Definition 9.4.80E1E539882AD

Definition 9.4.8.0E1E539882AD A complex space is unibranch if all its germs $(X,x)$ are irreducible. Equivalently, each local ring $\mathcal O_{X,x}$ is a domain whose integral closure in its field of fractions is a local ring. Indeed, in a local model $\mathcal O_{X,x}=\mathcal O_ N/I_{X,x}$, so by proposition 9.3.1 the germ $(X,x)$ is irreducible if and only if $\mathcal O_{X,x}$ is a domain. In this case, by proposition 9.4.3, the integral closure is $\widetilde{\mathcal O}_{X,x}$, and taking the limit at $x$, which exists by the same proposition, is a ring homomorphism $\widetilde{\mathcal O}_{X,x}\to \mathbb C$. An element with nonzero limit is a unit, because its reciprocal is holomorphic and bounded on $X_{\mathrm{reg}}$ near $x$, hence weakly holomorphic; an element with limit zero is not a unit. So the non-units form the ideal of elements with limit zero, and $\widetilde{\mathcal O}_{X,x}$ is a local ring. Thus this definition agrees with the definition of unibranch local rings in commutative algebra.