Functional Equations and the Generalised Elliptic Genus

Braden, H. W.; Feldman, K. E.

doi:10.2991/jnmp.2005.12.s1.7

Cited by 5 publications

(10 citation statements)

References 32 publications

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“…By expanding this relation up to z 6 we get the expressions c = 3(q 2 1 + q 2 ), −5q 4 = q 2 2 + 4q 1 q 3 , q 5 = q 2 q 3 , 35q 6 = 2q 3 2 − 12q 1 q 2 q 3 − 5q 2 3 , 5q 7 = 2q 3 (q 2 2 − q 1 q 3 ), −525q 8 = 9q 4 2 + 72q 1 q 2 2 q 3 − 56q 2 1 q 2 3 + 100q 2 q 2 3 , and the relation (q 2 1 + q 2 )q 2 3 = 0. Thus c = 0 or q 3 = 0.…”

Section: Elliptic Function Of Level Nmentioning

confidence: 99%

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Hirzebruch Functional Equation: Classification of Solutions

Bunkova

2018

Proc. Steklov Inst. Math.

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The Hirzebruch functional equation iswith constant c and initial conditions f (0) = 0, f ′ (0) = 1. In this paper we find all solutions of the Hirzebruch functional equation for n 6 in the class of meromorphic functions and in the class of series. Previously, such results were known only for n 4. The Todd function is the function determining the two-parametric Todd genus (i.e. the χ a,b genus). It gives a solution to the Hirzebruch functional equation for any n. The elliptic function of level N is the function determining the elliptic genus of level N . It gives a solution to the Hirzebruch functional equation for n divisible by N .A series corresponding to a meromorphic function f with parameters in U ⊂ C k is a series with parameters in the Zariski closure of U in C k , such that for parameters in U it coincides with the series expansion at zero of f . The main results are:Theorem. 10.2. Any series solution of the Hirzebruch functional equation for n = 5 corresponds to the Todd function or to the elliptic function of level 5.Theorem. 11.3. Any series solution of the Hirzebruch functional equation for n = 6 corresponds to the Todd function or to the elliptic function of level 2, 3 or 6.This gives a complete classification of complex genera that are fiber multiplicative with respect to CP n−1 for n 6.

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Section: Elliptic Function Of Level Nmentioning

confidence: 99%

“…Related works include classification results for systems of equations (1) taken for different n simultaneously (see [14,Section 4.6] and [19]). Other topics in the intersection of Baker-Akhiezer functions and functional equations include [1,2,4,9], and Lax equations [15]. Recent developments of ideas of [15] are given in [23]- [26].…”

Section: Introductionmentioning

confidence: 99%

Hirzebruch Functional Equation: Classification of Solutions

Bunkova

2018

Proc. Steklov Inst. Math.

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“…The structures J 1 and J 2 define the invariant almost complex structure on SU (4)/T 3 which is not integrable, as J 1 is not integrable. Formula (7) gives that Φ(SU (4)/T 3 , J) =…”

Section: Consider An Invariant Connection C On This Fibration Whose Hmentioning

confidence: 99%

“…The structures J 1 and J 2 define an invariant almost complex structure on G 2 /T 2 . Then formula (7) implies that…”

Section: Consider An Invariant Connection C On This Fibration Whose Hmentioning

confidence: 99%

“…Example 7.1.5.8. Let G be compact connected Lie group, H its closed connected subgroup such rk G = rk H. Assume that G/H admits an invariant almost complex structure J 1 and fix some invariant almost complex structure J 2 on H/T k , where T k is the common maximal torus for H and G. Following the notations from (7) we take X = G/H and F = H/T k endowed with the structures J 1 and J 2 and canonical that E and X are smooth manifolds with the actions of T k , π is a smooth map which is invariant under the action of T k and the tangent bundle τ F (E) along the fibers is equipped with T k -equivariant stable complex structure c π .…”

Section: Compatible Tangentially Stable Complex T K -Fibrationsmentioning

confidence: 99%

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Toric Genera of Homogeneous Spaces and their Fibrations

Buchstaber

Terzić

2012

International Mathematics Research Notices

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The aim of this paper is to study further the universal toric genus of compact homogeneous spaces and their homogeneous fibrations. We consider the homogeneous spaces with positive Euler characteristic. It is well known that such spaces carry many stable complex structures equivariant under the canonical action of the maximal torus T k . As the torus action in this case has only isolated fixed points it is possible to effectively apply localization formula for the universal toric genus. Using this we prove that the famous topological results related to rigidity and multiplicativity of a Hirzebruch genus can be obtained on homogeneous spaces just using representation theory. In that context for homogeneous SU -spaces we prove the well known result about rigidity of the Krichever genus. We also prove that for a large class of stable complex homogeneous spaces any T k -equivariant Hirzebruch genus given by an odd power series vanishes. Related to the problem of multiplicativity we provide construction of stable complex T k -fibrations for which the universal toric genus is twistedly multiplicative. We prove that it is always twistedly multiplicative for almost complex homogeneous fibrations and describe those fibrations for which it is multiplicative. As a consequence for such fibrations the strong relations between rigidity and multiplicativity for an equivariant Hirzebruch genus is established. The universal toric genus of the fibrations for which the base does not admit any stable complex structure is also considered. The main examples here for which we compute the universal toric genus are the homogeneous fibrations over quaternionic projective spaces. 1

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