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).

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 study how Gromov-Witten invariants of projective 3-folds transform under a standard flop and a small extremal transition in the algebro-geometric setting from the recent development of algebraic relative Gromov-Witten theory and its applications. This gives an algebro-geometric account of Witten's wall-crossing formula for correlation functions of the descendant nonlinear sigma model in adjacent geometric phases of a gauge linear sigma model and of the symplectic approach in an earlier work of An-Min Li and Yongbin Ruan on the same problem. A terse account from the stringy and the symplectic viewpoint is given in the appendix to complement and compare to the discussion in the main text.

We associate a half-integer number, called the quantum index, to algebraic curves in the real plane satisfying to certain conditions. The area encompassed by the logarithmic image of such curves is equal to π2 times the quantum index of the curve, and thus has a discrete spectrum of values. We use the quantum index to refine enumeration of real rational curves in a way consistent with the Block–Göttsche invariants from tropical enumerative geometry.

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.