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Let M and N be two compact complex manifolds. We show that if the tautological line bundle O_{T^∗}M(1) is not pseudo-effective and O_{T^∗}N(1) is nef, then there is no non-constant holomorphic map from M to N. In particular, we prove that any holomorphic map from a compact complex manifold M with RC-positive tangent bundle to a compact complex manifold N with nef cotangent bundle must be a constant map. As an application, we obtain that there is no non-constant holomorphic map from a compact Hermitian manifold with positive holomorphic sectional curvature to a Hermitian manifold with non-positive holomorphic bisectional curvature.

Given a smooth projective variety X and a smooth divisor D\subset X. We study relative Gromov-Witten invariants of (X,D) and the corresponding orbifold Gromov-Witten invariants of the r-th root stack X_{D,r}. For sufficiently large r, we prove that orbifold Gromov- Witten invariants of X_{D,r} are polynomials in r. Moreover, higher genus relative Gromov-Witten invariants of (X,D) are exactly the constant terms of the corresponding higher genus orbifold Gromov-Witten invari- ants of X_{D,r}. We also provide a new proof for the equality between genus zero relative and orbifold Gromov-Witten invariants, originally proved by Abramovich-Cadman-Wise. When r is sufficiently large and X=C is a curve, we prove that stationary relative invariants of C are equal to the stationary orbifold invariants in all genera.

We introduce the notion of completed F-crystals on the absolute prismatic site of a smooth p-adic formal scheme. We define a functor from the category of completed prismatic F-crystals to that of crystalline étale Zp-local systems on the generic fiber of the formal scheme and show that it gives an equivalence of categories. This generalizes the work of Bhatt and Scholze, which treats the case of a complete discrete valuation ring with perfect residue field.

Using the mirror theorem [CCIT15], we give a Landau-Ginzburg mirror description for the big equivariant quantum cohomology of toric Deligne-Mumford stacks. More precisely, we prove that the big equivariant quantum D-module of a toric Deligne-Mumford stack is isomorphic to the Saito structure associated to the mirror Landau-Ginzburg potential. We give a GKZ-style presentation of the quantum D-module, and a combinatorial description of quantum cohomology as a quantum Stanley-Reisner ring. We establish the convergence of the mirror isomorphism and of quantum cohomology in the big and equivariant setting.