Arithmetic theta lifting and Sturm-type bounds for modular forms of higher degrees

Arithmetic theta lifting provides a correspondence between automorphic forms and algebraic cycles on Shimura varieties, which is based on one of the central conjectures in Kudla's program predicting the modularity of the generating series of special cycles in Chow groups. This can be viewed as a higher-dimensional analogue of Gross--Zagier type results for modular curves and Heegner points, an also a geometric analogue of classical theta correspondences. I will review some of the recent advances and applications of the conjecture, and explain how certain Sturm-type bounds can help establish the convergence of the generating series in this conjecture. This is work in progress with Qiao He.