Komplexe elliptische Geschlechter und S^1-aequivariante Kobordismustheorie（复椭圆亏格与S^1-等变配边理论）
Komplexe elliptische Geschlechter und S^1-aequivariante Kobordismustheorie (Complex elliptic genera and S^1-equivariant cobordism theory)
AI总结：
本文引入万有复椭圆亏格并证明其S^1-作用刚性，刻画其核，构造基序列，并证明Level N亏格在特定余维数爆破下的不变性。
摘要：
我们引入万有复椭圆亏格phi_ell，它是从复配边环Omega^U到多项式环C[A,B,C,D]的环同态，对应于特征幂级数Q(x)=x/f(x)，其中f是微分方程(f'/f)'=S(f'/f)的解，S(y)=(y+A/2)^4-B/4*(y+A/2)^2+4C*(y+A/2)+B^2/64-2D。形式上，phi_ell作为流形环空间的Dolbeault算子的指标出现。对于第一陈类消失的流形，phi_ell成为全Jacobi群Z^2 x PSL_2(Z)的Jacobi形式F(z,τ)。我们证明了phi_ell对SU-流形上S^1-作用的刚性。在有理SU-配边环中，phi_ell的核被刻画为由具有固定类型t（整数）且t不等于0的S^1-作用的流形生成的理想。\n 当z是τ确定的椭圆曲线上的N-分点时，phi_ell特化为Level N亏格phi_N。我们引入第一陈类被N整除的稳定近复流形的配边环Omega^{U,N}，并在有理化环Omega^{U,N}中通过某些理想刻画phi_N的核。\n 在第1章中，我们构造有理配边环Omega^U的基序列W_1, W_2, W_3, ...，使得phi_ell在W_i（i>=5）上的值分别为A, B, C, D和0。在第2章中，研究phi_ell和phi_N，并证明主要结果。第3章包含进一步的结果：如果复流形X的子流形Y的余维数模N同余于1，则Level N亏格在沿Y的爆破下不变。
英文摘要：
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.
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