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Simple Grassmannian flops

Jiun-Cheng Chen, Hsian-Hua Tseng

TL;DR

The paper introduces simple Grassmannian flops, a class of crepant birational maps modeled on Grassmannian subbundle geometry, and establishes their foundational properties via semismall contraction structure. It proves a derived-category equivalence across the flop and derives crepant transformation results for projective local models, identifying cohomological and Gromov-Witten data before and after the birational map. For these local models, genus-zero Gromov-Witten theory is matched by an explicit symplectic transformation, extending to quantum cohomology via isomorphisms of the relevant rings, with higher-genus theory recovered through semisimplicity and reconstruction theorems. Overall, the work provides a concrete framework for analyzing crepant transformations beyond $\,P^r$-flops, linking derived categories, quantum cohomology, and higher-genus GW invariants in the Grassmannian-flop setting.

Abstract

We introduce a class of flops between projective varieties modelled on direct sums of universal subbundles of Grassmannians. We study basic properties of these flops.

Simple Grassmannian flops

TL;DR

The paper introduces simple Grassmannian flops, a class of crepant birational maps modeled on Grassmannian subbundle geometry, and establishes their foundational properties via semismall contraction structure. It proves a derived-category equivalence across the flop and derives crepant transformation results for projective local models, identifying cohomological and Gromov-Witten data before and after the birational map. For these local models, genus-zero Gromov-Witten theory is matched by an explicit symplectic transformation, extending to quantum cohomology via isomorphisms of the relevant rings, with higher-genus theory recovered through semisimplicity and reconstruction theorems. Overall, the work provides a concrete framework for analyzing crepant transformations beyond -flops, linking derived categories, quantum cohomology, and higher-genus GW invariants in the Grassmannian-flop setting.

Abstract

We introduce a class of flops between projective varieties modelled on direct sums of universal subbundles of Grassmannians. We study basic properties of these flops.

Paper Structure

This paper contains 12 sections, 6 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Background
  3. This paper
  4. Acknowledgment
  5. Foundations
  6. Local models and general definition
  7. Comparison with $\mathbb{P}^r$-flops
  8. Derived equivalence
  9. Discussion on embedding of normal bundles
  10. Gromov-Witten theory of projective local models
  11. Genus zero
  12. Higher genus

Key Result

Lemma 1.3

Notations as in Example example:local. The contraction $\bar{{\phi}}_{-}: \bar{X}_-\to \bar{X}_0$ is semismall (in the sense of intersection homology). Hence a simple Grassmannian flop $\psi$ in Definition def:sGr_flop arises from a semismall contraction.

Theorems & Definitions (17)

  • Conjecture 1
  • Example 1.1: Local models
  • Definition 1.2
  • Lemma 1.3
  • proof
  • Remark 1
  • Lemma 1.4
  • proof
  • Proposition 1.5
  • proof
  • ...and 7 more