[Paper Review] Special Lagrangians of cohomogeneity one in the resolved conifold

Marianty Ionel, Maung Min-Oo|ArXiv.org|Jun 16, 2005

Geometry and complex manifolds11 references3 citations

TL;DR

This paper constructs cohomogeneity one special Lagrangian submanifolds in the resolved conifold, a Calabi-Yau 3-fold, using T²- and SO(3)-symmetries. It proves the resolved conifold admits a foliation by T²×ℝ special Lagrangians and an SO(3)-invariant family topologically S²×ℝ, both asymptotically approaching the same special Lagrangian cones in the conifold limit, mirroring results from the deformed conifold in prior work.

ABSTRACT

In this paper, which is a natural continuation of our previous paper math.DG/0504557, we describe some special Lagrangians of cohomogeneity one in the resolved conifold. Our main result gives a foliation of the resolved conifold by T^2-invariant special Lagrangians, where the generic leaf is topologically T^2 X R. We also obtain a family of SO(3)-invariant special Lagrangians. These special Lagrangian families in both the deformed and the resolved conifold approach asymptotically the same special Lagrangian cones in the conifold.

Motivation & Objective

• To extend the classification of cohomogeneity one special Lagrangians from the deformed conifold to the resolved conifold.
• To construct explicit families of special Lagrangian submanifolds invariant under T² and SO(3) actions in the resolved conifold.
• To analyze the asymptotic behavior of these special Lagrangians and compare them to the special Lagrangian cones in the singular conifold limit.
• To establish that both the T²- and SO(3)-invariant families in the resolved conifold approach the same special Lagrangian cones as their counterparts in the deformed conifold.

Proposed method

• Utilizes the resolved conifold as a Calabi-Yau 3-fold given by the total space of the canonical bundle over ℂP¹, with a holomorphic volume form derived from local coordinates.
• Applies the moment map formalism for SO(3) actions to identify orbits in the zero level set, ensuring Lagrangian condition ω|L ≡ 0.
• Imposes the special Lagrangian condition Im(Ω)|L ≡ 0 by analyzing the holomorphic volume form in local coordinates (U, Y, V) and computing its contraction on tangent vectors.
• Solves the resulting differential condition Re(Y²) = c, leading to a family of hyperbolic curves in the complex Y-plane, parameterizing the special Lagrangian leaves.
• Analyzes the topology of the resulting special Lagrangians, showing T²×ℝ and S²×ℝ structures via symmetry orbits and parameter dependence.
• Compares asymptotic behavior of the constructed families to the special Lagrangian cones in the conifold, confirming identical limiting geometry.

Experimental results

Research questions

• RQ1 Can cohomogeneity one special Lagrangians be constructed in the resolved conifold using T² and SO(3) symmetries?
• RQ2 What is the topology and geometry of the special Lagrangian leaves in the resolved conifold under these symmetries?
• RQ3 How do the asymptotic behaviors of these special Lagrangians compare to those in the deformed conifold and the singular conifold?
• RQ4 Do the SO(3)-invariant special Lagrangians in the resolved conifold approach the same special Lagrangian cones as in the deformed conifold?

Key findings

• The resolved conifold admits a foliation by T²-invariant special Lagrangians, with generic leaves diffeomorphic to T²×ℝ.
• An SO(3)-invariant family of special Lagrangians is constructed, each diffeomorphic to S²×ℝ, parameterized by the real constant c in Re(Y²) = c.
• The SO(3)-invariant special Lagrangians are asymptotic to the same special Lagrangian cones in the conifold as the SO(3)-invariant families in the deformed conifold from prior work.
• The T²-invariant special Lagrangians in the resolved conifold are asymptotic to the same conical special Lagrangians as their counterparts in the deformed conifold.
• The construction confirms that both the T²- and SO(3)-invariant families in the resolved conifold approach the same limiting conical geometry in the conifold limit.
• The special Lagrangians do not intersect the zero-section (the bolt) S², as |Y|² is preserved under the SO(3) action.