The Laplace operator on the nth tensor power of a line bundle: eigenvalues which are uniformly bounded in n

Almost Complex Structures and Geometric Quantization

“…This is in fact a more natural generalization of the Kähler situation then it might seem. By the results of [10] and our Theorem 4.2, the first d k eigenvalues of ∆ • k can be bounded independently of k, while the rest of the spectrum drifts to the right with a gap of O(k). So as k → ∞ the first d k eigenvalues (which are zero in the integrable case) remain within some fixed interval around zero even in the non-integrable case.…”

“…Let Π k : L 2 (X, L ⊗k ) → H k be the orthogonal projection. Then [10] shows that the Π k are the components of a projector of Szegö type. That is, they define a generalized Toeplitz structure in the sense of [4].…”

“…It turns out that the semiclassical theory of this quantization method, which we'll refer to as "almost Kähler quantization," is as good as that of Kähler quantization. This is based on the results of [10] and the calculus of Hermite Fourier integral operators. In fact the methods of proof of the main results of [1], [2] and [3] generalize directly to this setup.…”

Nearly Kählerian embeddings of symplectic manifolds

“…In [11] it was observed that Mellin's inequality implies the existence of constants C 1 , C 2 > 0 such that the spectrum of B k is contained in…”

“…We will prove here Theorem 3.1 and Proposition 3.4, that is, that the projector Π : L 2 (Z) → H is an Hermite FIO of the same form as the Szego projector. Our starting point is the following result from [11]:…”