[Paper Review] Special Lagrangian Fibrations I: Topology

Mark Gross|arXiv (Cornell University)|Oct 6, 1997

Geometry and complex manifolds14 references62 citations

TL;DR

This paper proposes a topological framework for special Lagrangian fibrations on Calabi-Yau manifolds, assuming their mirrors arise via dualization of such fibrations. It establishes a cohomological mirror map via monodromy actions and proves that the monodromy weight filtration and Leray filtration coincide in the threefold case, aligning with predictions from mirror symmetry and homological mirror symmetry.

ABSTRACT

In 1996, Strominger, Yau and Zaslow made a conjecture about the geometric relationship between two mirror Calabi-Yau manifolds. Roughly put, if X and Y are a mirror pair of such manifolds, then X should possess a special Lagrangian torus fibration $f:X o B$ such that Y is obtained by dualizing the fibration f. This leaves a huge amount to be done to verify such conjectures. This paper takes a speculative point of view, in that it assumes that a special Lagrangian torus fibration exists on X. We address a number of questions of a topological nature: what is the relationship between the cohomology of X and the cohomology of the dual fibration? what kind of information does the Leray spectral sequence for f contain? what is the relationship between the topological (1,1) couplings of the dual of f and the (1,n-1)-couplings of X in the large complex structure limit? These questions are shown to have nice answers if a key conjecture about the monodromy diffeomorphisms about a large complex structure limit point holds. Roughly put, this conjecture says that monodromy about a large complex structure limit point can be described as a very natural generalization of a Dehn twist for an elliptic curve. Given this conjecture, we show, among other results, that the large complex radius limit of the (1,n-1) couplings on X coincide with the topological (1,1) couplings on Y, and if dim X=3, the Leray filtration and weight filtrations of the mixed Hodge structure coincide, as conjectured by myself and P.M.H. Wilson, and independently by D. Morrison.

Motivation & Objective

• To investigate the topological properties of special Lagrangian torus fibrations on Calabi-Yau manifolds under the assumption that their mirrors arise from dualizing such fibrations.
• To understand the monodromy action around large complex structure limit points in the complex moduli space, particularly its role in mirror symmetry.
• To construct a natural cohomological mirror map between $ H^{\text{even}}(\check{X},\mathbb{Q}) $ and $ H^{\text{odd}}(X,\mathbb{Q}) $, motivated by Kontsevich's homological mirror symmetry conjecture.
• To verify that the monodromy action on cohomology matches the expected behavior of $(1,n-1)$ Yukawa couplings and the Leray filtration in the threefold case.

Proposed method

• Analyzes the Leray spectral sequence for a special Lagrangian fibration $ f: X \to B $, deriving consequences for dual fibrations and cohomological structures.
• Introduces a conjecture (Conjecture 3.7) that monodromy about a boundary divisor through a large complex structure limit point acts as translation by a section, generalizing the Dehn twist.
• Computes the action of translation monodromy on cohomology, showing agreement with topological Yukawa couplings on the mirror.
• Uses the monodromy action to define a cohomological mirror map $ \phi_2 $, mapping Mukai vectors on $ \check{X} $ to cohomology classes on $ X $.
• Verifies the compatibility of the mirror map with intersection theory and the Fourier-Mukai-type structure via $ T_D $-action and $ e^D $-twisting.
• Employs mirror Riemann-Roch and intersection theory to check consistency of the map across all degrees, particularly in $ H^4 $ and $ H^2 $.

Experimental results

Research questions

• RQ1 What topological constraints must special Lagrangian fibrations on Calabi-Yau manifolds satisfy to support a mirror construction via dualization?
• RQ2 How does monodromy about a large complex structure limit point act on the cohomology of the fibration, and can it be described as a section translation?
• RQ3 Can a natural cohomological mirror map be constructed between $ H^{\text{even}}(\check{X},\mathbb{Q}) $ and $ H^{\text{odd}}(X,\mathbb{Q}) $, and is it compatible with known invariants?
• RQ4 Do the monodromy weight filtration and the Leray filtration coincide in the Calabi-Yau threefold case, as conjectured?
• RQ5 Is the proposed mirror map compatible with intersection numbers and the action of $ e^D $-twists on cohomology classes?

Key findings

• The monodromy action about a boundary divisor through a large complex structure limit point is conjectured to act as translation by a section, generalizing the Dehn twist in elliptic fibrations.
• The action of this monodromy on cohomology reproduces the expected $(1,n-1)$ Yukawa couplings on the mirror, confirming consistency with topological mirror symmetry.
• In the threefold case, the monodromy weight filtration and the Leray filtration on cohomology coincide, verifying a conjecture from [11] and [18].
• A cohomological mirror map $ \phi_2 $ is constructed from $ H^*(\check{X},\mathbb{Q}) $ to $ H^*(X,\mathbb{Q}) $, with explicit formulas for $ \phi_2(1,0,0,0) $, $ \phi_2(D) $, and $ \phi_2(e^D C) $.
• The map $ \phi_2 $ satisfies all required intersection properties: $ \phi_2(D) \cdot \phi_2(C) = -D.C $, $ \phi_2(1,0,0,0) \cdot \phi_2(D) = 0 $, and $ \phi_2(D) \cdot \phi_2(E) = 0 $, as required by mirror symmetry.
• The compatibility of $ \phi_2 $ with the $ e^D $-twist is verified via $ T_{-D} \circ \phi_2 = \phi_2 \circ e^D $, confirming consistency with the Fourier-Mukai-type structure.