Elliptic genera of level $N$ for complete intersections

TL;DR

This paper investigates the elliptic genera of level N for complete intersections, expressing them through generalized binomial coefficients and analyzing their values in relation to the first Chern class, with implications for classical genera.

Contribution

It provides a new description of elliptic genera of level N for complete intersections using generalized binomial coefficients and explores their values based on the sign of the first Chern class.

Findings

Explicit formulas for elliptic genera of level N in terms of binomial coefficients.

Connections established between elliptic genera and classical genera like Todd and A-hat.

Analysis of elliptic genera values for different signs of the first Chern class.

Abstract

We study the elliptic genera of level $N$ at the cusps of $\Gamma_1(N)$ for any complete intersection. These genera are described as the summations of generalized binomial coefficients, where each generalized binomial coefficient is related to the dimension and multi-degree of complete intersection. For complete intersection $X_n(\underline{d})$, write $c_1(X_n(\underline{d}))=c_1x$, where $x\in H^2(X_n(\underline{d});\mathbb{Z})\cong\mathbb{Z}$ is a generator. We mainly discuss the values of the elliptic genera of level $N$ for $X_n(\underline{d})$ in the case of $c_1>0, =0$ or $<0$. In particular, the values about the Todd genus, $\hat{A}$-genus and $A_k$-genus of $X_n(\underline{d})$ can be derived from the elliptic genera of level $N$.