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].

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].

Reviewer: Gheorghe Gussi (Bucureşti)

##### MSC:

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 |