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Harmonic analysis on toric varieties. (English) Zbl 1041.32004
Bland, John (ed.) et al., Explorations in complex and Riemannian geometry. A volume dedicated to Robert E. Greene. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3273-5/pbk). Contemp. Math. 332, 267-286 (2003).
The authors study harmonic analysis on toric varieties, as a model setting for harmonic analysis on Kähler manifolds. This means the following. Let \(L \to M\) be a hermitian holomorphic line bundle, with \(c_1(L) = \omega\), and \(\mathcal{H} =\bigoplus_{n = 0}^\infty H^\circ(M, L^N)\) and consider its Hilbert completion (here \(H^\circ(M, L^N)\) is the space of holomorphic sections of \(L^{\otimes N}\). The torus action \(M\) give rise to a natural Fourier analysis on \(\mathcal{H}\), which extends to all \(L^2\) sections.
The main results are, for \((M_p, \omega_p)\) a Kähler toric variety, the following: there exists a Fourier multiplier \(\mathcal{M}\) such that \(M_N^{M_p} (x, y) = \mathcal{M} \langle i(x), \overline{i(x)} \rangle^N\) where \(i : X_p \to S^{2d + 1}\) is the lift of the monomial embedding \(\phi\) for which \(\phi^\psi \omega_{FS} = \omega_p\). More over \(\prod{M_p} \mathcal{M} \prod^{M_p}\) is a Toeplitz operator. Here \(\prod_N = \prod_N^M\) is the Szegő kernel: \(\mathcal{L}^2(M, L^N) \to H^\circ(M, L^N), P\) is a convex integral polytope in \(\mathbb{R}^m\), the lattice points in \(P\), \(P \cap \mathbb{Z}^m = \{ \alpha(1), \ldots, \alpha (\# P)\}\), \(\phi_p^c\) is the map \((\mathbb{C}^*)^m \to \mathbb{C} \mathbb{P}^{\# p - 1}\) defined by \(\phi_p^c (z) = [c_{\alpha(1)} z^{\alpha(1)}, \ldots, c_{\alpha (\# P)} z^{\alpha(\# p)}]\) and if \(P\) is Delzant \(\phi_P^c\) is an embedding, and \(\mathbb{C}^{*m}\) with its image (the open orbit) in \(M_P^c\) and the resulting embedding (with \(c\) and \(P\) is fixed) is denoted by \(\phi : M \hookrightarrow \mathbb{C} \mathbb{}^d\), \( d = \# P - 1\) (and is refered as the monomial embedding). \(i : X_{\mathbb{P}} \to S^{2d + i}\) is the lift of the monomial embedding \(\phi\) for which \(\phi^* \omega_{FS} = \omega_p\) (\(FS\) denotes Fubini-Study).
The analysis is based on the identification of \(\mathcal{H}\) with the Hardy space of CR functions \(\mathcal{H}^2 (X) = \{ F + L^2(X), \bar \partial_G F = 0\}\) (Here \(X\) is the unit circle bundle in the dual line bundle \(F^{-1}\)). There \(\prod \mathcal{M} \prod \) is Toeplitz means that there is a smooth symbol \(\sigma\) so that \(\prod \mathcal{M} \prod = \prod \sigma M\) modulo smoothing operators. The above result gives a relation between the Szegő kernel of a toric variety and the kernel obtained by pulling back the (very simple) of the projective space Szegő kernel under a holomorphic embedding. This Fourier multiplier \(\mathcal{M}\) has thus simple asymptotic properties, i.e. it is polyhomogeneous along rays of lattice points.
The authors obtain as a sequence of the main result an interesting integral formula for the equivariant characters \(\chi_{NP}\) on \((\mathbb{C}^*)^n\) defined as \(\chi_{NP} (e^{i \varphi}) = \sum_{\alpha \in NP} e^{i \langle \varphi, \alpha \rangle}\), \( \varphi = (\varphi_1, \ldots, \varphi_n)\), namely \[ \chi_{NP} (e^{i \varphi}) \sim \int_{MP} e^{N \psi (n, \varphi)} A_N (x) \,dV(x) \] where \(\psi\) is a non degenerate complex phase function of positive type, \(A_N\) is a polyhomogeneous function of \(N\), “\(\sim\)” means modulo a smoothing operator, and the integral is a complex oscillatory integral.
For the entire collection see [Zbl 1023.00007].

32A50 Harmonic analysis of several complex variables
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
43A85 Harmonic analysis on homogeneous spaces
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
32Q20 Kähler-Einstein manifolds