Kudla-Millson lift of toric cycles and restriction of Hilbert modular forms

Let $V$ be quadratic space of even dimension and of signature $(p,q)$ with $p\ge q>0$. We show that the Kudla-Millson lift of toric cycles - attached to algebraic tori - is a cusp form that is the diagonal restriction of a Hilbert modular form of parallel weight one. We deduce a formula relating the dimension of the span of such diagonal restrictions and the dimension of the span of toric and special cycles.