[Paper Review] Special Lagrangian Fibrations II: Geometry

Mark Gross|ArXiv.org|Sep 14, 1998

Geometry and complex manifolds24 references43 citations

TL;DR

This paper investigates the geometry of special Lagrangian fibrations on Calabi-Yau manifolds, focusing on constructing mirror manifolds via dual fibrations. By assuming the existence of a special Lagrangian fibration, it uses symplectic and complex geometry to define a dual fibration and constructs a mirror K3 surface without relying on the Torelli theorem, leveraging Yau's theorem and harmonic form analysis on elliptic fibrations to achieve mirror symmetry in dimension two.

ABSTRACT

We continue the study of the Strominger-Yau-Zaslow mirror symmetry conjecture. Roughly put, this states that if two Calabi-Yau manifolds X and Y are mirror partners, then X and Y have special Lagrangian torus fibrations which are dual to each other. Much work on this conjecture is necessarily of a speculative nature, as in dimension 3 it is still a very difficult problem of how to construct such fibrations. Nevertheless, assuming the existence of such fibrations there are many things one can prove. This paper covers a number of issues. First it applies results from the theory of completely integrable hamiltonian systems to understand some aspects of the geometry of such fibrations. From this, using reasonable regularity assumptions on the fibrations, one can understand how the cohomology of dual fibrations are related. We then study the question of how, given one such fibration, one would put a symplectic and complex structure on the dual fibrations, generalising work of Hitchin. While this question cannot be answered at this stage, these results should give insight into the nature of the problem. We sum up these ideas in a refined version of the Strominger-Yau-Zaslow conjecture. Finally, to give evidence for this conjecture, we prove it explicitly for K3 surfaces. One finds a construction of mirror symmetry for K3 surfaces which does not require the use of Torelli theorems, and is much more differential geometric in nature than previous constructions.

Motivation & Objective

• To explore the geometric properties of special Lagrangian fibrations on Calabi-Yau manifolds, particularly in the context of the Strominger-Yau-Zaslow mirror symmetry conjecture.
• To establish a method for constructing the mirror manifold of a Calabi-Yau threefold via dual fibrations when a special Lagrangian fibration exists.
• To demonstrate that for K3 surfaces, the mirror can be constructed directly from a special Lagrangian fibration without invoking the Torelli theorem.
• To verify that the constructed mirror satisfies the required cohomological mirror symmetry conditions, including Hodge number exchange and Kähler class positivity.
• To show that the mirror complex structure and Kähler form can be uniquely determined via harmonic forms and cohomological data modulo exceptional divisors.

Proposed method

• Assumes the existence of a special Lagrangian fibration $ f: X \to B $, focusing on the geometry of the regular locus and its symplectic structure.
• Applies Duistermaat's theory of global action-angle coordinates to construct canonical coordinates on the regular part of the fibration using the cotangent bundle of the base $ B $.
• Uses the fact that special Lagrangian submanifolds are volume-minimizing to guide expectations about the regularity and behavior of the fibration.
• Constructs the dual fibration $ \check{X}_0 \to B_0 $ via the sheaf $ R^1f_{0*}\mathbb{R}/R^1f_{0*}\mathbb{Z} $, and compactifies it to obtain $ \check{X} $.
• Applies Yau's theorem to construct a Ricci-flat Kähler metric on the mirror by ensuring the cohomology class of the real part of the holomorphic 2-form is positive and of type (1,1).
• Uses the Hodge decomposition and duality to relate cohomology classes of $ \Omega $, $ \check{\Omega} $, $ \omega $, and $ \check{\omega} $, and resolves ambiguity in the $ B $-field via section choice.

Experimental results

Research questions

• RQ1 How can a special Lagrangian fibration on a Calabi-Yau manifold be used to construct its mirror manifold without relying on the Torelli theorem?
• RQ2 What conditions ensure that the dual fibration constructed from a special Lagrangian fibration yields a well-defined complex structure on the mirror?
• RQ3 How do the cohomological data of the holomorphic and Kähler forms on the original and mirror manifolds satisfy the mirror symmetry conditions?
• RQ4 To what extent can the $ B $-field ambiguity be resolved by choosing a different zero-section in the dual fibration?
• RQ5 What role do harmonic forms and volume minimization play in ensuring the existence and uniqueness of the mirror Kähler structure?

Key findings

• The mirror K3 surface $ \check{S}_K $ is constructed directly from a special Lagrangian fibration on $ S $, without appealing to the Torelli theorem.
• The cohomology class of the holomorphic 2-form $ \Omega $ on the mirror is determined modulo the exceptional divisor $ E $, and satisfies $ [\Omega]^2 = 0 $, which fixes its class up to normalization.
• The Kähler class $ [\mathop{\rm Re}\check{\Omega}] $ is shown to be positive and of type (1,1), ensuring the existence of a unique Ricci-flat Kähler metric via Yau's theorem.
• The construction resolves the ambiguity in the $ B $-field by redefining the zero-section $ \sigma_0 $, thereby ensuring uniqueness of the mirror structure.
• The mirror symmetry relation $ \mathop{\rm Im}\check{\Omega} \propto \omega $, $ \mathop{\rm Im}\Omega \propto \check{\omega} $, and $ \mathop{\rm Re}\check{\Omega} - \sigma_0 = \mathbf{B} $ holds up to cohomological equivalence modulo $ E $.
• The volume of the mirror fiber satisfies $ \mathop{\rm Vol}(\check{S}_b) = 1/\mathop{\rm Vol}(S_b) $, confirming the expected duality in fiber volumes.