Khler Geometry of Toric Manifolds

Introduction

Khler geometry of toric manifolds sits at the intersection of several important areas of differential geometry, algebraic geometry, and symplectic geometry. Toric manifolds provide a rich class of examples that are both geometrically beautiful and accessible due to their high degree of symmetry. This page explores the fundamental concepts, constructions, and applications of Khler geometry in the toric setting.

The study of toric manifolds has a rich history dating back to the 1970s, with foundational work by Demazure, Mumford, and others in the algebraic setting, and by Atiyah, Guillemin-Sternberg, and others in the symplectic setting. The marriage of these perspectives has led to deep insights and powerful techniques that continue to influence research in geometry today.

Background: Khler Geometry

Khler geometry is the study of complex manifolds equipped with a Riemannian metric and a symplectic structure that are compatible in a specific way. A Khler manifold is a triple (M, J, g, ) where:

• M is a smooth manifold
• J is an almost complex structure satisfying J = -Id
• g is a Riemannian metric
• is a symplectic (closed, non-degenerate) 2-form

These structures are required to satisfy the compatibility condition g(X,Y) = (JX,Y) for all vector fields X, Y on M.

A theorem of Newlander-Nirenberg tells us that the integrability condition [JX,JY] = J[JX,Y] + J[X,JY] - [X,Y] ensures that the almost complex structure comes from an honest complex structure on M, making M a complex manifold.

Notable examples include complex Euclidean space , complex projective space P, and complex tori. Khler manifolds enjoy properties from all three of their defining structures, such as:

• Complex analytic properties (like Hodge theory)
• Riemannian geometric properties (like curvature theory)
• Symplectic topological properties (like Hamiltonian group actions)

The Hodge decomposition of cohomology is particularly elegant on Khler manifolds, with the Khler identities providing powerful tools for calculations.

Toric Manifolds

Toric manifolds are compact complex manifolds equipped with an effective holomorphic action of a complex torus (*) having an open dense orbit. More concretely, a toric manifold of complex dimension n is a 2n-dimensional manifold with a Hamiltonian action of the n-torus T = (S).

These manifolds can be completely encoded by certain combinatorial objects:

• A convex polytope in
• A fan in

This combinatorial description makes toric manifolds particularly tractable and provides a bridge between algebraic/differential geometry and combinatorics. The polytope determines the moment map image of the torus action, while the fan encodes the local structure of the complex manifold.

One of the most elegant aspects of toric geometry is the way topological invariants can be computed from the combinatorial data. For instance, the cohomology ring of a toric manifold is determined by its polytope, and intersection numbers can be computed via explicit combinatorial formulas.

Khler Structure on Toric Manifolds

The torus action on a toric manifold admits a moment map : M whose image is a convex polytope . This relationship between symplectic geometry and convex geometry is at the heart of many developments in the field.

The famous Atiyah-Guillemin-Sternberg convexity theorem states that the image of is indeed a convex polytope whose faces correspond to fixed point sets of sub-tori. This theorem provides the foundation for the combinatorial approach to toric symplectic geometry.

Any smooth Khler metric on a toric manifold corresponds (via the moment map) to a smooth strictly convex function on , called a symplectic potential or Khler potential. The function encodes the metric data, and the Khler form can be recovered from through a specific formula.

This correspondence allows one to study Khler metrics on toric manifolds using analytic and variational methods in the setting of convex polytopes. In particular, the moment map identifies:

• Complex coordinates on M with parameters describing points in
• The Khler metric with the Hessian of
• The volume form of M with the determinant of the Hessian of

This identification shows that the geometry of (M, g) is captured by the Hessian of on .

Extremal Khler Metrics on Toric Manifolds

A fundamental problem in Khler geometry is the existence of canonical metrics. An extremal Khler metric is a critical point of the L-norm of the scalar curvature within its Khler class. These include constant scalar curvature (cscK) metrics and Khler-Einstein metrics as special cases.

In the toric setting, extremal Khler metrics correspond to solutions of a real Monge-Ampre type equation on the polytope . This equation is known as the Abreu equation:

det(u_{ij} + u_{ijkl}^i^j) = f()

where u is the symplectic potential, and f is determined by the scalar curvature functional. The existence of extremal metrics on toric varieties is related to stability conditions from algebraic geometry, leading to deep conjectures about the equivalence of differential-geometric and algebro-geometric stability.

For example, the Yau-Tian-Donaldson conjecture (proven in many cases) relates the existence of constant scalar curvature Khler metrics to a suitable notion of K-stability, which can be checked combinatorially in the toric case.

Toric Degenerations and Mirror Symmetry

Toric manifolds serve as building blocks for more general Khler manifolds through toric degenerations. A toric degeneration is a flat family of complex varieties that degenerate a general member to a toric variety. This technique has proven valuable in studying:

• Complex structure deformations
• Aspects of mirror symmetry
• Gromov-Witten theory computations
• Integrable systems on the moduli space of flat connections

In the context of mirror symmetry, toric manifolds provide the A-model side for many examples. Their mirror partners can be explicitly constructed using combinatorial data from the polytope or fan, leading to rich dualities between different geometric pictures.

The Givental-Hori-Vafa construction provides an explicit method for constructing mirror partners for toric complete intersections, while Batyrev's mirror construction uses reflexive polytopes to create mirror pairs of Calabi-Yau toric varieties.

The Monge-Ampre Equation and Toric Geometry

The study of extremal metrics leads to deep connections with the theory of Monge-Ampre equations. The Calabi conjecture, solved by Yau, states that on a compact Khler manifold, the Ricci form determines the Khler form uniquely up to a constant. In the toric case, this reduces to solving a real Monge-Ampre equation on the polytope.

This connection between Khler geometry and partial differential equations on convex domains has led to significant cross-fertilization between complex geometry and the theory of plurisubharmonic functions, optimal transport, and convex analysis.

Applications and Current Research

The study of Khler geometry of toric manifolds has found applications in diverse areas:

• Gauge Theory: Through connections with vortex equations and moment maps
• Mathematical Physics: In string theory and quantum field theory
• Geometric Analysis: As test cases for general existence and regularity results
• Symplectic Topology: Through Hamiltonian group actions and moment polytopes
• Algebraic Geometry: Via toric varieties and their deformations

Current research directions include:

• The study of stability conditions for toric varieties
• Canonical Khler metrics on possibly singular toric varieties
• Toric degenerations of Fano varieties and applications to the Yau-Tian-Donaldson conjecture
• Hamiltonian torus actions beyond the toric case
• Connections to optimal transport and convex geometry
• Symplectic reduction techniques for more general Hamiltonian group actions
• Toric Khler-Einstein metrics and their moduli
• Higher-dimensional analogues of classical toric geometry

Conclusion

Khler geometry of toric manifolds provides a fertile ground where complex, symplectic, and Riemannian geometry interact with combinatorial structures in remarkable ways. The correspondence between Khler metrics and convex potentials continues to inspire developments in geometric analysis, while the combinatorial nature of toric geometry makes it an accessible yet rich setting to explore deep questions from differential geometry and mathematical physics.

As the field continues to evolve, new connections emerge with neighboring areas, reinforcing the central role of toric geometry as both an intrinsic subject of study and a bridge between different mathematical landscapes. The interplay between algebraic and differential-geometric perspectives continues to yield deep insights, while applications to physics and other areas of mathematics demonstrate the far-reaching implications of this beautiful subject.