Minimal model program for log canonical pairs on complex analytic spaces

We study the minimal model program for lc pairs on projective morphism between complex analytic spaces. More precisely, we generalize the results by Birkar and the second author to the setup by Fujino.

💡 Research Summary

This paper develops a comprehensive Minimal Model Program (MMP) for log‑canonical (lc) pairs in the setting of complex analytic spaces, extending the earlier work on Kawamata log‑terminal (klt) pairs by Birkar, Cascini, Hacon, McKernan, and Fujino. The authors work with a projective morphism π : X → Y between complex analytic spaces, where Y is a Stein space and W ⊂ Y is a compact Stein subset satisfying a collection of finiteness conditions (denoted (P)). These conditions replace the Noetherian property of the structure sheaf that is available in the algebraic category and guarantee that the relative Picard number over W is finite, allowing the MMP to be run locally around W.

The paper is organized as follows. Section 2 collects preliminaries on complex analytic varieties, stable base loci, Iitaka fibrations, the negativity lemma, singularities of pairs, and bimeromorphic maps, all adapted to the analytic context. Lemma 2.6 shows that under (P) the pre‑image π⁻¹(W) has only finitely many connected components intersecting W, which is essential for controlling the geometry during the program.

Section 3 defines the notion of a sequence of steps of an MMP in the analytic setting, introduces scaling of an auxiliary ample divisor, and proves fundamental results such as special termination, lifting of MMP steps, and the existence of nef thresholds. The authors adapt the “scaling” technique to R‑divisors, defining λ_i = inf{μ ≥ 0 | K_{X_i}+Δ_i+μA_i is nef over W} and showing that λ_i decreases to zero.

Section 4 establishes the analytic analogues of asymptotic vanishing order and the Nakayama–Zariski decomposition. These tools allow the authors to define “log abundant” pairs in the analytic context and to give a criterion for the existence of a log minimal model.

The main results are presented in Section 5. Theorem 1.1 (Theorem 5.9) treats an lc pair (X, B + A) where A is π‑ample. After taking a dlt blow‑up f : \tilde X → X and setting \tilde Γ = \tilde B + f^*A, the authors run a (K_{\tilde X}+ \tilde Γ)‑MMP with scaling of a suitable ample divisor \tilde H. After possibly shrinking Y around W, the process terminates with either a log minimal model or a Mori fibre space for (X, B + A) over Y.

Theorem 1.2 (Theorem 5.10) handles the case where A is an effective ℝ‑Cartier divisor such that K_X + B + A is ℝ‑linearly trivial over Y. With an auxiliary π‑ample divisor H making (X, B + H) lc and K_X + B + H nef over W, a (K_X + B)‑MMP with scaling of H is run. Again, after shrinking Y, the outcome is a log minimal model or a Mori fibre space for (X, B).

The most technical result is Theorem 1.3 (Theorem 5.8). For an lc pair (X, Δ) and an effective ℝ‑divisor A with K_X + Δ + A nef over W, the authors prove that any infinite sequence of (K_X + Δ)‑MMP steps with scaling of A satisfies two crucial properties: (i) the scaling numbers λ_i tend to zero, and (ii) infinitely many steps produce log abundant pairs. The proof combines the asymptotic vanishing order theory, Nakayama–Zariski decomposition, special termination, and the equi‑dimensional reduction technique of