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A non-vanishing conjecture for cotangent bundles on elliptic surfaces

Haesong Seo

TL;DR

This work resolves the non-vanishing conjecture for cotangent bundles on isotrivial elliptic surfaces, establishing that pseudoeffectivity of $\Omega_S^1$ forces the existence of nonzero symmetric differential sections $H^0(S, S^m\Omega_S^1)$ for some $m>0$. The authors adapt Höring–Peternell's framework by passing to a birational model with a finite cover removing orbifold divisors and then perform a detailed analysis of logarithmic and local obstructions arising from singular fibres of Kodaira types, including $II$, $III$, $IV$ and their stars. Key steps involve bounding the poles of logarithmic symmetric differentials along exceptional curves, embedding global sections into $(C\times E)$-models, and leveraging ramification bounds together with Riemann–Hurwitz, to deduce pseudoeffectivity of $\Omega_S^1$ if non-vanishing fails. Consequently, together with HP2020, the result completes the non-vanishing question for all surfaces with Kodaira dimension $\kappa(S)\le 1$, providing a full picture for cotangent bundles in this regime.

Abstract

In this paper, we prove the non-vanishing conjecture for cotangent bundles on isotrivial elliptic surfaces. Combined with the result by Höring and Peternell, it completely solves the question for surfaces with Kodaira dimension at most $1$.

A non-vanishing conjecture for cotangent bundles on elliptic surfaces

TL;DR

This work resolves the non-vanishing conjecture for cotangent bundles on isotrivial elliptic surfaces, establishing that pseudoeffectivity of forces the existence of nonzero symmetric differential sections for some . The authors adapt Höring–Peternell's framework by passing to a birational model with a finite cover removing orbifold divisors and then perform a detailed analysis of logarithmic and local obstructions arising from singular fibres of Kodaira types, including , , and their stars. Key steps involve bounding the poles of logarithmic symmetric differentials along exceptional curves, embedding global sections into -models, and leveraging ramification bounds together with Riemann–Hurwitz, to deduce pseudoeffectivity of if non-vanishing fails. Consequently, together with HP2020, the result completes the non-vanishing question for all surfaces with Kodaira dimension , providing a full picture for cotangent bundles in this regime.

Abstract

In this paper, we prove the non-vanishing conjecture for cotangent bundles on isotrivial elliptic surfaces. Combined with the result by Höring and Peternell, it completely solves the question for surfaces with Kodaira dimension at most .
Paper Structure (6 sections, 11 theorems, 48 equations, 3 figures, 1 table)

This paper contains 6 sections, 11 theorems, 48 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Elliptic surfaces
  3. The non-vanishing conjecture for cotangent bundles
  4. Logarithmic symmetric differentials
  5. Local obstructions
  6. The case of $II^{\ast}$, $III^{\ast}$ and $IV^{\ast}$

Key Result

Theorem 1.2

Let $f:S \rightarrow B$ be a relatively minimal isotrivial elliptic surface. If $\Omega_S^1$ is pseudoeffective, then $H^0(S, S^m \Omega_S^1) \neq 0$ for some $m > 0$.

Figures (3)

  • Figure 2.1: Blowing-down procedures. The notation $a(-b)$ indicates that the corresponding curve has multiplicity $a$ and self-intersection $-b$. The curves contracted in each procedure are colored red.
  • Figure 3.1: Coordinate charts on the $f'$-fibre of type $III$.
  • Figure 3.2: Coordinate charts on the $f'$-fibre of type $III^{\ast}$.

Theorems & Definitions (16)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3: cf. HP2020
  • Corollary 1.4
  • Lemma 2.1: PS2020
  • Lemma 3.1: cf. HP2020
  • proof
  • Lemma 3.2: cf. BTVA2022
  • proof : Proof of Lemma \ref{['lem:log-symmetric-differential-on-III']}
  • Lemma 3.3
  • ...and 6 more