[Paper Review] A note on Bogomolov-Gieseker type inequality for Calabi-Yau 3-folds

Yukinobu Toda|arXiv (Cornell University)|Jan 24, 2012

Algebraic Geometry and Number Theory8 references4 citations

TL;DR

This paper proves a conjectural Bogomolov-Gieseker-type inequality for slope stable sheaves with minimal first Chern class on certain Calabi-Yau 3-folds, including quintic 3-folds in ℙ⁴. Using universal extensions and the classical Bogomolov-Gieseker inequality, the authors establish the inequality for sheaves with c₁ = [H] and ch₂(E)H > 0, showing ch₃(E) ≤ ch₂(E)H/(3ch₀(E)) with equality only for 𝒪_X(H).

ABSTRACT

The conjectural Bogomolov-Gieseker (BG) type inequality for tilt semistable objects on projective 3-folds was proposed by Bayer, Macri and the author. In this note, we prove our conjecture for slope stable sheaves with the smallest first Chern class on certain Calabi-Yau 3-folds, e.g. quintic 3-folds.

Motivation & Objective

• To establish a conjectural Bogomolov-Gieseker-type inequality for tilt semistable objects on Calabi-Yau 3-folds with minimal first Chern class.
• To verify Conjecture 1.1 for Calabi-Yau 3-folds where Pic(X) is generated by an ample divisor H.
• To extend the known Castelnuovo-type inequality for curves to higher rank sheaves via cohomological techniques.
• To provide a foundational step toward proving the full conjectural BG inequality in [3, Conjecture 1.3.1] via induction on c₁.
• To demonstrate that ch₃(E) is bounded above by a rational function of ch₀(E) and ch₂(E)H for slope stable sheaves with c₁ = [H].

Proposed method

• Construct the universal extension of a torsion-free sheaf E with c₁(E) = [H] to obtain a new sheaf F′ with controlled Chern characters.
• Apply the classical Bogomolov-Gieseker inequality to F′ to derive a bound on ext¹(F,𝒪_X), linking it to ch₂(E)H and H³.
• Use Serre duality and dimension bounds on |H| to estimate cohomological invariants and constrain ch₃(E).
• Leverage the assumption dim|H| ≥ (7/6)H³ − 3 and χ(𝒪_C) ≥ (1/6)H³ − C·H to derive the final inequality.
• Utilize the fact that ch₃(E) ≤ 0 unless E ≅ 𝒪_X(H), based on cohomological bounds and the structure of the extension sequence.
• Reduce the problem to known Castelnuovo-type inequalities for curves via geometric constraints on H and C.

Experimental results

Research questions

• RQ1 Does the conjectural Bogomolov-Gieseker inequality hold for slope stable sheaves with c₁ = [H] on Calabi-Yau 3-folds?
• RQ2 Can the inequality ch₃(E) ≤ ch₂(E)H/(3ch₀(E)) be proven for higher rank sheaves when ch₂(E)H > 0?
• RQ3 What geometric conditions on the ample divisor H ensure the validity of the BG-type inequality in the minimal c₁ case?
• RQ4 Is the inequality sharp, and when does equality occur?
• RQ5 Can the method of universal extensions and cohomological bounds be generalized to higher c₁ classes?

Key findings

• The conjectural Bogomolov-Gieseker inequality (1) holds for all torsion-free slope stable sheaves E with c₁(E) = [H] and ch₂(E)H > 0 on Calabi-Yau 3-folds satisfying the dimension and Euler characteristic assumptions.
• For quintic 3-folds in ℙ⁴, the inequality is proven unconditionally, as they satisfy the required geometric conditions.
• Equality in the inequality ch₃(E) ≤ ch₂(E)H/(3ch₀(E)) holds only when E ≅ 𝒪_X(H), confirming a uniqueness condition.
• The third Chern character ch₃(E) is bounded above by 0 for all such E except 𝒪_X(H), implying strong constraints on higher rank sheaves.
• The method provides a new cohomological approach that avoids hypersurface restriction, differing from prior bounds in [8].
• The result supports the broader conjecture in [3, Conjecture 1.3.1] and enables partial verification for c₁(E) = 2[H] via induction.