Integrals of Borcherds forms

In his Inventiones papers in 1995 and 1998, Borcherds constructed holomorphic automorphic forms $\Psi(F)$ with product expansions on bounded domains $D$ associated to rational quadratic spaces $V$ of signature (n,2). The input $F$ for his construction is a vector valued modular form of weight $1-n/2$ for $SL_2(Z)$ which is allowed to have a pole at the cusp and whose non-positive Fourier coefficients are integers $c_\mu(-m)$, $m\ge0$. For example, the divisor of $\Psi(F)$ is the sum over $m>0$ and the coset parameter $\mu$ of $c_\mu(-m) Z_\mu(m)$ for certain rational quadratic divisors $Z_\mu(m)$ on the arithmetic quotient $X = \Gamma D$. In this paper, we give an explicit formula for the integral $\kappa(\Psi(F))$ of $-\log||\Psi(F)||^2$ over $X$, where $||.||^2$ is the Petersson norm. More precisely, this integral is given by a sum over $\mu$ and $m>0$ of quantities $c_\mu(-m) \kappa_\mu(m)$, where $\kappa_\mu(m)$ is the limit as $Im(\tau) -> \infty$ of the $m$th Fourier coefficient of the second term in the Laurent expansion at $s= n/2$ of a certain Eisenstein series $E(\tau,s)$ of weight $n/2 + 1$ attached to $V$. It is also shown, via the Siegel--Weil formula, that the value $E(\tau, n/2)$ of the Eisenstein series at this point is the generating function of the volumes of the divisors $Z_\mu(m)$ with respect to a suitable K\"ahler form. The possible role played by the quantity $\kappa(\Psi(F))$ in the Arakelov theory of the divisors $Z_\mu(m)$ on $X$ is explained in the last section.