On Shokurov's rational connectedness conjecture

By Christopher D. Hacon, J. McKernan. 2005

240 citations recorded by the source corpus.

We prove a conjecture of V. V. Shokurov which in particular implies that the fibers of a resolution of a variety with divisorial log terminal singularities are rationally chain connected.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0f522a844953c9fd4d5649cc53a9335f2c905aa7

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In Computational Theory and Mathematics

1. Chi Li
2. C. Birkar
3. De-Qi Zhang
4. Yuri Prokhorov
5. Constantin Shramov
6. J. McKernan
7. S. Druel
8. Meng Chen

In Geometry and Topology

1. Chi Li
2. C. Birkar
3. Yuri Prokhorov
4. H. Blum
5. De-Qi Zhang
6. Constantin Shramov
7. Sheng Meng
8. J. McKernan

Citations in the corpus

• On pluricanonical systems of algebraic varieties of general type (2008, background): “By theorems of Shokurov [16] and Hacon-Mkernan [11], each fiber of πY : Y ′ −→ Y is rationally chain connected.”
• Differential forms in the h-topology (2013, background): “[GKKP11] Theorem 5.1 for reflexive differentials. 5.2. Proof of Theorem 5.2. Let π : X˜ →X be a resolution of singularities. The main result of [GKKP11] states that Ω[p](X) = Ωp(X˜). On the other hand, [HM07] Corollary 1.5 asserts that all fibers of πare rationally chain connected so that Ωp h(X) = Ω p”
• A geometric characterization of toric varieties (2016, background): “ be a log terminal model of (X,A+∆ 0). Replacing X by Y we may assume that KX +A+∆ 0 is nef. In this case KX +A+∆ 0 is semiample. Let f: X −→ W be the induced model. Then −(KX + ∆ 0) is ample over W. [7] implies that the fibres of f are rationally connected. Thus f factors through the maximal rationa”
• Codimension 1 foliations with numerically trivial canonical class on singular spaces (2018, background): “es that G is uniruled as well since F is general. But this contradicts Proposition 4.22, and shows that B= 0. By Proposition 4.22 applied to H , we see that H is canonical. Since π 1(F) is finite (see [HM07]), there is an ´etale cover f: F 1 → F such that f−1H is given by a closed rational 1-form (po”
• Building blocks of polarized endomorphisms of normal projective varieties (2016, background): “ is a point and Xi−1 99K Xi is either a divisorial contraction or a flip for each i < r. Clearly, Xr−1 is then a Fano variety of Picard number one, so Xr−1 and hence X are rationally connected; see [17]. By Theorem 1.8(4), f∗ is a scalar. Thus Proposition 8.2 is applicable. (2) By taking pullback,”
• Special test configurations and $K$-stability of Fano varieties (2011, methodology): “1], which was originally obtained by applying Hacon-McKernan’s extension theorem as in [HM07].”
• Jordan property for Cremona groups (2012, methodology): “Actually, it has been essentially proved in [13, Corollary 1.7(1)].”
• Terminal valuations and the Nash problem (2014, background): “rjective whenever there exists a nonsingular minimal model over Xwith exceptional locus of pure codimension one. Since every exceptional divisor over a variety with terminal singularities is uniruled [HM07], Theorem 1.1 also implies, as a byproduct, that every divisorial valuation defined by a diviso”
• Anti-pluricanonical systems on Fano varieties (2016, background): “Since X ′ is of Fano type, the general fibres of f ′ are also of Fano type, hence they are rationally connected which in turn implies the general fibres of f are rationally connected [16][46].”