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Quantum cohomology of blowups

Hiroshi Iritani

Abstract

We prove a decomposition theorem of the quantum cohomology D-module of the blowup of a smooth projective variety X along a smooth subvariety Z. The main tools we use are shift operators and Fourier analysis for equivariant quantum cohomology.

Quantum cohomology of blowups

Abstract

We prove a decomposition theorem of the quantum cohomology D-module of the blowup of a smooth projective variety X along a smooth subvariety Z. The main tools we use are shift operators and Fourier analysis for equivariant quantum cohomology.
Paper Structure (49 sections, 38 theorems, 254 equations, 5 figures, 1 table)

This paper contains 49 sections, 38 theorems, 254 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Decomposition theorem
  3. Equivariant quantum cohomology and Fourier transformation
  4. Global Kähler moduli space for blowups
  5. Plan of the paper
  6. Preliminaries
  7. Gromov-Witten invariants
  8. Formal power series rings
  9. Quantum cohomology and quantum connection
  10. Equivariant quantum cohomology and quantum connection
  11. Equivariant curve classes
  12. Extended shift operators
  13. Givental cone
  14. Twisted Gromov-Witten invariants
  15. Quantum Riemann-Roch theorem
  16. ...and 34 more sections

Key Result

Theorem 1.1

There exist a formal invertible change of variables $H^*({\widetilde{X}}) \to H^*(X) \oplus H^*(Z)^{\oplus (r-1)}$, ${\tilde{\tau}} \mapsto \left(\tau({\tilde{\tau}}), \{\varsigma_j({\tilde{\tau}})\}_{0\le j\le r-2}\right)$ defined over $\mathbb{C}(\!(\mathfrak{q}^{-\frac{1}{r-1}})\!)[\![Q]\!]$ and that commutes with the quantum connection. Moreover, $\Psi$ intertwines the pairing $P_{\widetilde{

Figures (5)

  • Figure 1: The GIT fan on the $T$-ample cone $C_T(W)$ and the associated "toric variety" $\mathfrak{M} = \mathfrak{M}_X \cup \mathfrak{M}_{{\widetilde{X}}}$. See also Figure \ref{['fig:Mori_cones']}. The variable $\mathfrak{q}$, which is written also as $y S^{-1}$ in the main body of the text, comes from a shift operator.
  • Figure 2: $W = \operatorname{Bl}_{Z\times \{0\}} (X\times \mathbb{P}^1)$ and a moment map $\mu \colon W\to \mathbb{R}$
  • Figure 3: A schematic picture of the cones $C_X^\vee$ and $C_{\widetilde{X}}^\vee$ in $N_1^T(W)$.
  • Figure 4: Integration contour on the $\lambda$-plane
  • Figure 5: Euler eigenvalues near $Q={\tilde{\tau}}=0$ for $r=4$. The eigenvalues of $E_Z\star_{\varsigma_j({\tilde{\tau}})}$, $j=0,\dots,r-2$ surround those of $E_X\star_{\tau({\tilde{\tau}})}$ as "satellites". Here we assume the convergence of quantum cohomology and the maps $\tau({\tilde{\tau}}), \varsigma_j({\tilde{\tau}})$. We also choose $\mathfrak{q}$ to be positive real.

Theorems & Definitions (97)

  • Theorem 1.1: see Theorem \ref{['thm:decomposition_fd']} for the details
  • Corollary 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7: related works
  • Conjecture 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 87 more