Chin-Lung Wang

equivalence in birational geometry and characterizations of complex elliptic genera

We show that for smooth complex projective varieties the most general combinations of chern numbers that are invariant under the K-equivalence relation consist of the complex elliptic genera. Combined with a recent result of Totaro, we deduce that up to complex cobordism any K-equivalence can be decomposed into a sequence of classical flops.

Invariance of Gromov-Witten theory under a simple flop

Abstract In this work, we continue our study initiated in [11]. We show that the generating functions of Gromov–Witten invariants with ancestors are invariant under a simple flop, for all genera, after an analytic continuation in the extended Kähler moduli space. The results presented here give the first evidence, and the only one not in the toric category, of the invariance of full Gromov–Witten theory under the K-equivalence (crepant transformation).

Invariance of Quantum Rings under Ordinary Flops II: A quantum Leray--Hirsch theorem

This is the second of a sequence of papers proving the quantum invariance for ordinary flops over an arbitrary smooth base. In this paper, we complete the proof of the invariance of the big quantum rings under ordinary flops of splitting type. To achieve that, several new ingredients are introduced. One is a quantum Leray--Hirsch theorem for the local model (a certain toric bundle) which extends the quantum D module of Dubrovin connection on the base by a Picard--Fuchs system of the toric fibers. Nonsplit flops as well as further applications of the quantum Leray--Hirsch theorem will be discussed in subsequent papers. In particular, a quantum splitting principle is developed in Part III which reduces the general ordinary flops to the split case solved here.