Geometric quantization of generalized Hirzebruch fibrations
Andrea Piccirilli
TL;DR
The paper constructs generalized Hirzebruch fibrations as toric symplectic manifolds by realizing them as projectivizations $\mathcal{H}^d_n=\text{Proj}(\mathcal{O}\oplus \mathcal{O}(-n))$ over $\mathbb{C}\mathbb{P}^d$, embeds them into a product of projective spaces to obtain a toric action, and computes their moment polytopes. Central to the work is the toric geometric principle that the dimension of geometric quantization equals the number of lattice points in the moment polytope, $\dim \mathcal{Q}(M,\omega)=\#(\Delta\cap \mathbb{Z}^n)$, which the authors exploit to derive an explicit quantization formula $\mathcal{Q}_{a,b,d}(n)$ as a lattice-point count in the polytope $\Delta^d_{a,b,n}$. They obtain a recurrence $\sum_{k=0}^{d+1}(-1)^k\binom{d+1}{k}\mathcal{Q}_{a,b,d}(n+d+1-k)=0$, show that $\mathcal{Q}_{a,b,d}(n)$ is a polynomial of degree $d$ in $n$, and analyze the large-twisting limit $n\to\infty$, where $\mathcal{Q}_{a,b,d}(n)$ is asymptotically the symplectic volume $\text{Vol}(\mathcal{H}^d_{a,b,n})$, with explicit $1/b$ corrections governed by Bernoulli numbers. Special cases $n=0$ and $n=1$ recover product and blow-up geometries, with closed-form quantization in those instances. Collectively, these results provide a concrete link between toric geometry, quantization, and asymptotic lattice-point counts for a new infinite class of fibrations.
Abstract
Hirzebruch surfaces, defined as the projectivization of line bundles over $\C\mathbb{P}^1$, support a toric action and thus represent an infinite class of symplectic toric manifolds of complex dimension 2. In this paper, an infinite class of toric manifolds given as projective bundles over $\mathbb{C}\mathbb{P}^d$ will be constructed for every complex dimension $d$ and it will be shown that each manifold supports a symplectic structure. With the toric and symplectic structure of the manifolds at our disposal, we then study their geometric quantization and how it relates to different values of the twisting parameter of the fibrations.