[Paper Review] Quasitoric Manifolds with Invariant Almost Complex Structure

Andrei Kustarev|ArXiv.org|Feb 2, 2009

Geometry and complex manifolds10 references3 citations

TL;DR

This paper establishes that a quasitoric manifold $M^{2n}$ admits a $T^n$-invariant almost complex structure if and only if it admits a positive omniorientation. The existence of such a structure is entirely determined by the cohomology of the underlying polytope, with all obstructions vanishing trivially, and the canonical weakly complex structure is equivariantly equivalent to any such invariant almost complex structure.

ABSTRACT

We prove that any quasitoric manifold $M^{2n}$ admits a $T^n$-invariant almost complex structure if and only if $M$ admits a positive omniorientation. In particular, we show that all obstructions to existence of $T^n$-invariant almost complex structure on $M^{2n}$ arise from cohomology of underlying polytope - and hence are trivial.

Motivation & Objective

• To determine the necessary and sufficient condition for the existence of a $T^n$-invariant almost complex structure on a quasitoric manifold.
• To resolve Problem 7.6 from Davis and Januszkiewicz (1991), which asks for a criterion in terms of the characteristic map $\lambda$.
• To show that all obstructions to the existence of such structures are cohomological and vanish when the omniorientation is positive.
• To establish that any $T^n$-invariant almost complex structure is equivariantly equivalent to the canonical weakly complex structure.

Proposed method

• Use induction on the skeleta of the orbit polytope $P$ to extend an almost complex structure from lower-dimensional strata.
• Define obstruction classes $\sigma^i_J \in C^i(P, \pi_{i-1}(SO(2i)/U(i)))$ for extending the structure across $i$-skeleta.
• Prove that these obstructions are cocycles using a homotopy-theoretic argument involving tubular neighborhoods and boundary spheres.
• Show that the cocycle $\sigma^i_J$ is a coboundary due to the triviality of the homology of $P$, allowing modification of the structure to eliminate obstructions.
• Leverage the isomorphism of homotopy groups $\pi_{i-1}(SO(2i-2)/U(i-1)) \to \pi_{i-1}(SO(2i)/U(i))$ for $i > 1$ to ensure consistency across dimensions.
• Use the sign of fixed points, defined via $\det(e_j) \cdot \det(\lambda_v)$, to characterize positive omniorientation and link it to the existence of the structure.

Experimental results

Research questions

• RQ1 Under what conditions does a quasitoric manifold $M^{2n}$ admit a $T^n$-invariant almost complex structure?
• RQ2 How are the obstructions to constructing such a structure related to the topology of the orbit polytope $P$?
• RQ3 Is the canonical weakly complex structure the only $T^n$-invariant almost complex structure up to equivariant equivalence?
• RQ4 Can a quasitoric manifold admit a $T^n$-invariant almost complex structure without being a toric variety or symplectic manifold?
• RQ5 What role does positive omniorientation play in the existence of $T^n$-invariant almost complex structures?

Key findings

• A quasitoric manifold $M^{2n}$ admits a $T^n$-invariant almost complex structure if and only if it admits a positive omniorientation.
• All obstructions to the existence of such a structure arise from the cohomology of the orbit polytope $P$, and since $P$ has trivial homology, all obstructions vanish.
• The canonical weakly complex structure on $M$ is equivariantly equivalent to any $T^n$-invariant almost complex structure.
• For $k$ odd, $\mathbb{C}P^2_k$ admits a $T^2$-invariant almost complex structure despite not being a toric variety or symplectic manifold.
• For $k$ even, $\mathbb{C}P^2_k$ does not admit any almost complex structure, as shown by the non-integrality of the Todd genus $td = (k+1)/2$.
• The existence of a $T^n$-invariant almost complex structure does not imply the manifold is algebraic or symplectic, as demonstrated by $\mathbb{C}P^2_k$ with $k \geq 3$ odd.