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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

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  • $\begingroup$ It seems that Prop 5.1 of arxiv.org/pdf/0902.0648.pdf by Koll\'ar-Kov\'acs gives a precise answer to this question in the LC case (after running an mmp you get a Mori fiber space $X'->S$ with general fiber $\mathbb P ^1$ such that there are 2 sections $E_1,E_2$ corresponding to the non-klt locus of $(X',D')$; moreover $(X',D')$ is plt. If you are indeed interested in the non-lc case, one can start by taking a dlt model and then running an mmp as in the proof of Prop 5.1 of arxiv.org/pdf/0902.0648.pdf. Does this help? $\endgroup$
    – Hacon
    Commented Oct 10, 2017 at 23:06

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