[Paper Review] Compactifications of moduli spaces inspired by mirror symmetry

David R. Morrison|ArXiv.org|Apr 23, 1993

Geometric and Algebraic TopologyMathematics39 references115 citations

TL;DR

This paper proposes a mirror-symmetry-inspired compactification of moduli spaces for Calabi-Yau varieties, extending Looijenga’s semi-toric compactifications to non-arithmetic quotients. It introduces a minimal partial compactification analogous to Satake-Baily-Borel, conjecturing that maximally unipotent boundary points correspond to mirror Calabi-Yau manifolds with specific Kähler cones and arithmetic data, providing a geometric framework for mirror symmetry in moduli compactifications.

ABSTRACT

We study moduli spaces of nonlinear sigma-models on Calabi-Yau manifolds, using the one-loop semiclassical approximation. The data being parameterized includes a choice of complex structure on the manifold, as well as some ``extra structure'' described by means of classes in H^2. The expectation that this moduli space is well-behaved in these ``extra structure'' directions leads us to formulate a simple and compelling conjecture about the action of the automorphism group on the Kähler cone. If true, it allows one to apply Looijenga's ``semi-toric'' technique to construct a partial compactification of the moduli space. We explore the implications which this construction has concerning the properties of the moduli space of complex structures on a ``mirror partner'' of the original Calabi-Yau manifold. We also discuss how a similarity which might have been noticed between certain work of Mumford and of Mori from the 1970's produces (with hindsight) evidence for mirror symmetry which was available in 1979. [The author is willing to mail hardcopy preprints upon request.]

Motivation & Objective

• To develop a geometric compactification of moduli spaces for Calabi-Yau varieties using insights from mirror symmetry.
• To extend Looijenga’s semi-toric compactifications beyond arithmetic quotients of symmetric domains to general moduli spaces.
• To formulate a minimal partial compactification of the moduli space that reflects the duality between complex structure and Kähler moduli under mirror symmetry.
• To conjecture a correspondence between maximally unipotent boundary points and mirror Calabi-Yau manifolds with specified automorphism groups and Kähler cones.
• To clarify the role of infinite discrete structures (e.g., Kähler cones) modulo automorphisms via a conjectured Kähler cone property.

Proposed method

• Adapts Looijenga’s semi-toric compactification framework to non-arithmetic moduli spaces arising in mirror symmetry.
• Introduces a flat connection on the holomorphic cotangent bundle of the moduli space to analyze degenerations.
• Analyzes boundary points of moduli spaces using degenerating variations of Hodge structure, especially non-normal crossing types.
• Applies mirror symmetry duality: complex structure moduli of one Calabi-Yau correspond to Kähler moduli of its mirror partner.
• Proposes a minimal compactification analogous to Satake-Baily-Borel, built from the Satake-Baily-Borel decomposition of the Kähler cone's rational hull.
• Uses the conjectural Kähler cone property to relate infinite discrete data (e.g., ample divisors) to finite data modulo automorphisms.

Experimental results

Research questions

• RQ1 How can moduli spaces of Calabi-Yau varieties be compactified in a way that reflects mirror symmetry duality?
• RQ2 What is the geometric meaning of maximally unipotent boundary points in the moduli space of Calabi-Yau varieties?
• RQ3 Can a minimal partial compactification be constructed for non-arithmetic moduli spaces analogous to the Satake-Baily-Borel compactification?
• RQ4 How do the Kähler cones of mirror Calabi-Yau manifolds relate to the structure of the moduli space boundary?
• RQ5 What is the role of the automorphism group and integral cohomology in defining the compactification data at boundary points?

Key findings

• A minimal partial compactification of the moduli space of Calabi-Yau varieties is conjectured to exist, analogous to the Satake-Baily-Borel compactification, with distinguished maximally unipotent boundary points.
• Each maximally unipotent boundary point corresponds to a mirror Calabi-Yau manifold $X_j$, equipped with its Kähler cone ${\cal K}_j$ and automorphism group $\mathop{\rm Aut}(X_j)$.
• The arithmetic data $\Gamma_j$ is defined as the quotient of $H^2(X_j,\mathbb{Z})$ by torsion, modulo $\mathop{\rm Aut}(X_j)$, forming a discrete group acting on the moduli space.
• The locally rational polyhedral decomposition ${\cal P}_j$ at each boundary point is identified with the Satake-Baily-Borel decomposition of the cone $({\cal K}_j)_+$, the rational convex hull of the closure of the Kähler cone.
• The conjectured Kähler cone property clarifies that infinite discrete structures (e.g., ample divisors) are finite modulo automorphisms, supported by verification in a nontrivial case with A. Grassi.
• A historical connection is drawn between Mori’s 1979 example of an abelian surface with real multiplication and Mumford’s figure 1, suggesting mirror symmetry could have been anticipated in 1979 had the duality been recognized.