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Counting surfaces on Calabi-Yau 4-folds I: Foundations

Younghan Bae, Martijn Kool, Hyeonjun Park

TL;DR

This paper develops a foundational framework for counting surfaces on Calabi–Yau 4-folds by introducing three moduli spaces of 2-dimensional objects (the Hilbert scheme of 2D subschemes and PT_q stable-pair moduli for q ∈ {−1,0,1}) and relating them via GIT wall-crossing and derived-category stability. It constructs reduced Oh-Thomas virtual cycles using non-degenerate cosections from semi-regularity, proving deformation invariance along Hodge loci and establishing a reduced virtual dimension rvd = $n-\tfrac{1}{2}\gamma^2$ + $\tfrac{1}{2}\rho_\gamma$, which enables a virtual-generalized version of the variational Hodge conjecture in this setting. The work develops moduli theory for PT_q pairs, connects these to complexes and Bayer’s polynomial Bridgeland stability, and provides relative and derived-algebraic-geometry perspectives that underpin a robust, deformation-invariant apparatus for counting surfaces. These foundations pave the way for sequels that prove DT/PT-type correspondences, compute invariants in concrete geometries, and explore physics-inspired connections such as G_4-flux and instanton frameworks. The integration of cosection localization, algebraic twistor methods, and (-2)-shifted symplectic enhancements offers a versatile toolkit for studying surface counts on CY 4-folds and their Hodge-theoretic ramifications.

Abstract

This is the first part in a series of papers on counting surfaces on Calabi-Yau 4-folds. Besides the Hilbert scheme of 2-dimensional subschemes, we introduce \emph{two} types of moduli spaces of stable pairs. We show that all three moduli spaces are related by GIT wall-crossing and parametrize stable objects in the bounded derived category. We construct \emph{reduced} Oh-Thomas virtual cycles on the moduli spaces via Kiem-Li cosection localization and prove that they are deformation invariant along Hodge loci. As an application, we show that the variational Hodge conjecture holds for any family of Calabi-Yau 4-folds supporting a non-zero reduced virtual cycle.

Counting surfaces on Calabi-Yau 4-folds I: Foundations

TL;DR

This paper develops a foundational framework for counting surfaces on Calabi–Yau 4-folds by introducing three moduli spaces of 2-dimensional objects (the Hilbert scheme of 2D subschemes and PT_q stable-pair moduli for q ∈ {−1,0,1}) and relating them via GIT wall-crossing and derived-category stability. It constructs reduced Oh-Thomas virtual cycles using non-degenerate cosections from semi-regularity, proving deformation invariance along Hodge loci and establishing a reduced virtual dimension rvd = + , which enables a virtual-generalized version of the variational Hodge conjecture in this setting. The work develops moduli theory for PT_q pairs, connects these to complexes and Bayer’s polynomial Bridgeland stability, and provides relative and derived-algebraic-geometry perspectives that underpin a robust, deformation-invariant apparatus for counting surfaces. These foundations pave the way for sequels that prove DT/PT-type correspondences, compute invariants in concrete geometries, and explore physics-inspired connections such as G_4-flux and instanton frameworks. The integration of cosection localization, algebraic twistor methods, and (-2)-shifted symplectic enhancements offers a versatile toolkit for studying surface counts on CY 4-folds and their Hodge-theoretic ramifications.

Abstract

This is the first part in a series of papers on counting surfaces on Calabi-Yau 4-folds. Besides the Hilbert scheme of 2-dimensional subschemes, we introduce \emph{two} types of moduli spaces of stable pairs. We show that all three moduli spaces are related by GIT wall-crossing and parametrize stable objects in the bounded derived category. We construct \emph{reduced} Oh-Thomas virtual cycles on the moduli spaces via Kiem-Li cosection localization and prove that they are deformation invariant along Hodge loci. As an application, we show that the variational Hodge conjecture holds for any family of Calabi-Yau 4-folds supporting a non-zero reduced virtual cycle.
Paper Structure (42 sections, 80 theorems, 469 equations, 1 figure)

This paper contains 42 sections, 80 theorems, 469 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Counting surfaces
  4. Moduli spaces of stable pairs
  5. Reduced virtual cycles
  6. Deformation invariance
  7. Moduli stack of semi-stable sheaves
  8. Derived algebraic geometry
  9. Relations to physics
  10. Sequels
  11. Related work
  12. Notation and conventions
  13. Stable pairs
  14. $\mathrm{PT}_q$-stability
  15. Special cases
  16. ...and 27 more sections

Key Result

Theorem 1.3

Let $X$ be a smooth projective variety and $v\in H^*(X,\mathbb{Q})$. Then, for each $q \in \{-1,0,1\}$, there exists a fine moduli space ${\curly P}^{(q)}_v(X)$ of $\mathrm{PT}_q$-stable pairs on $X$ with $\operatorname{ch}(F)=v$ as a projective scheme. Moreover, there exists a projective scheme $\m In particular, ${\curly P}^{(q)}_v(X)$ is the GIT quotient $\mathcal{M}/\!/_{\mathcal{L}^{(q)}}\mat

Theorems & Definitions (221)

  • Definition 1.1: Definition \ref{["def:stabilityconditionsofpairs'"]}
  • Example 1.2: Proposition \ref{['Lem.PT0=PT1']}
  • Theorem 1.3: Theorem \ref{['Thm:GIT']}
  • Theorem 1.4: Theorem \ref{['Thm:PairtoPerf']}, Proposition \ref{['prop:PTq=Bayerstab']}
  • Example 1.5
  • Theorem 1.6: Theorem \ref{['Thm:RVFC']}, Proposition \ref{['prop:redSOT']}
  • Theorem 1.7: Theorem \ref{['Thm:SR=smoothofrvd']}
  • Example 1.8
  • Proposition 1.9: Proposition \ref{['prop:4.6.1']}
  • Corollary 1.10: Corollary \ref{['cor:cicy4']}, \ref{['cor:4.6.5']}
  • ...and 211 more