1 + ( e 1 − e 3 ) f Kr ( u ) 2  and f ′ Bc ( u ) 2 = 1 − a 1 f Bc ( u ) 2 + a 2 f Bc ( u ) 4 − a 3 f Bc ( u ) 6 w e get a 3 = 0 . Coro llary 2.1. The formal grou p F Bc ( u, v ) corresponds to a Hi rze bruc h genus Bc : Ω U → Z  1 2 , a 1 , a 2 , a 3  , whi ch we ca ll t he Buc hstaber genus, where deg a i = − 4 i, i = 1 , . . . , 3 . In [ B V24 ], it was established that the exponential of the univ ersal formal group of c omplex c obordism is given b y the series f U ( u ) = u +  n ≥ 1 [ Θ n ] u n + 1 ( n + 1 ) ! , where [ Θ n ] denotes the c obordism class of a smooth theta divisor (of comp lex dimension n ) on a general princip ally polarised abelian v ariety A n + 1 . 9 It is w ell known that Ω U ⊗ Q  Q [ [ C P 1 ] , [ C P 2 ] , ..., [ C P n ] , ... ] , see [ Nov62 ]. In [ BV26 , f ormula (4)] the fo llowing identity in Ω U ⊗ Q was estab lished: [ C P n ] = ( n + 1 ) L n ( τ 1 , ..., τ n ) , τ k = [ Θ k ] ( k + 1 ) ! , where L n denotes the Lagrange in version po lynomial. Hence, Ω U ⊗ Q  Q [ [ Θ 1 ] , ..., [ Θ n ] , ... ] . The ring Z [ [ Θ 1 ] , ..., [ Θ n ] , ... ] is a proper subring of Ω U . By a famous result of Noviko v [ Nov60 , Theorem 1] and Milnor (unpublished), Ω U is a po lynomial ring Z [ a 1 , ..., a n , ... ] with one generator a n in every dimension 2 n (where n ≥ 1 ). The problem of nding good geometric representativ es for the generators a n remains open. From the univ ersality of the exponential f U , w e obtain the identity: f Bc ( u ) = u +  n ≥ 1 Bc ( Θ n ) u n + 1 ( n + 1 ) ! . From the oddness of f Bc ( u ) it f ollo ws that Bc ( Θ 2 n + 1 ) = 0 f or all n ≥ 0 . W e hav e: Bc ( Θ 2 ) = − a 1 , Bc ( Θ 4 ) = a 2 1 + 12 a 2 , Bc ( Θ 6 ) = − a 3 1 − 132 a 1 a 2 − 360 a 3 . (13) Observ e that the rst three nonzero v alues Bc ( Θ n ) lie in the ring Z [ a 1 , a 2 , a 3 ] . It is not a c oincidence. Proposition 4. The va lues Bc ( Θ n ) be long to the ri ng Z [ a 1 , a 2 , a 3 ] . Proof. F or brevity , write f : = f Bc . From Theorem 2 , point (iii) , w e know that f ′ ( x ) =  1 − a 1 f ( x ) 2 + a 2 f ( x ) 4 − a 3 f ( x ) 6 . (14) Squaring and di erentiating ( 14 ) gives: f ′ ′ = − a 1 f + 2 a 2 f 3 − 3 a 3 f 5 , where f ( 0 ) = 0 , f ′ ( 0 ) = 1 . (15) Denote b n = Bc ( Θ 2 n ) , n > 0 and b 0 = 1 . Then we ha ve: f ( x ) =  n ≥ 0 b n x 2 n + 1 ( 2 n + 1 ) ! . (16) 10 Using ( 15 ) and ( 16 ), the coecient c omp arison implies b n + 1 = − a 1 b n + 2 a 2  i + j + k = n − 1  2 n + 1 2 i + 1 , 2 j + 1 , 2 k + 1  b i b j b k − 3 a 3  i 1 +· · ·+ i 5 = n − 2  2 n + 1 2 i 1 + 1 , . . . , 2 i 5 + 1  5 Ö r = 1 b i r , n ≥ 2 . Henc e, b n ∈ Z [ a 1 , a 2 , a 3 ] . Now w e compute the v alues of the Buchstaber genus on C P n ’s. Applying the genus to Mishchenko’s logarithm f or the formal group F U ( u, v ) , we get: g Bc ( u ) =  n ≥ 0 Bc ( C P n ) u n + 1 n + 1 . Henc e, Bc ( C P 2 n + 1 ) = 0 f or all n ≥ 0 . W e obtain: Bc ( C P 2 ) = a 1 2 , Bc ( C P 4 ) = 3 a 2 1 − 4 a 2 8 , Bc ( C P 6 ) = 5 a 3 1 − 12 a 1 a 2 + 8 a 3 16 . Rec all that the Krichev er genus corresponds to a formal group with the exponential ( 3 ) giv en by the Baker–Akhiezer function Φ ( x, z ) . This genus depends on 4 parameters: b 1 = α, b 2 = ℘ ( z ) , b 3 = ℘ ′ ( z ) , and b 4 = g 2 . As it was pro v ed in [ Buc90 ], an y formal group of the f orm ( 2 ) has f Kr as its exponential. The Krichev er genus Kr : Ω U → A , where A is the ring of c oecients of the series λ 1 , λ 2 from ( 2 ) , is an isomorphism of graded abelian groups in real dimensions less than 10 [ BPR10 , Corollary 6.11]. There exists a class K ∈ Ω − 12 U on which the Krichev er genus tensored b y Q vanishes, while the Buchstaber genus remains nonzero , even for a 1 = a 2 = 0 . By ( 13 ) , degree 12 is minimal f or the existenc e of such a class. Proposition 5. (i) Let Θ n : = [ Θ n ] denote the c lass of a smooth t heta divisor of comp lex dimensi on n in Ω − 2 n U . Introdu ce a n element K ∈ Ω − 12 U : K = Θ 6 + 9 Θ 6 1 − 15 Θ 4 1 Θ 2 − 3 Θ 3 1 Θ 3 − 13 Θ 2 1 Θ 2 2 + 3 Θ 2 1 Θ 4 + 29 Θ 1 Θ 2 Θ 3 + 10 Θ 3 2 − 11 Θ 2 Θ 4 − 10 Θ 2 3 . (17) 11 Then Bc ( K ) = − 360 a 3 in Z [ a 1 , a 2 , a 3 ] a nd Kr ( K ) = 0 in A ⊗ Q  Q [ α, ℘ ( z ) , ℘ ′ ( z ) , g 2 ] , where A is the ri ng of coeffi cients of t he series λ 1 , λ 2 from ( 2 ) . (ii) The Q -vector space Ker ( Kr : Ω U ⊗ Q → Q [ α, ℘ ( z ) , ℘ ′ ( z ) , g 2 ] ) is 2 -dimensi ona l and gener ated by K and L = Θ 5 Θ 1 − 3 Θ 4 Θ 2 1 − 11 Θ 3 Θ 2 Θ 1 + 12 Θ 3 Θ 3 1 + 22 Θ 2 2 Θ 2 1 − 30 Θ 2 Θ 4 1 + 9 Θ 6 1 . Proof. (i) Let u = ℘ ( z ) and v = ℘ ′ ( z ) . From the exp ansion f Kr ( u ) , we obtain: Kr ( Θ 1 ) = 2 α, Kr ( Θ 2 ) = 3 ( α 2 + u ) , Kr ( Θ 3 ) = 4 ( α 3 + 3 αu − v ) , Kr ( Θ 4 ) = 5 α 4 + 30 α 2 u − 20 αv + 45 u 2 − 3 g 2 , Kr ( Θ 5 ) = 6 α 5 + 60 α 3 u − 60 α 2 v + 270 αu 2 − 18 αg 2 − 132 uv, Kr ( Θ 6 ) = 7 α 6 − 63 α 2 g 2 + 105 α 4 u − 99 g 2 u + 945 α 2 u 2 + 1215 u 3 − 140 α 3 v − 924 αuv + 160 v 2 . Combining it with the earlier c omputations ( 13 ) , we nd: Bc ( K ) = − 360 a 3 and Kr ( K ) = 90  4 u 3 − g 2 u − g 3 − v 2  = 0 . (ii) This part f ollo ws from direct c omputations. Rec all that the Witten genus [ W it88 ] corresponds to a f ormal group with the exponential f W t ( u ) = σ ( u ; g 2 , g 3 ) giv en b y the W eierstrass σ -function. Set a 1 = 0 and comp are the Buchstaber f Bc ( u ) = 1 √ ℘ ( u ; g 2 , − g 3 ) and W itten f W t ( u ) = σ ( u ; g 2 , g 3 ) exponentials: f Bc ( u ) = u − 3 g 2 u 5 5! + 90 g 3 u 7 7! + 189 g 2 2 u 9 9! − 4 3740 g 2 g 3 u 11 11! +  − 68 607 g 3 2 + 2 673 000 g 2 3  u 13 13! + O ( u 15 ) f W t ( u ) = u − e g 2 u 5 5! − e g 3 u 7 7! − 9 e g 2 2 u 9 9! − 6 e g 2 e g 3 u 11 11! +  69 e g 3 2 − 6 e g 2 3  u 13 13! + O ( u 15 ) , (18) where e g 2 = g 2 2 and e g 3 = 6 g 3 . Rec all that a f ormal pow er series u + Í n ≥ 1 c n u n + 1 ( n + 1 ) ! , where c n ∈ R , is c alled a Hurwitz seri es ov er a ring R . It is known (see, e.g., [ BL05 ]) that f W t ( z ) is a Hurwitz series ov er the ring Z  g 2 2 , 2 g 3  . In [ Bun17 ], Bunkov a c onjectured that f W t ( z ) = σ ( z ; g 2 , g 3 ) is a Hurwitz series o ver the ring Z  g 2 2 , 6 g 3  . 12 Proposition 6. For a 1 = 0 , t he exponential f Bc ( u ) = 1 √ ℘ ( u ; g 2 , − g 3 ) of t he Buc hstaber genus is a Hurwitz seri es over t he ring Z [ g 2 , g 3 ] . Proof. From ( 12 ) , we hav e g 2 = − 4 a 2 , and g 3 = − 4 a 3 . Let f ( u ) = Í n ≥ 0 b n u n n ! , b 0 = 0 , b 1 = 1 . Arguing as in the proof of Proposition 4 , we nd the re currence relation: b n + 2 = − g 2 2 C n + 3 g 3 4 D n , (19) where C n =  i + j + k = n  n i, j, k  b i b j b k , D n =  i 1 +· · ·+ i 5 = n  n i 1 , . . . , i 5  5 Ö r = 1 b i r . From f being odd, w e know that b n = 0 , f or each even n ≥ 0 . If n is odd, then every multinomial in C n has three odd parts. 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