We realize the moduli spaces of cubic fourfolds with specified group actions as arithmetic quotients of complex hyperbolic balls or type IV symmetric domains, and study their compactifications. We prove the geometric (GIT) compactifications are naturally isomorphic to the Hodge theoretic (Looijenga, in many cases Baily–Borel) compactifications. The key ingredients of the proof are the global Torelli theorem by Voisin, the characterization of the image of the period map given by Looijenga and Laza independently, and the functoriality of Looijenga compactifications proved in the Appendix.

This is an expanded version of the third author's lecture in String-Math 2015 at Sanya. It summarizes some of our works in quantum cohomology. After reviewing the quantum Lefschetz and quantum Leray--Hirsch, we discuss their applications to the functoriality properties under special smooth flops, flips and blow-ups. Finally, for conifold transitions of Calabi--Yau 3-folds, formulations for small resolutions (blow-ups along Weil divisors) are sketched.

Let Y be the zero loci of a regular section of a convex vector bundle Y over Y . We provide a proof of a conjecture of Cox, Katz and Lee for the virtual class of the genus zero moduli of stable maps to Y . This in turn yields the expected relationship between Gromov-Witten theories of Y and Y which together with Mirror Theorems allows for the calculation of enumerative invariants of Y inside of Y .

We study the limit of Calabi-Yau modular forms, and in particular, those resulting in classical modular forms. We then study two parameter families of elliptically fibred Calabi-Yau fourfolds and describe the modular forms arising from the degeneracy loci. In the case of elliptically fibred Calabi-Yau threefolds our approach gives a mathematical proof of many observations about modularity properties of topological string amplitudes starting with the work of Candelas, Font, Katz and Morrison. In the case of Calabi-Yau fourfolds we derive new identities not computed before.

Let X be an equivariant embedding of a connected reductive group X over an algebraically closed field X of positive characteristic. Let X denote a Borel subgroup of X . A X -Schubert variety in X is a subvariety of the form $\diag (G)\cdot V $, where X is a X -orbit closure in X . In the case where X is the wonderful compactification of a group of adjoint type, the X -Schubert varieties are the closures of Lusztig's X -stable pieces. We prove that X admits a Frobenius splitting which is compatible with all X -Schubert varieties. Moreover, when X is smooth, projective and toroidal, then any X -Schubert variety in X admits a stable Frobenius splitting along an ample divisors. Although this indicates that X -Schubert varieties have nice singularities we present an example of a non-normal X -Schubert variety in the wonderful compactification of a group of type X . Finally we also extend the Frobenius splitting results to the more general class of X -Schubert varieties.