Komplexe elliptische Geschlechter und S^1-aequivariante Kobordismustheorie (Complex elliptic genera and S^1-equivariant cobordism theory)

TL;DR

This work constructs and analyzes the universal complex elliptic genus φ_ell within the framework of S^1-equivariant cobordism for stably almost complex (U) and SU-manifolds, introducing N-structures c1(X)≡0 mod N and a hierarchy of equivariant ideals I_*^N, I_*^{N,t}, J_*^N (and their SU analogues) to capture fixed-point data. It proves key structural results: Ω_*^{U,N}⊗Q ≅ Ω_*^U⊗Q and a universal basis {W_i} with φ_ell(W_1,...,W_4) = (A,B,C,D) and φ_ell(W_i)=0 for i≥5; characterizes kernels of φ_ell and φ_N via twisted CP-bundles and their associated ideals, and shows that φ_N is controlled by modular curves C_N with explicit elimination theory for the corresponding curves. The thesis establishes rigidity of φ_ell under S^1-actions on SU-manifolds, relates φ_N to Level-N modular forms, and proves that blowing up along submanifolds of complex codimension ≡1 (mod N) leaves φ_N invariant, linking geometric operations to modular and index-theoretic invariants. These results tie complex elliptic genera to modular curves, twisted projective bundles, and equivariant cobordism, providing a geometrical and algebraic framework for understanding elliptic genera at level N and their rigidity properties.

Abstract

We introduce the universal complex elliptic genus phi_ell as the ring homomorphism from the complex cobordism ring Omega^U to the polynomial ring C[A,B,C,D] associated to the characteristic power series Q(x)=x/f(x), where f is the solution of the differential equation (f'/f)'=S(f'/f), S(y) = (y+A/2)^4-B/4*(y+A/2)^2+4C*(y+A/2)+B^2/64-2D. Formally, phi_ell arises as the index of the Dolbeault operator of the loop space of a manifold. For manifolds with vanishing first Chern class, phi_ell becomes a Jacobi form F(z,τ) for the full Jacobi group Z^2 x PSL_2(Z). We prove the rigidity of phi_ell for S^1-actions on SU-manifolds. The kernel of phi_ell in the rational SU-cobordism ring is characterized as the ideal generated by manifolds with S^1-action of fixed type t (an integer) different from 0. For z to be an N-division point on the elliptic curve determined by τ, phi_ell specializes to the Level N genus phi_N. We introduce the cobordism ring Omega^{U,N} of stably almost complex manifolds with first Chern class divisible by N and characterize the kernel of phi_N by certain ideals in the rationalized ring Omega^{U,N}. In Chapter 1, we construct a base sequence W_1, W_2, W_3, ... of the rational cobordism ring Omega^U on which phi_ell has the values A, B, C, D, and 0 for W_i with i>=5. In Chapter 2, phi_ell and phi_N are investigated and the main results are proven. Chapter 3 contains the further result that the level N genus is invariant under the blow up along a submanifold Y of a complex manifold X if the codimension of Y in X is congruent 1 modulo N.