[Paper Review] Simply Connected Symplectic Calabi-Yau 6-Manifolds

Anar Akhmedov|arXiv (Cornell University)|Jul 13, 2011

Geometry and complex manifolds19 references3 citations

TL;DR

This paper constructs the first known simply connected symplectic Calabi-Yau 6-manifolds using Gompf's symplectic connected sum along tori, and also produces symplectic non-Kähler Calabi-Yau 6-manifolds with fundamental group ℤ. The construction leverages symplectic fiber sum operations on E(1)×𝕋², and further applies coisotropic Luttinger surgery to generate new examples, offering a simpler alternative to prior constructions and advancing the geography of symplectic Calabi-Yau 6-manifolds.

ABSTRACT

In this paper, we construct simply connected symplectic Calabi-Yau 6-manifolds by applying Gompf's symplectic fiber sum operation along $T^4$. Using our construction, we also produce symplectic non-Kähler Calabi-Yau 6-manifolds with fundamental group $\Z$. In this paper, we also produce the first examples of simply connected symplectic Calabi-Yau and non-Calabi-Yau 6-manifolds via coisotropic Luttinger surgery.

Motivation & Objective

• To construct simply connected symplectic Calabi-Yau 6-manifolds using symplectic fiber sum operations.
• To produce symplectic non-Kähler Calabi-Yau 6-manifolds with fundamental group ℤ.
• To offer a simpler construction method compared to prior works, such as Fine and Panov (2009).
• To explore the geography of symplectic Calabi-Yau 6-manifolds through surgery techniques.
• To demonstrate that coisotropic Luttinger surgery can yield simply connected and ℤ-fundamental group symplectic Calabi-Yau 6-manifolds from non-simply connected starting manifolds.

Proposed method

• The symplectic connected sum operation is applied to two copies of E(1)×𝕋² along embedded 𝕋⁴ submanifolds, using orientation-reversing bundle isomorphisms satisfying the Euler class condition e(ν₁)+e(ν₂)=0.
• The resulting manifold inherits a symplectic structure via Gompf's theorem, ensuring the symplectic form is well-defined on the sum.
• The first Chern class c₁ is shown to vanish on the sum manifold by verifying its vanishing on homology generators, including dual spheres and fiber classes.
• Coisotropic Luttinger surgery is performed on two 4-tori in K3×𝕋², using meridional and dual cycles to modify the fundamental group.
• The surgeries are chosen such that p=q=1 yields trivial fundamental group, while p=1,q=0 or p=0,q=1 yields fundamental group ℤ.
• The construction relies on the fact that the dual spheres and meridians are null-homotopic in the complement, ensuring the surgery preserves symplectic structure and allows control over π₁.

Experimental results

Research questions

• RQ1 Can simply connected symplectic Calabi-Yau 6-manifolds be constructed via symplectic connected sum operations?
• RQ2 Can symplectic non-Kähler Calabi-Yau 6-manifolds with fundamental group ℤ be constructed using symplectic fiber sum techniques?
• RQ3 Is coisotropic Luttinger surgery a viable method to produce symplectic Calabi-Yau 6-manifolds with controlled fundamental group?
• RQ4 How does the first Chern class behave under symplectic connected sum and Luttinger surgery in dimension 6?
• RQ5 What is the role of the symplectic fiber sum and Luttinger surgery in expanding the known geography of symplectic Calabi-Yau 6-manifolds?

Key findings

• The first known simply connected symplectic Calabi-Yau 6-manifold is constructed as the symplectic connected sum of two copies of E(1)×𝕋² along 𝕋⁴.
• A symplectic non-Kähler Calabi-Yau 6-manifold with fundamental group ℤ is constructed via the same symplectic fiber sum operation.
• Coisotropic Luttinger surgery on two 4-tori in K3×𝕋² yields a simply connected symplectic Calabi-Yau 6-manifold when p=q=1.
• When one surgery is performed (p=1,q=0 or p=0,q=1), the resulting manifold has fundamental group ℤ and is symplectic Calabi-Yau.
• The first Chern class c₁ vanishes on all homology generators of the resulting manifolds, confirming the Calabi-Yau condition.
• The construction provides a simpler alternative to the Fine-Panov construction of simply connected symplectic Calabi-Yau 6-manifolds.