Volumes of Canonical Fano Threefolds II

In this proposal, the PI continues on the study of the upper bound of the anticanonical volumes V(X) of a canonical Fano threefold X. We propose two approaches: 1. Consider Fano threefolds with a conic bundle structure and study the geometry of the anticanonical system. We will use the method of un-projection. 2. The PI proposes a Bogomolov-Miyaoka-Yau type inequality for chern numbers on a varieties with nef tangent bundle. As an easy corollary, the anticanonical volume is at most 72 for canonical Fano threefolds. We consider the stability of tangent sheaves and plan to use the theory of foliated minimal model program. First year: We study the anti-canonical system of Weil divisors. The key is to understand the base scheme of |-K_X| and singularities of the pair (X,|-K_X|). When |-K_X| is mobile and (X,|-K_X|) is canonical, it can be shown that V(X) is at most 72. This depends on early work of Iskovskih-Prokhorov-Karzhemanov on Gorenstein canonical threefolds and the geometry of the linear system |-K_X|. Our goal is to extend their results to non-Gorenstein case. We may assume X is terminal Q-factorial to start with, and hence the method of the explicit geometry for terminal threefolds can be applied. The first question is to determine if there is a lower bound of V(X) to guarantee that |-K_X| is mobile. Assume that X now equips a conic bundle structure, as an evidence, one can show that the anti-canonical map can not be a rational pencil if V(X)>8. Second year: We focus on the foliation theory. If the tangent sheaf is unstable, then the maximal destablizing subsheaf defines a foliation F. We consider smooth Fano manifold of Picard number one. It can be proved that c_1(F) is ample and hence F is a Fano foliation. There are partial classification result of Fano foliations by Araujo-Druel-Kovacs unless c_1(F) is primitive. In the latter case, we only have to prove the proposed BMY inequality and there are many established geometric results for use. On the other hand, possibly generalizing an argument of J. Wahl, we suspect this work can be extended to unipolar Fano varieties with mild singularities.Third year: We consider Fano varieties with Picard rank bigger than one. Then c_1(F) is only big. To improve the positivity, we run a suitable foliated MMP. This is possible in dimension three due to recent advances by Cascini-Spicer-Svaldi. The first question is to understand the singularities of the pair (X,F) and see if one can run a foliated MMP in the category of varieties with terminal/canonical singularities. Finally, the proposed BMY inequality holds for smooth surfaces with nef tangent bundles. Hence there are chances to extend our approach to this setting and find applications in future research.