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On the quantum cohomology of blow-ups of four-dimensional quadrics

Jianxun Hu, Huazhong Ke, Changzheng Li, Lei Song

TL;DR

The paper analyzes the small quantum cohomology of blow-ups of the four-dimensional quadric $Q^4$ along a point and along a plane, with the goal of validating Conjecture $\mathcal{O}$ and Galkin's lower bound in these new Fano manifolds. It combines geometric blow-up descriptions, explicit two-point Gromov–Witten invariants, and carefully chosen cohomology bases to compute the quantum multiplication by $c_1$, reducing to genus-zero two-point data via the divisor axiom. The main results show that the spectral radius $\rho(\hat{c}_1)$ increases under these blow-ups and that both Conjecture $\mathcal{O}$ and the refined lower bound hold for $X_0$ and $X_2$, with clear positive matrix behaviour after taking suitable powers. The methods provide evidence for a broader principle that blow-ups along higher-codimension centers raise $\rho(\hat{c}_1)$ beyond the base variety, offering a blueprint for extending these techniques to other blow-ups of Grassmannians or related Fano manifolds.

Abstract

We propose a conjecture relevant to Galkin's lower bound conjecture, and verify it for the blow-ups of a four-dimensional quadric at a point or along a projective plane. We also show that Conjecture $\mathcal{O}$ holds in these two cases.

On the quantum cohomology of blow-ups of four-dimensional quadrics

TL;DR

The paper analyzes the small quantum cohomology of blow-ups of the four-dimensional quadric along a point and along a plane, with the goal of validating Conjecture and Galkin's lower bound in these new Fano manifolds. It combines geometric blow-up descriptions, explicit two-point Gromov–Witten invariants, and carefully chosen cohomology bases to compute the quantum multiplication by , reducing to genus-zero two-point data via the divisor axiom. The main results show that the spectral radius increases under these blow-ups and that both Conjecture and the refined lower bound hold for and , with clear positive matrix behaviour after taking suitable powers. The methods provide evidence for a broader principle that blow-ups along higher-codimension centers raise beyond the base variety, offering a blueprint for extending these techniques to other blow-ups of Grassmannians or related Fano manifolds.

Abstract

We propose a conjecture relevant to Galkin's lower bound conjecture, and verify it for the blow-ups of a four-dimensional quadric at a point or along a projective plane. We also show that Conjecture holds in these two cases.

Paper Structure

This paper contains 14 sections, 22 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conjecture $\mathcal{O}$
  4. Quantum cohomology of $Q^4$
  5. Quantum cohomology of $Bl_{\rm \mathbb{P}^0}Q^4$
  6. Geometric construction
  7. Classical cohomology
  8. Comparison of two bases of $H^*(X_0)$
  9. Comparison in $H^2(X_0)$ and $H^6(X_0)$
  10. Comparison in $H^4(X_0)$
  11. Two-point invariants
  12. Quantum cohomology of $Bl_{{\mathbb P}^2}Q^4$
  13. Classical cohomology
  14. Two-point invariants

Key Result

Theorem 1.2

Conjecture $\mathcal{O}$, Conjecture conj and Galkin's lower bound conjecture all hold for the blow-ups $Bl_{\mathbb{P}^0} {Q}^4$ and $Bl_{\mathbb{P}^2} {Q}^4$.

Theorems & Definitions (45)

  • Conjecture 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Corollary 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 35 more