Let (X, ) be a pair. We study how the condition (X, ) causes surjectivity or birationality of the Albanese map and the Albanese morphism of (X, ) in both characteristic (X, ) and characteristic (X, ) . In particular in characteristic (X, ) we generalize Kawamata's result to the cases of log canonial pairs, and in characteristic (X, ) we generalize a result of Hacon-Patakfalvi to the cases of log pairs. Moreover we show that if (X, ) is a normal projective threefold in characteristic (X, ) , the coefficients of the components of (X, ) are (X, ) and (X, ) is semiample, then the Albanese morphism of (X, ) is surjective under reasonable assumptions on (X, ) and the singularities of the general fibers of the Albanese morphism. This is a positive characteristic analog in dimension 3 of a result of Zhang on a conjecture of Demailly-Peternell-Schneider.

We use the mirror theorem for toric Deligne-Mumford stacks, proved recently by the authors and by Cheong-Ciocan-Fontanine-Kim, to compute genus-zero Gromov-Witten invariants of a number of toric orbifolds and gerbes. We prove a mirror theorem for a class of complete intersections in toric Deligne-Mumford stacks, and use this to compute genus-zero Gromov-Witten invariants of an orbifold hypersurface.

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We show that Q-Fano varieties of fixed dimension with anti-canonical degrees and alpha-invariants bounded from below form a bounded family. As a corollary, K-semistable Q-Fano varieties of fixed dimension with anti-canonical degrees bounded from below form a bounded family.

Let X_\Sigma be a toric surface and let (\check {X}, W) be its Landau-Ginzburg (LG) mirror where W is the Hori-Vafa potential as shown in their preprint. We apply asymptotic analysis to study the extended deformation theory of the LG model (\check {X}, W), and prove that semi-classical limits of Fourier modes of a specific class of Maurer-Cartan solutions naturally give rise to tropical disks in X with Maslov index 0 or 2, the latter of which produces a universal unfolding of W. For X = \mathbb{P}^2, our construction reproduces Gross' perturbed potential W_n [Adv. Math. 224 (2010), pp. 169-245] which was proven to be the universal unfolding of W written in canonical coordinates. We also explain how the extended deformation theory can be used to reinterpret the jumping phenomenon of W_n across walls of the scattering diagram formed by Maslov index 0 tropical disks originally observed by Gross in the same work (in the case of X = \mathbb{P}^2).