Let $X$ be a normal variety with $\mathbb Q$-Cartier divisor $D$, such that $K_X+D$ is $\mathbb Q$-Cartier. Let $(X,D)$ is log Calabi-Yau pair, i.e, $K_X+D\sim_\mathbb Q0$. (For example take $ X$ be a toric variety and $D$ be the complement of the embedded torus, then (X,D) is log CY pair).

Take $$\mathfrak C(X,D)=\bigcup\{W\subset X\:| \exists \; {\text{birational morphism}}\;\tilde g:\tilde X\to X \; \text{and prime divisor E on }\;\tilde X\; {\text{with discrepancy }}\alpha\leq -1 , g(E)=W\}$$

As motivation: If $\mathfrak C(X,D)$ be empty then the pair $(X,D)$ has klt singularities.

Shokurov showed that it has one or two connected component. My question is when it has exactly one connected component. Is there any reference about additional information on this component?

Reference : V. V. Shokurov, 3-fold log flips, Izv. Akad. Nauk. SSSR Ser. Mat. 56 (1992), 105–201, 57 (1993), 141– 175 (Russian); English transl., Math. USSR-Izv. 40 (1993), 93–202, 43 (1994), 527–558

Let $X$ be a normal variety with $\mathbb Q$-Cartier divisor $D$, such that $K_X+D$ is $\mathbb Q$-Cartier. Let $(X,D)$ is log Calabi-Yau pair, i.e, $K_X+D\sim_\mathbb Q0$. (For example take $ X$ be a toric variety and $D$ be the complement of the embedded torus, then (X,D) is log CY pair).

Take $$\mathfrak C(X,D)=\bigcup\{W\subset X\:| \exists \; {\text{birational morphism}}\;\tilde g:\tilde X\to X \; \text{and prime divisor E on }\;\tilde X\; {\text{with discrepancy }}\alpha\leq -1 , g(E)=W\}$$

As motivation: If $\mathfrak C(X,D)$ be empty then the pair $(X,D)$ has klt singularities. In fact it is related to existance of rational curve and conjecture of Mumford-Demailly

Shokurov showed that it has one or two connected component. My question is when it has exactly one connected component. Is there any reference about additional information on this component?

Reference : V. V. Shokurov, 3-fold log flips, Izv. Akad. Nauk. SSSR Ser. Mat. 56 (1992), 105–201, 57 (1993), 141– 175 (Russian); English transl., Math. USSR-Izv. 40 (1993), 93–202, 43 (1994), 527–558

Let $X$ be a normal variety with $\mathbb Q$-Cartier divisor $D$, such that $K_X+D$ is $\mathbb Q$-Cartier. Let $(X,D)$ is log Calabi-Yau pair, i.e, $K_X+D\sim_\mathbb Q0$. For example take $ X$ be a toric variety and $D$ be the complement of the embedded torus.

Take $$\mathfrak C(X,D)=\bigcup\{W\subset X\:| \exists \; {\text{birational morphism}}\;\tilde g:\tilde X\to X \; \text{and prime divisor E on }\;\tilde X\; {\text{with discrepancy }}\alpha\leq -1 , g(E)=W\}$$

As motivation: If $\mathfrak C(X,D)$ be empty then the pair $(X,D)$ has klt singularities.

Shokurov showed that it has one or two connected component. My question is when it has exactly one connected component. Is there any reference about additional information on this component?

Reference : V. V. Shokurov, 3-fold log flips, Izv. Akad. Nauk. SSSR Ser. Mat. 56 (1992), 105–201, 57 (1993), 141– 175 (Russian); English transl., Math. USSR-Izv. 40 (1993), 93–202, 43 (1994), 527–558

Let $X$ be a normal variety with $\mathbb Q$-Cartier divisor $D$, such that $K_X+D$ is $\mathbb Q$-Cartier. Let $(X,D)$ is log Calabi-Yau pair, i.e, $K_X+D\sim_\mathbb Q0$. (For example take $ X$ be a toric variety and $D$ be the complement of the embedded torus, then (X,D) is log CY pair).

Take $$\mathfrak C(X,D)=\bigcup\{W\subset X\:| \exists \; {\text{birational morphism}}\;\tilde g:\tilde X\to X \; \text{and prime divisor E on }\;\tilde X\; {\text{with discrepancy }}\alpha\leq -1 , g(E)=W\}$$

As motivation: If $\mathfrak C(X,D)$ be empty then the pair $(X,D)$ has klt singularities.

Shokurov showed that it has one or two connected component. My question is when it has exactly one connected component. Is there any reference about additional information on this component?

Reference : V. V. Shokurov, 3-fold log flips, Izv. Akad. Nauk. SSSR Ser. Mat. 56 (1992), 105–201, 57 (1993), 141– 175 (Russian); English transl., Math. USSR-Izv. 40 (1993), 93–202, 43 (1994), 527–558