Remarks on generating series for special cycles

TL;DR

This paper investigates the modularity of generating series for special algebraic cycles on certain Shimura varieties, proposing a conjectural extension of known results to all codimensions under the Bloch-Beilinson conjecture.

Contribution

It proves the modularity of Chow group valued generating series for special cycles in a broad setting, extending prior results limited to specific cases, assuming the Bloch-Beilinson conjecture.

Findings

Modularity of generating series in cohomology established.

Conditional proof of modularity for Chow groups for all n.

Extension of Yuan-Zhang-Zhang's results to higher codimensions.

Abstract

In this note, we consider special algebraic cycles on the Shimura variety S associated to a quadratic space V over a totally real field F, |F:\Q|=d, of signature ((m,2)^{d_+},(m+2,0)^{d-d_+}), 1\le d_+<d. For each n, 1\le n\le m, there are special cycles Z(T) in S, of codimension nd_+, indexed by totally positive semi-definite matrices with coefficients in the ring of integers O_F. The generating series for the classes of these cycles in the cohomology group H^{2nd_+}(S) are Hilbert-Siegel modular forms of parallel weight m/2+1. One can form analogous generating series for the classes of the special cycles in the Chow group CH^{nd_+}(S). For d_+=1 and n=1, the modularity of these series was proved by Yuan-Zhang-Zhang. In this note we prove the following: Assume the Bloch-Beilinson conjecture on the injectivity of Abel-Jacobi maps. Then the Chow group valued generating series for special cycles of codimension nd_+ on S is modular for all n with 1\le n\le m.