Simple connectedness of the general leaf ,
As above let F (resp. Z) be the Zariski closure of a general leaf of F (resp. of G ). Then both F reg and Z reg are simply connected ,
, This follows easily from Lemma ?.?.?. In fact
we have ? 1 (Y 0 × F reg ) {1}, which implies that ? 1 (Y 0 ) ? 1 (F reg ) {1}. Again by Lemma ?.?.?, we see that Y 0 can be regarded as a Zariski open of Z (and thus of Z reg since Y 0 is smooth), p.1 ,
, As in the §?.?.?, in this step we will reduce the proof of Theorem C to the Q-factorial case. Assume that Theorem C for X with terminal Q-factorial singularities, let us prove that it holds for general X, Step ?: Reduction to the Q-factorial terminal case
, By construction X term is equipped with an effective Q-divisor ? term on X term such that K X term + ? term ? Q g * (K X + ?)
, By our assumption, the MRC fibration of X term induces a decomposition X term Z term ×F term with K Z term ? 0 and F term rationally connected, But by Lemma ?.?.? the irregularity of F term is zero, hence by [Dru??a, Lemma ?.?] we get a decomposition X Z × F, and we have K Z ? 0 and F rationally connected
, In the sequel we always assume that X has Q-factorial terminal singularities. As pointed above, F and G are weakly regular foliations. By construction F is an algebraically integrable foliation, we intend to apply Theorem ?.?.?? ([Dru??b, Theorem ?.?]) to prove that F is induced by an equidimensional fibre space, Step ?: Weak Regularity of the foliations and everywhere-definedness of the MRC fibration
Let everything as above, then the foliation F has canonical singularities (c.f ,
1 ) is pseudoeffective. By Proposition ?.?.?, up to multiplying A by a integer divisible by r, we can assume that f * D A,1 is an integral Cartier divisor (noting that Pic 0 (X) is an Abelian variety, thus divisible). In consequence, by replacing A by A?f * D A,1 , we get an integral Cartier divisor A on X ,
, ? A is f -very ample
, ? for general w ? W and for any k ? Z >0 the natural morphism Sym k H 0
,
, Since ? is birational, ? * A is ?-big and by [Deb??, Lemma ?.??] the natural morphism Sym k H 0 (M y , O M y (? * A)) H 0 (M y , O M y
, is surjective for all k ? Z >0 . Then by Proposition ?.?.? we have that
, But the augmented irregularity of X is zero, its Albanese variety Alb X is trivial, then a fortiori ? 1 (X reg ) {1}, in particular ? 1 (X reg ) is finite. Thus we proved the proposition. By the proposition above, we see that Conjecture ? implies Conjecture ?; moreover, since varieties of Fano type have vanishing augmented irregularity (every quasi-étale cover of a projective variety of Fano type remains Fano type
,
The Gurjar-Zhang conjecture is first proved for del Pezzo surfaces in ,
for weak Fano surfaces) and the question is explicitly raised in [Zha??, Introduction] for log Fano varieties (c.f. also [Sch??, Question ?.??]) and in [Zha??] the conjecture is proved for canonical (klt) Fano threefolds under some additional assumption that X has isolated singularities ([Zha??, Theorem ?]) or that the index of X is dimX ? 2 ([Zha??, Theorem ?]). The three-dimensional Fano case is fully confirmed by ,
, As for Conjecture ?, the question is raised in [GGK??] and it is proved therein that for X klt projective with trivial canonical divisor and vanishing augmented irregularity the fundamental group of X reg has only finitely many k-dimensional complex representations for every k ? Z >0 , and that the image of each finite dimensional representation of ? 1 (X reg ) is finite. It is also proved that the étale fundamental group of X reg is finite for X an irreducible holomorphic symplectic variety or an even-dimensional Calabi
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