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Extension theorems, non-vanishing and the existence of good minimal models. (English) Zbl 1278.14022
Let $$X$$ be a complex projective manifold (or normal complex projective variety with mild singularities). The aim of the minimal model program is to construct a birational model $$X \dashrightarrow X'$$ such that either $$X'$$ admits a fibration with general fibre a Fano variety or $$X'$$ is a good minimal model, that is some positive multiple of the canonical divisor $$K_{X'}$$ defines a morphism. If $$X$$ is covered by rational curves or $$X$$ is of general type (that is some positive multiple of $$K_X$$ defines a birational map) the minimal model program is completed in the landmark paper by C. Birkar et al., [J. Am. Math. Soc. 23, No. 2, 405–468 (2010; Zbl 1210.14019)]. Thus the main challenge is now to study projective manifolds $$X$$ that are not covered by rational curves and not of general type. By a fundamental result of S. Boucksom et al. [J. Algebr. Geom. 22, No. 2, 201–248 (2013; Zbl 1267.32017)], the canonical divisor $$K_X$$ is then pseudoeffective, that is $$K_X$$ is a limit of effective divisor. However the nonvanishing conjecture claims that some positive multiple of the canonical divisor is actually effective. Once we know that there exists at least one effective pluricanonical divisor $$D$$ one can hope to establish the existence of good minimal models inductively by proving that the restriction morphism $H^0(X, \mathcal O_X(mK_X)) \rightarrow H^0(D, \mathcal O_D(mK_X))$ is surjective for $$m \gg 0$$. A similar extension result played a crucial role in the proof of the existence of flips by C. D. Hacon and J. McKernan [J. Am. Math. Soc. 23, No. 2, 469–490 (2010; Zbl 1210.14021)]. In the paper under review the authors realise an important step of this strategy by proving the following “plt” extension theorem:
Let $$X$$ be a projective manifold and $$S+B$$ a $$\mathbb Q$$-divisor with simple normal crossings such that
1) $$(X,S+B)$$ is plt (i.e. $$S$$ is a prime divisor with $$\mathrm{mult}_S(S+B)=1$$ and $$\lfloor B \rfloor =0$$), and
2) there exists an effective $$\mathbb Q$$-divisor $$D\sim _{\mathbb Q}K_X+S+B$$ such that $S\subset \mathrm{Supp} (D)\subset \mathrm{Supp} (S+B),$ and
3) for any ample divisor $$A$$ and any rational number $$\epsilon >0$$, there is an effective $$\mathbb Q$$-divisor $$D\sim _{\mathbb Q}K_X+S+B+\epsilon A$$ whose support does not contain $$S$$).
Consider $$\pi: \tilde X\to X$$ a log-resolution of $$(X, S+B)$$, so that we have $K_{\tilde X}+ \tilde S+ \tilde B= \pi^*(K_X+S+B)+ \tilde E$ where $$\tilde S$$ is the strict transform of $$S$$. Let $$m$$ be an integer, such that $$m(K_X+S+B)$$ is Cartier, and let $$u$$ be a section of $$m(K_X+S+B)|_S$$, such that $Z_{\pi^*(u)}+ m\tilde E|_{\tilde S}\geq m\Xi,$ where $$Z_{\pi^*(u)}$$ is the zero divisor of the section $$\pi^*(u)$$ and $$\Xi$$ the extension obstruction divisor (cf. [Zbl 1210.14021]). Then $$u$$ extends to $$X$$.
The main achievement of this theorem compared to earlier extension results is that one does not assume $$B$$ to be strictly positive (i.e. ample or big). The authors conjecture that their statement also holds under the weaker assumption that the pair $$(X,S+B)$$ is dlt. This stronger extension result would then reduce the minimal model conjecture to the nonvanishing problem. More precisely the authors prove the following theorem: Suppose that the “dlt” extension theorem holds in dimension $$n$$. Suppose also that the non-vanishing conjecture holds for semi-log-canonical pairs of dimension $$n$$. Then every $$n$$-dimensional projective manifold that is not covered by rational curves has a good minimal model.

##### MSC:
 14E30 Minimal model program (Mori theory, extremal rays) 14J40 $$n$$-folds ($$n>4$$) 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
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