... asymptotic expansion as ε → 0 of the integration of B(z,z)oneachfiberΩε(x )of :Ωε→ X in terms of the Bergman kernel on the diagonal Bm(x )of 514 KENGO HIRACHIH0(X, E⊗m). The formulas (5.3) ... perturbations of the domain (Theorem 1). We also show that the same invariant appears as the coefficient of the logarithmic term of the volume expansion of the domain with respect to the Bergman ... variational formula of the Szeg¨o kernel. After these preparations, wegive in Section 4 the proofs of the main theorems. Finally in Section 5, we proveProposition 3 by relating Vol(Ωε) to the...