Definition 6.12.3B5514FFD42D1

Definition 6.12.3.B5514FFD42D1 Let $E\subset Y$ be a prime divisor. Its center on $X$ is $\pi (E)$, which is an analytic subset of $X$ by theorem 9.7.2; it is connected, because $E$ is. Let $f$ be a holomorphic function on an open set $V\subset X$ containing the center, and let $V_0$ be the connected component of $V$ containing it. If $f$ vanishes identically on $V_0$, for instance if $f=0$, we set $\operatorname {ord}_ Ef=+\infty $. Otherwise $f\circ \pi $ does not vanish identically on any nonempty open subset of $\pi ^{-1}(V_0)$, because such a set contains a nonempty open set mapped biholomorphically onto an open subset of $V_0$, on which $f$ does not vanish identically by the identity theorem; since $E\subset \pi ^{-1}(V_0)$, the order of $f$ along $E$ is $\operatorname {ord}_ Ef=\operatorname {ord}_ E(f\circ \pi )<\infty $. This order along a divisor differs from the order at a point of proposition 6.7.9, although the two agree for the exceptional divisor of the blowup of a point (example 6.12.19). For a holomorphic germ $f$ at a point $x$, this defines $\operatorname {ord}_ Ef$ for the prime divisors $E$ on smooth models of open neighborhoods of $x$ on which $f$ is defined; this is how orders of germs enter the results below. The log discrepancy of $E$ is

where $\operatorname {ord}_ EK_{Y/X}=\operatorname {ord}_ EJ_\pi $ is the coefficient of $E$ in $K_{Y/X}$. Since $J_\pi $ is holomorphic, $A_ X(E)\geq 1$, and a prime divisor on $X$ itself, with $\pi $ the identity, has log discrepancy one. The center, the order and the log discrepancy depend only on the prime divisor over $X$ represented by $E$.