Transversality on Orbifolds and Counting Holomorphic Curves

Transversality does not play well with symmetry as symmetric objects are typically not in "general position". As a result, one may not be able to achieve transversality on orbifolds. This feature is often accompanied with another feature of orbifolds: Poincar\'e duality only holds over rational numbers but not integers.

Inspired by an old proposal of Fukaya-Ono in symplectic topology, we define a new notion of transversality for sections of complex orbifold vector bundles. For sections satisfying this transversality condition, its zero locus is a Thom-Mather stratified space and each stratum defines a cycle in integer coefficients. We also obtained several applications in symplectic topology, including the definition of integer-valued Gromov-Witten invariants and proved the Arnold conjecture over integers. This is a joint work with Shaoyun Bai.