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Continuous and Discrete Asymptotic Behaviours of the J-function of a Fano Manifold

ChunYin Hau

TL;DR

The paper introduces the asymptotically Mittag-Leffler (aML) condition for cohomology-valued J-functions of Fano manifolds and proves that, when $t^{\frac{1}{2}\dim X} J(t)$ is $(T,\theta,A)$-scaled aML, the J-function admits the exponential growth $J(t) = \frac{1}{r} t^{-\frac{1}{2}\dim X} e^{T t} (A + o(1))$ as $t\to+\infty$, enabling computation of the principal asymptotic class from $J$-function coefficients. The authors establish that the aML property is preserved under products and under taking hypersurfaces (via quantum Lefschetz), and they verify the projective space case explicitly, showing the framework aligns with known Gamma conjecture I predictions. They further develop a calculus for aML series, including a Riemann-Liouville extension to cohomology and a product rule that yields aML behavior for products of manifolds, thereby enabling the construction of aML $J$-functions for a broad class of Fano manifolds. Overall, the work provides a practical, coefficient-based method to extract principal asymptotic data from the $J$-function, with implications for understanding the Gamma class and its conjectural equality (Gamma conjecture I) in various geometric settings.

Abstract

In this paper, we propose a condition on the coefficients of a cohomology-valued power series, which we call ``asymptotically Mittag-Leffler''. We show that if the $J$-function of a Fano manifold is asymptotically Mittag-Leffler, then it has the exponential growth as $t\to +\infty$. This provides an alternative method to compute the principal asymptotic class of a Fano manifold using the coefficients of $J$-function. We also verify that the $J$-function of the projective space is asymptotically Mittag-Leffler, and the property of having an asymptotically Mittag-Leffler $J$-function is preserved when taking product and hypersurface.

Continuous and Discrete Asymptotic Behaviours of the J-function of a Fano Manifold

TL;DR

The paper introduces the asymptotically Mittag-Leffler (aML) condition for cohomology-valued J-functions of Fano manifolds and proves that, when is -scaled aML, the J-function admits the exponential growth as , enabling computation of the principal asymptotic class from -function coefficients. The authors establish that the aML property is preserved under products and under taking hypersurfaces (via quantum Lefschetz), and they verify the projective space case explicitly, showing the framework aligns with known Gamma conjecture I predictions. They further develop a calculus for aML series, including a Riemann-Liouville extension to cohomology and a product rule that yields aML behavior for products of manifolds, thereby enabling the construction of aML -functions for a broad class of Fano manifolds. Overall, the work provides a practical, coefficient-based method to extract principal asymptotic data from the -function, with implications for understanding the Gamma class and its conjectural equality (Gamma conjecture I) in various geometric settings.

Abstract

In this paper, we propose a condition on the coefficients of a cohomology-valued power series, which we call ``asymptotically Mittag-Leffler''. We show that if the -function of a Fano manifold is asymptotically Mittag-Leffler, then it has the exponential growth as . This provides an alternative method to compute the principal asymptotic class of a Fano manifold using the coefficients of -function. We also verify that the -function of the projective space is asymptotically Mittag-Leffler, and the property of having an asymptotically Mittag-Leffler -function is preserved when taking product and hypersurface.

Paper Structure

This paper contains 10 sections, 11 theorems, 145 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Principal Asymptotic Class
  4. Riemann-Liouville Integral
  5. Asymptotically Mittag-Leffler
  6. Definition
  7. Properties
  8. Example: Projective Spaces
  9. Product
  10. Hypersurface

Key Result

Theorem 1

If the $J$-function of a Fano manifold $X$ is asymptotically Mittag-Leffler eq:disAsym, then:

Figures (3)

  • Figure : Figure of the distribution of the eigenvalues of $c_1(X_3)\star_0$
  • Figure : Figures of the domains $D$, $D'$ and $D"$
  • Figure : Figure of $\mathop{\mathrm{Polytope}}\limits \left( 0, \xi_{r_X}^k S_X, \xi_{r_Y}^\ell S_Y, \xi_{r_X}^k S_X+\xi_{r_Y}^\ell S_Y \right) \subseteq \mathbb{C}$

Theorems & Definitions (32)

  • Definition
  • Theorem : Main Results
  • Remark
  • Definition 2.1: Gromov-Witten Invariant MR1677117, MR1291244
  • Definition 2.2: Quantum Product MR1677117
  • Definition 2.3: Quantum Differential Equations MR1677117
  • Definition 2.4: Simple Rightmost Eigenvalue MR3536989counterexamples
  • Definition 2.5: Principal Asymptotic Class MR3536989
  • Definition 2.6: $J$-function MR1677117
  • Proposition 2.7: MR3536989
  • ...and 22 more