Abstract
We investigate the frame set of regular multivariate Gaussian Gabor frames using methods from Kähler geometry such as Hörmander’s \({\overline{\partial }}\)-\(L^2\) estimate with singular weight, Demailly’s Calabi–Yau method for Kähler currents and a Kähler-variant generalization of the symplectic embedding theorem of McDuff–Polterovich for ellipsoids. Our approach is based on the well-known link between sets of interpolation for the Bargmann-Fock space and the frame set of multivariate Gaussian Gabor frames. We state sufficient conditions in terms of a certain extremal type Seshadri constant of the complex torus associated to a lattice to be a set of interpolation for the Bargmann-Fock space, and give also a condition in terms of the generalized Buser-Sarnak invariant of the lattice. In particular, we obtain an effective Gaussian Gabor frame criterion in terms of the covolume for almost all lattices, which is the first general covolume criterion in multivariate Gaussian Gabor frame theory. The recent Berndtsson–Lempert method and the Ohsawa–Takegoshi extension theorem also allow us to give explicit estimates for the frame bounds in terms of certain Robin constant. In the one-dimensional case we obtain a sharp estimate of the Robin constant using Faltings’ theta metric formula for the Arakelov Green functions.
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1 Introduction
Let \(\Lambda \) be a lattice in \({\mathbb {R}}^{2n}\) and
be the Siegel upper half-space. Fix \(\Omega \in {\mathfrak {H}}\), a Gaussian Gabor system, denoted by \((g_\Omega ,\Lambda )\), is the family of functions \(\{\pi _\lambda g_\Omega \}_{\lambda \in \Lambda }\), where \(\Lambda \) is a lattice in \({\mathbb {R}}^{2n}\) and
denotes a time-frequency shift of the Gaussian \(g_\Omega (t):=\overline{e^{\pi i t^T\Omega t}}\). Letting I denote the identity matrix, we note that \(g_{iI}(t)\) is the standard Gaussian \(e^{-\pi |t|^2}\). The frame set of \(g_\Omega \) is the set of all lattices \(\Lambda \) in \({\mathbb {R}}^{2n}\) such that \(\{\pi _\lambda g_\Omega \}_{\lambda \in \Lambda }\) is a frame for \(L^2({\mathbb {R}}^n)\). Here, by a frame, we mean there exist positive constants A, B (called frame bounds) such that
where \((\cdot , \cdot )\) denotes the \(L^2\)-inner product. Using the Bargmann transform, one may reformulate the frame property for Gaussian Gabor frames in terms of sampling property of the Bargmann–Fock space in complex analysis. In the one-dimensional case there is a density criterion for the interpolation problem in the Bargmann-Fock space due to Lyubarskii and Seip-Wallstén, which implies a seminal result in the theory of Gaussian Gabor frames. Namely, that \(\{\pi _\lambda g_{iI}\}_{\lambda \in \Lambda }\) is a Gabor frame if and only if the covolume \(|\Lambda |<1\), [Lyu92, Sei92, SW92]. Recent progress on the description of the frame set of a Gabor atom has been made for totally positive functions [GS13, GRS18] and for rational functions [BKL21]. Note that all aforementioned results on frame sets for Gabor systems are for uniformly discrete point sets in the plane. The generalization to the higher-dimensional case has been one of the most intriguing problems in the study of Gaussian Gabor frames since the methods in [Lyu92, Sei92, SW92] do not have natural counterparts in the theory of several complex variables. The reason being that the theory of sampling and interpolation in several complex variables [MT00, Lin01, Gro11, GL20, LM09] is far more intricate than in the one-dimensional case (see [PR13, section 3]) and despite considerable effort not well understood. In particular, the following central problem in multivariate Gaussian Gabor frame theory is still open.
Problem A
Is there an equivalent Gabor frame criterion for \((g_{iI}, \Lambda )\) only in terms of the covolume \(|\Lambda |\) for almost all lattices \(\Lambda \) in \({\mathbb {R}}^{2n}\) (\(n>1\))?
We obtain the following partial result, which is a direct consequence of Proposition 1.4 and our Hörmander criterion in Sect. 1.2.
Theorem 1.1
(First main theorem). Fix \(\Omega \in {\mathfrak {H}}\), if \(|\Lambda |<\frac{n!}{n^n}\) then \((g_\Omega ,\Lambda )\) is a Gabor frame for almost all \(\Lambda \) in \({\mathbb {R}}^{2n}\). More precisely, \((g_\Omega ,\Lambda )\) is a Gabor frame if \(|\Lambda |<\frac{n!}{n^n}\) and \((\Omega , \Lambda )\) is a transcendental pair (see the definition and the remark below for explicit examples of the transcendental pairs).
Definition 1.1
Let \( \Lambda ^\circ :=\{(\eta , y)\in {\mathbb {R}}^n \times {\mathbb {R}}^n: \xi ^T y -x^T\eta \in {\mathbb {Z}}, \ \forall \ (\xi ,x) \in \Lambda \} \) denote the symplectic dual of \(\Lambda \) (also known as the adjoint lattice of \(\Lambda \)). Put
where \((\textrm{Im}\, \Omega )^{-1/2}\) denotes the unique positive definite matrix whose square equals \((\textrm{Im}\, \Omega )^{-1}\). We call \((\Omega , \Lambda )\) a transcendental pair if the complex torus \( {\mathbb {C}}^n/ \Gamma _{\Omega , \Lambda ^\circ } \) has no analytic subvariety of dimension \(1\le d <n\) (see Definition 1.4 for a related notion).
Remark
From the definition we know that all \((\Omega , \Lambda )\) are transcendental in case \(n=1\). In case \(n=2\), by [Sha13, p. 161] we know that \((\Omega , \Lambda )\) is a transcendental pair if \({\mathbb {C}}^2/ \Gamma _{\Omega , \Lambda ^\circ }\) is biholomorphic to \({\mathbb {C}}^2/\Gamma \) for some lattice
with the set \(\{1, a, b, c, d, ad-bc\}\) being linearly independent over \({\mathbb {Z}}\) (for example, if
then \(\Gamma \) is transcendental). In fact, the argument in [Sha13, page 160–163] can also be used to prove that \((\Omega , \Lambda )\) is transcendental for almost all lattices \(\Lambda \) in \({\mathbb {C}}^n\). The proof of our first main theorem also suggests the following conjecture, which would answer our Problem A (just take \(\Omega =iI\)).
Conjecture A
Let \((\Omega , \Lambda )\) be a transcendental pair (see Definition 1.1). Then \((g_\Omega ,\Lambda )\) is a Gabor frame for \(L^2({\mathbb {R}}^n)\) if and only if \(|\Lambda |<\frac{n!}{n^n}\).
Remark
Since all \((\Omega , \Lambda )\) are transcendental in case \(n=1\), the above conjecture is precisely the Lyubarskii-Seip-Wallstén theorem [Lyu92, Sei92, SW92] in the one dimensional case. In case \(n>1\), the only known necessary condition is \(|\Lambda |<1\), which is a consequence of the Balian-Low type theorems (see [AFK14, GHO19] or a complex analysis proof of \(|\Lambda |\le 1\) by Lindholm [Lin01]).
Our second main result is a multivariate Gaussian Gabor frame criterion for general lattices (see the remark after Theorem A in Sect. 1.2 for the proof and Corollary 1.8 for applications).
Theorem 1.2
(Second main theorem). Fix \(\Omega \in {\mathfrak {H}}\) and a lattice \(\Lambda \) in \(\mathbb R^{2n}\). Assume that there exist \(r>1\), \(\beta =(\beta _1,\cdots , \beta _n)\in {\mathbb {R}}^n\) with
and a holomorphic injection f from the ellipsoid
to the torus \(X:={\mathbb {C}}^n/ \Gamma _{\Omega , \Lambda ^\circ }\) (see (1.2)) such that
for some smooth function \(\phi \) on X, where \(\omega :=\frac{i}{2} \sum _{j=1}^n dw_j \wedge d{\bar{w}}_j\) is the Euclidean Kähler form on X (note that the holomorphic cotangent bundle of X is trivial with global frame \(\{dw_j\}\)). Then \((g_\Omega ,\Lambda )\) is a Gabor frame.
Remark
In case \((\Omega , \Lambda )\) is transcendental the above theorem is equivalent to our first main theorem. In order to prove this equivalence, we generalize (see Theorem A in Sect. 1.2) McDuff–Polterovich’s result [MP94] (see Theorem 3.4) to all Kähler ellipsoid embeddings (see Sect. 3.1).
In the one-dimensional case, we also obtain the following frame bound estimates (see Theorem B in Sect. 1.3 for more results), which can be seen as an effective version of [BGL10, Theorem 1.1].
Theorem 1.3
(Third main theorem). Let \(\Lambda \) be a lattice in \({\mathbb {R}}\times {\mathbb {R}}\). Suppose that the lattice
in \({\mathbb {C}}\) is generated by \(\{1, \tau \}\) with \(\textrm{Im}\, \tau >1\). Then for all \(f\in L^2({\mathbb {R}})\) with \(||f||=1\), we have
where \(\eta (\tau ):=e^{\pi i\tau /12} \,\Pi _{n=1}^\infty (1-e^{2\pi i n \tau })\) is the Dedekind eta function.
The above three main theorems are special cases of the Hörmander criterion (Theorem 1.7), Theorems A and B in Sect. 1.2. The whole paper is organized as follows.
1.1 Background
Our starting point is the following observation, which is an extension of a well-known duality result for \(\Omega =iI\) [Jan82, Lyu92, Sei92].
Proposition 1.4
\((g_\Omega , \Lambda )\) is a Gabor frame for \(L^2({\mathbb {R}}^n)\) if and only if \(\Gamma _{\Omega , \Lambda ^\circ }\) (see (1.2)) is a set of interpolation for the Bargmann–Fock space \({\mathcal {F}}^2\) (see (1.4) and Definition 1.2 in Sect. 1.2).
Our first main theorem is an extension of the approach by Berndtsson–Ortega Cerdà [BO95] for one-dimensional Gaussian Gabor frames to the multivariate case by utilizing the theory of Hörmander’s \(L^2\)-estimates for \({\overline{\partial }}\) in the higher-dimensional case, which has been developed during the past two decades and has received quite some attention [OT87, Ohs94, Dem92, Ber10, Ber06, Ber09, Blo13, GZ15, BL16]. The new idea is to apply Demailly’s mass concentration technique [Dem93] (see [Tos16] for a nice survey).
A crucial notion in the proof of our second main theorem is a generalized extremal type Seshadri constant (see Definition 1.5 and Theorem 3.3 for the extremal property), which replaces the covolume of a lattice in the one-dimensional case. Note also that the Seshadri constant has a quite different flavor than the (Beurling) densities used in the discussion of Gabor frames, since it actually takes into account the volumina of all subvarieties of the complex torus (see Theorem 3.9) and not just of the whole complex torus.
Our third main theorem is based on a recent result of Berndtsson–Lempert [BL16], which also yields explicit estimates for the multivariate Gaussian Gabor frame bounds (see Theorems 5.1 and 5.2).
A short account of our other related results is given in Sects. 1.2 and 1.3 below (the Hörmander criterion, Theorems A, B are among the most crucial ones).
1.2 Part I: Interpolation in Bargmann–Fock space
Consider the Bargmann–Fock space
where \({\mathcal {O}}({\mathbb {C}}^n)\) denotes the space of holomorphic functions on \({\mathbb {C}}^n\) and we omit the Lebesgue measure in the integral. Let \(\Gamma \) be a lattice in \({\mathbb {C}}^n\).
Definition 1.2
We call \(\Gamma \) a set of interpolation for \({\mathcal {F}}^2\) if there exists a constant \(C>0\) such that for every sequence of complex numbers \(a=\{a_\lambda \}_{\lambda \in \Gamma }\) with \(\sum _{\lambda \in \Gamma } |a_\lambda |^2 e^{-\pi |\lambda |^2}=1, \) there exists \(F\in {\mathcal {F}}^2\) such that \(F(\lambda )=a_\lambda \) for all \(\lambda \in \Gamma \) and \(||F||^2 \le C\).
Denote by \(|\Gamma |\) (the covolume of \(\Gamma \)) the volume of the torus \( X:={\mathbb {C}}^n/\Gamma \) with respect to the Lebesgue measure. In the one-dimensional case, Lyubarskii, Seip and Wallstén [Lyu92, Sei92, SW92] (see [OS98] for the most general one-dimensional generalization) proved that
Theorem 1.5
A lattice \(\Gamma \) in \({\mathbb {C}}\) is a set of interpolation for \({\mathcal {F}}^2\) if and only if \(|\Gamma |>1\).
Another proof of the “sufficient" part of the above theorem was given by Berndtsson and Ortega Cerdà in [BO95] using the Hörmander \({\overline{\partial }}\) theory. In order to make best use of the Hörmander theory, we shall introduce the following notion of the Hörmander constant, which is an analogue of the Seshadri constant introduced by Demailly in [Dem92].
Definition 1.3
Let \(\Gamma \) be a lattice in \({\mathbb {C}}^n\). The Hörmander constant of \(\Gamma \) is defined by
where “\(\Gamma \)-invariant" means that \(\psi (z+\lambda ) =\psi (z)\) for all \(z\in {\mathbb {C}}^n\) and \(\lambda \in \Gamma \); “psh" means subharmonic on each embedded disc.
By using of a result of Tosatti (see [Tos18, Theorem 4.6]), we can prove the following:
Proposition 1.6
Assume that the only positive dimensional analytic subvariety of \(X:={\mathbb {C}}^n/\Gamma \) is X itself, then
The above result suggests the following definition.
Definition 1.4
A lattice \(\Gamma \) in \({\mathbb {C}}^n\) is said to be transcendental if the only positive dimensional analytic subvariety of \(X:={\mathbb {C}}^n/\Gamma \) is X itself.
By Proposition 1.4 and Proposition 1.6, we know that the following criterion implies our first main result — Theorem 1.1.
Theorem 1.7
(The Hörmader criterion). Let \(\Gamma \) be a lattice in \({\mathbb {C}}^n\). If \(\iota _\Gamma >1\) then \(\Gamma \) is a set of interpolation for \({\mathcal {F}}^2\). In particular, if \(\Gamma \) is transcendental and \(|\Gamma |>\frac{n^n}{n!}\), then \(\Gamma \) is a set of interpolation for \({\mathcal {F}}^2\).
In order to use the above criterion for non-transcendental lattices we have to investigate the Hörmander constant in more depth. Our main idea is to introduce the following definition.
Definition 1.5
Let \((X, \omega )\) be an n-dimensional compact Kähler manifold. Fix \(x\in X\) and take a holomorphic coordinate chart \(z=\{z_j\}\) near x such that \(z(x)=0\). Put
The \(\beta \)-Seshadri constant of \((X, \omega )\) at \(x\in X\) is defined by
where “\(\omega \)-psh" means that \(\psi \) is upper semi continuous on X with
in the sense of currents on X.
Remark
In case \(X={\mathbb {C}}^n/\Gamma \) and \(\omega = dd^c(\pi |z|^2)\), we know that \(\epsilon _x(\omega ;\beta )\) does not depend on \(x\in X\). Comparing the above definition with Definition 1.3, we further get
In case \(\beta _1=\cdots =\beta _n=1/n\), we know that \(n \epsilon _x(\omega ;\beta )\) equals the classical Seshadri constant of Demailly (see [Tos18, section 4.4]). A famous result of McDuff–Polterovich is a symplectic embedding formula [MP94] for the classical Seshadri constant. Theorem 1.2 follows from the following generalization of McDuff–Polterovich’s result to all \(\beta \)-Seshadri constants.
Theorem A
Let \((X, \omega )\) be a compact Kähler manifold. Denote by \({\mathcal {K}}_\omega \) the space of Kähler metrics for the cohomology class \([\omega ]\). Fix \(x\in X\) and \(\beta _j >0\), \(1\le j\le n\) with \(\beta _1+\cdots +\beta _n=1\). Then the \(\beta \)-Seshadri constant \(\epsilon _x(\omega ;\beta )\) is equal to the following \(\beta \)-Kähler width
where “" means that there exists a holomorphic injection \(f: B^\beta _r \rightarrow X\) such that \( f(0)=x\) and \(f^*({\tilde{\omega }})=\frac{i}{2} \sum _{j=1}^n dz_j \wedge d{\bar{z}}_j\).
Remark
The above theorem implies Theorem 1.2. In fact, by the above theorem and (1.7), the assumption in Theorem 1.2 implies that \(\iota _\Gamma >1\). Hence our second main theorem follows from the Hörmander criterion, Theorem 1.7 and Proposition 1.4.
In case \(X={\mathbb {C}}^n/\Gamma \) and \(\omega =dd^c(\pi |z|^2)\) is the Euclidean Kähler form, we know that \(B_r^\beta \) is included in X if and only if \( (\gamma + B_r^\beta ) \cap B_r^\beta =\emptyset , \ \ \forall \ 0\ne \gamma \in \Gamma , \) which is equivalent to that
Hence Theorem A gives
Put
then (1.7) gives
Apply the Hörmander criterion above we get:
Corollary 1.8
Let \(\Gamma \) be a lattice in \({\mathbb {C}}^n\). If \( \sup _{\beta \in {\mathcal {B}}} \inf _{0\ne z\in \Gamma } \sum _{j=1}^n\beta _j |z_j|^2 > \frac{4}{\pi }, \) then the Hörmander constant \(\iota _\Gamma >1\) (see Definition 1.3) and \(\Gamma \) is a set of interpolation for \({\mathcal {F}}^2\).
Remark
In case all \(\beta _j\) are equal to 1/n,
is known as the Buser–Sarnak invariant \(m(\Gamma )\) (see [BS94, Laz96, Laz04, Theorem 5.3.6]) of \(\Gamma \). For general \(\beta \in {\mathcal {B}}\) we call
the \(\beta \)-Buser–Sarnak invariant of \(\Gamma \). In general, we have that \(\sup _{\beta \in {\mathcal {B}}}m_\beta (\Gamma )>m(\Gamma )/n\). For example, if
then a direct computation gives
1.3 Part II: Gaussian Gabor frames
In this section we shall show how to apply the preceding results on sets of interpolation in \({\mathcal {F}}^2\) in Gabor analysis. We use the same symbol \(\Gamma _{\Omega , \Lambda ^\circ }\) in (1.2) to denote the underlying lattice
in \({\mathbb {R}}^n\times {\mathbb {R}}^n\). By a direct computation, we know that the symplectic dual of \(\Gamma _{\Omega , \Lambda ^\circ }\) is equal to
Hence Proposition 1.4 gives the following:
Corollary 1.9
\((g_\Omega ,\Lambda )\) defines a frame in \(L^2({\mathbb {R}}^n)\) if and only if \((g_{iI}, \Gamma _{\Omega , \Lambda })\) does.
Remark
Notice that \(\Gamma _{\Omega , \Lambda }\) is equal to \(f_\Omega (\Lambda )\), where
is a linear mapping preserving the standard symplectic form \( \omega :=d\xi ^T\wedge dx\) on \({\mathbb {R}}^n\times {\mathbb {R}}^n\). In dimension one, we know that \((g_\Omega , \Lambda )\) defines a frame in \(L^2({\mathbb {R}})\) if and only if \((g_{iI}, \Lambda )\) defines a frame in \(L^2({\mathbb {R}})\) by the Theorem of Lyubarskii-Seip-Wallstén. However, the following result implies that this is not the case for multivariate Gabor systems.
Theorem 1.10
The Gabor system \((g_{iI}, ({\mathbb {Z}}\oplus \frac{1}{2}{\mathbb {Z}})^2)\) does not give a frame in \(L^2({\mathbb {R}}^2)\). Note that we have
-
(1)
There exists an \({\mathbb {R}}\)-linear isomorphism f of \({\mathbb {R}}^4\) preserving the standard symplectic form \(d\xi ^T\wedge dx\) on \({\mathbb {R}}^2\times {\mathbb {R}}^2\) such that \(( g_{iI}, f({\mathbb {Z}}\oplus \frac{1}{2}{\mathbb {Z}})^2)\) does give a frame in \(L^2({\mathbb {R}}^2)\);
-
(2)
There exists \(\Omega \in {\mathfrak {H}}\) such that \(( g_\Omega , ({\mathbb {Z}}\oplus \frac{1}{2}{\mathbb {Z}})^2)\) does give a frame in \(L^2({\mathbb {R}}^2)\). Moreover, the set \( \{\Omega \in {\mathfrak {H}}: ( g_\Omega , ({\mathbb {Z}}\oplus \frac{1}{2}{\mathbb {Z}})^2)\text { is not a frame}\} \) is included in a closed analytic subset of the Siegel upper half-space \({\mathfrak {H}}\).
In the one-dimensional case, we obtain the following estimates thanks to Faltings’ Green function formula [Fal84] (see also Sect. 5.3).
Theorem B
Let \(\Lambda \) be a lattice in \({\mathbb {R}}\times {\mathbb {R}}\). Put
Then \(|\Lambda |^{-1}=|\Gamma | \ge C\) and we have the following estimates:
-
(1)
If \( C \ge 2\) then for all \(f\in L^2({\mathbb {R}})\) with \(||f||=1\), we have
$$\begin{aligned} \frac{e}{4|\Lambda |} \le \sqrt{2}\cdot \sum _{\lambda \in \Lambda } |(f, \pi _\lambda g_{iI})|^2 \le \frac{1}{(1-e^{-C})|\Lambda |}. \end{aligned}$$(1.11) -
(2)
If \( 1<C< 2\) then for all \(f\in L^2({\mathbb {R}})\) with \(||f||=1\), we have
$$\begin{aligned} \frac{(C-1) e}{C^2|\Lambda |} \le \sqrt{2}\cdot \sum _{\lambda \in \Lambda } |(f, \pi _\lambda g_{iI})|^2 \le \frac{1}{(1-e^{-C})|\Lambda |}. \end{aligned}$$(1.12) -
(3)
The most general case. Suppose that \(\Lambda \)’s symplectic dual lattice \(\Gamma \) in \({\mathbb {C}}\) is generated by \(\{a, \tau \}\) with \(a>0\) and \(a\,\textrm{Im}\, \tau >1\). Then for all \(f\in L^2({\mathbb {R}})\) with \(||f||=1\), we have
$$\begin{aligned} \frac{4\pi (a\, \textrm{Im}\, \tau -1)|\eta (\tau /a)|^6 }{\left( \sum _{n\in {\mathbb {Z}}} e^{-\pi n^2 \textrm{Im}\, \tau /a}\right) ^2} \le \sqrt{2}\cdot \sum _{\lambda \in \Lambda } |(f, \pi _\lambda g_{iI})|^2 \le \frac{a\,\textrm{Im}\, \tau }{1-e^{-C}}, \end{aligned}$$(1.13)where \(\eta (\tau ):=e^{\pi i\tau /12} \,\Pi _{n=1}^\infty (1-e^{2\pi i n \tau })\) is the Dedekind eta function.
Remark
With the notation in (3), we have
thus one may look at (1.13) as an effective version of [BGL10, Theorem 1.1] (or the corresponding Theorem 1.5). By our definition (1.9) of the Buser–Sarnak constant, we have
The lower bound estimate in (1.13) is based on a precise Robin constant estimate (see Theorem 5.5). For the upper bound, note that by [Sie89, Theorem 37], we have
where equality holds if and only if \(\Gamma \) is the hexagonal lattice. Hence the best upper bound in (1.13) (for fixed \(a\,\textrm{Im}\,\tau \)) is attached if and only if \(\Gamma \) is the hexagonal lattice. This fact is compatible with the Strohmer–Beaver conjecture (see [FS17, section 2.4]). However, the upper bound in (1.13) is not optimal in general.
2 Preliminaries
2.1 Gabor transform and Bargmann transform
For \(\Omega \in {\mathfrak {H}}\) (see 1.1) we identify \({\mathbb {R}}^n\times {\mathbb {R}}^n\) with \({\mathbb {C}}^n\) via \((\xi ,x)\mapsto z:=\xi +\Omega x\). We call the short-time Fourier transform of \(f\in L^2({\mathbb {R}}^n)\) with respect to
the Gabor transform of f:
The latter can be related to the \(\Omega \)-Bargmann transform
as follows:
By Moyal’s identity we have (note than \(||g_\Omega ||^2=(2^n \det (\textrm{Im}\,\Omega ))^{-1/2}\))
Since \(z=\xi +\Omega x\) implies that
we know that (2.1) gives
where
Sometimes we shall omit the Lebesgue volume form \(\left( \frac{i}{2} \partial {\overline{\partial }}|z|^2\right) ^n\) in the above. Now we are ready to introduce the following definition.
Definition 2.1
Fix \(\Omega \in {\mathfrak {H}}\) and put (notice that \((\textrm{Im}\, \Omega ) x= \textrm{Im}\, z\))
We call the space of holomorphic functions F on \({\mathbb {C}}^n\) with
the \(\Omega \)-Bargmann–Fock space and denote it by \({\mathcal {F}}^2_\Omega \).
Remark
In case \(\Omega = i I_n\), where \(I_n\) denotes the identity matrix, \({\mathcal {F}}^2_\Omega \) is precisely the following classical Bargmann–Fock space (see [Mum79, page 7] for the related Von-Neumann–Stone theorem)
In time-frequency analysis, the Bargmann–Fock space \({\mathcal {F}}^2\) in (1.4) is more widely used. But these two spaces are naturally isomorphic to each other since
Let us briefly recall the basics of Gaussian Gabor frames: We associate to the Gabor system \((g_\Omega ,\Lambda )\) the following operators:
-
analysis operator: \(C^\Lambda _{g_\Omega }\) is a map from \(L^2({\mathbb {R}}^n)\) to \(l^2(\Lambda )\) defined by \(f\mapsto \{(f, \pi _\lambda g_\Omega )\}_{\lambda \in \Lambda }\);
-
synthesis operator: \(D_{g_\Omega }^\Lambda \) is a map from \(l^2(\Lambda )\) to \(L^2({\mathbb {R}}^n)\) given by \(D_{g_\Omega }^\Lambda c=\sum _{\lambda \in \Lambda } c_\lambda \pi _\lambda {g_\Omega }\);
-
frame operator: \(S_{g_\Omega }^\Lambda :=D_{g_\Omega }^\Lambda \circ C^\Lambda _{{g_\Omega }}\) is an operator on \(L^2({\mathbb {R}}^n)\):
$$\begin{aligned} S_g^\Lambda f=\sum _{\lambda \in \Lambda } (f, \pi _\lambda g_\Omega ) \pi _\lambda g_\Omega . \end{aligned}$$
An elementary computation shows that \((C^\Lambda _{g_\Omega })^*=D_{g_\Omega }^\Lambda \) and thus \(S_{g_\Omega }^\Lambda :=(C_{g_\Omega }^\Lambda )^*\circ C^\Lambda _{g_\Omega }\) is a selfadjoint operator. The following result is well known, see [Gro01].
Lemma 2.1
For an arbitrary lattice \(\Lambda \) in \({\mathbb {R}}^{n} \times {\mathbb {R}}^n\), the coefficient operator \(C^\Lambda _{g_\Omega }\) from \(L^2({\mathbb {R}}^n)\) to \(l^2(\Lambda )\) is bounded.
Proof
We shall give a proof for readers’ convenience. Using the \(\Omega \)-Bargmann transform, it suffices to show that
where \(\Gamma :=\{z=\xi +\Omega x: (\xi ,x)\in \Lambda \}\). Since for \(w:=(\textrm{Im}\, \Omega )^{-1/2}z\), we have
we know the above inequality is equivalent to that
where \(\Gamma ':=\{w=(\textrm{Im}\, \Omega )^{-1/2}(\xi +\Omega x): (\xi ,x)\in \Lambda \}\). By the submean inequality, we know that the above inequality is true for
hence the lemma follows. \(\square \)
There is a fundamental duality theory (see [DLL95, Jan95, RS97, CKL04] and [JL20, Theorem 4.22]) that links the Gabor system \((g_\Omega ,\Lambda )\) with another Gabor system associated to the symplectic dual lattice/adjoint lattice defined in Definition 1.1.
Theorem 2.2
(Duality Theorem). \((g_\Omega ,\Lambda )\) is a Gabor frame for \(L^2({\mathbb {R}}^n)\) with bounds A and B if and only if \(\{\pi _{\lambda ^\circ }g_\Omega \}_{\lambda ^\circ \in \Lambda ^\circ }\) is a Riesz sequence with bounds \(A|\Lambda |\) and \(B|\Lambda |\), i.e. we have
for all \(c\in \ell ^2(\Lambda ^\circ )\).
There is an intricate link between Gabor analysis and Bargmann-Fock spaces: a Gaussian Gabor system \((g_\Omega ,\Lambda )\) is a frame if and only if \(\Lambda \) is a set of sampling for \({\mathcal {F}}^2_\Omega \), and \((g_\Omega ,\Lambda ^\circ )\) is a Riesz basis for its closed linear span if and only if \(\Lambda ^\circ \) is a set of interpolation for \({\mathcal {F}}^2_\Omega \), see [GL20] for the standard case \(\Omega =iI\).
Motivated by this we introduce the following well known notions:
Definition 2.2
Let \(T: H_1 \rightarrow H_2\) be a bounded \({\mathbb {C}}\)-linear map between two complex Hilbert spaces. Then
-
(1)
T is called sampling if there exist constants \(A, B>0\) such that
$$\begin{aligned} A\,||f||^2 \le ||Tf||^2 \le B\,||f||^2, \ \ \forall \ f\in H_1; \end{aligned}$$(2.4) -
(2)
T is referred to as interpolating if T is surjective and there exist constants \(A, B>0\) such that
$$\begin{aligned} A\,||f_c||^2 \le ||c||^2 \le B\, ||f_c||^2, \ \ \ \forall \ c\in H_2, \end{aligned}$$(2.5)where \(f_c\) denotes the (unique) solution of \(T(\cdot )=c\) with minimal norm.
The constants A, B above are called the sampling (interpolating) bounds.
Proposition 2.3
T is sampling with (2.4) if and only if \(T^*\) is interpolating with (2.5).
Proof
Assume that T is sampling with (2.4). Then the eigenvalues of \(T^*T\) lie in [A, B], thus \(T^*T\) has an inverse, say \(S:=(T^*T)^{-1}\), which implies that \(T^* T S f=f, \ \ \forall \ c\in H_1\). Thus \(T^*: H_2\rightarrow H_1\) is surjective and the minimal solution of \(T^*(\cdot )=f\) is TSf (note that TSf is minimal since \(TSf \bot \ker \, T^*\)). We need to show that
In fact, \(S^{-1}\ge A\,I\) implies that
thus \(A||TSf||^2 \le ||f||^2\). Moreover, \(f=T^*TSf\) implies
which gives \(||f||^2 \le B ||TSf||^2\). This establishes one direction and the other direction may be deduced in a similar manner. \(\square \)
The following theorem follows directly from Theorem 2.2 and Definition 2.2.
Theorem 2.4
Let \(\Lambda \) be a lattice in \({\mathbb {R}}^{n} \times {\mathbb {R}}^n\). Then \(C_{g_\Omega }^\Lambda \) is sampling with
if and only if \(D_{g_\Omega }^{\Lambda ^\circ }\) is sampling with
Notice that \(C_{g_\Omega }^\Lambda \) is sampling if and only \((g_\Omega ,\Lambda )\) defines a frame in \(L^2({\mathbb {R}}^n)\). The density theorem for Gabor frames states that if \(C_{g_\Omega }^\Lambda \) is sampling, then \(|\Lambda | \le 1\). Furthermore, a Balian-Low type theorem (see [AFK14, Theorem 1.5] or [GHO19] for related results associated to general Fock spaces) further gives:
Theorem 2.5
Given a lattice \(\Lambda \) in \({\mathbb {R}}^{n} \times {\mathbb {R}}^n\). If \(C_{g_\Omega }^\Lambda \) is sampling then \(|\Lambda |<1\).
The above two theorems and Proposition 2.3 imply
Corollary 2.6
Given a lattice \(\Lambda \) in \({\mathbb {R}}^{n} \times {\mathbb {R}}^n\). Then \(C_{g_\Omega }^\Lambda \) is sampling if and only if \(C_{g_\Omega }^{\Lambda ^\circ }\) is interpolation. Moreover, the interpolation bounds are a scalar multiple of the sampling bounds. In particular, \(C_{g_\Omega }^\Lambda \) can not be both sampling and interpolation.
Proof
The first part follows directly from Theorem 2.4 and Proposition 2.3. For the second part, notice that if \(C_{g_\Omega }^\Lambda \) is both sampling and interpolation, we must have
which is a contradiction since \(|\Lambda | \cdot |\Lambda ^\circ |=1\). \(\square \)
Remark
In case \(n=1\) and
for some complex number a with \(\textrm{Im} \,a>0\), we known that \(C_{g}^\Lambda \) is sampling if and only if \(|\Lambda | <1\) (see [Lyu92, Sei92, SW92]). For general n, \(g_\Omega (t):= \overline{e^{\pi i t^T\Omega t}}\), Theorem 1.10 implies that there exists a lattice \(\Lambda \) in \({\mathbb {R}}^{n} \times {\mathbb {R}}^n\) such that \(C_{g_{\Omega }}^\Lambda \) is sampling (resp. interpolation) for some \(\Omega \in {\mathfrak {H}}\) but not for all \(\Omega \in {\mathfrak {H}}\). On the other hand, if \(C_{g_{\Omega _0}}^\Lambda \) is sampling for some \(\Omega _0\in {\mathfrak {H}}\) then by Theorem 1.3 in [AFK14], we know that \(C_{g_\Omega }^\Lambda \) is sampling if \(\Omega \) is very close to \(\Omega _0\).
2.2 Proof of Proposition 1.4
Proof of Proposition 1.4
By our definition, \((g_\Omega ,\Lambda )\) defines a frame in \(L^2({\mathbb {R}}^n)\) if and only if \(C_{g_\Omega }^\Lambda \) is sampling, which is equivalent to that \(C_{g_\Omega }^{\Lambda ^\circ }\) is interpolation (see Corollary 2.6). Using the \(\Omega \) Bargmann transform, we know that \(C_{g_\Omega }^{\Lambda ^\circ }\) is interpolation if and only if \(C_{g_\Omega }^{\Lambda ^\circ }\) is bounded and
is a set of interpolation for \({\mathcal {F}}^2_\Omega \). By Lemma 2.1, we know that \(C_{g_\Omega }^{\Lambda ^\circ }\) is always bounded, hence \((g_\Omega ,\Lambda )\) defines a frame in \(L^2({\mathbb {R}}^n)\) if and only if \(\Gamma \) is a set of interpolation for \({\mathcal {F}}^2_\Omega \). Notice that
implies that
defines an isomorphism from \({\mathcal {F}}^2_\Omega \) to \({\mathcal {F}}^2\). Thus \((g_\Omega ,\Lambda )\) defines a frame in \(L^2({\mathbb {R}}^n)\) if and only if
is a set of interpolation for \({\mathcal {F}}^2\). \(\square \)
The duality principle Theorem 2.4 further implies:
Theorem 2.7
With the notation in the above proof, the following statements are equivalent:
-
(1)
\(\Gamma _{\Omega , \Lambda ^\circ } \) is a set of interpolation for \({\mathcal {F}}^2\) and for all \(F\in {\mathcal {F}}^2\) with
$$\begin{aligned} \sum _{\gamma \in \Gamma _{\Omega , \Lambda ^\circ }} |F(\gamma )|^2e^{-\pi |\gamma |^2} =1, \end{aligned}$$we have
$$\begin{aligned} A\le \inf _{F' \in {\mathcal {F}}^2,\, F'=F \, \text {on} \,\Gamma _{\Omega , \Lambda ^\circ } } ||F'||^2 \le B; \end{aligned}$$ -
(2)
\(\Gamma \) is a set of interpolation for \({\mathcal {F}}_\Omega ^2\) and for all \(F\in {\mathcal {F}}_\Omega ^2\) with
$$\begin{aligned} \sum _{\gamma \in \Gamma } |F(\gamma )|^2e^{-2\pi \phi _\Omega (\gamma )} =1, \end{aligned}$$we have
$$\begin{aligned} A \cdot \det (\textrm{Im}\,\Omega )\le \inf _{F' \in {\mathcal {F}}_\Omega ^2,\, F'=F \, \text {on} \,\Gamma } ||F'||^2 \le B \cdot \det (\textrm{Im}\,\Omega ); \end{aligned}$$ -
(3)
\((\Lambda , g_\Omega )\) defines a frame in \(L^2({\mathbb {R}}^n)\) and for all \(f\in L^2({\mathbb {R}}^n)\), \(||f||=1\),
$$\begin{aligned} \frac{(B\cdot |\Lambda |)^{-1}}{\sqrt{2^n\det (\textrm{Im}\,\Omega )}} \le \sum _{\lambda \in \Lambda } |(f, \pi _\lambda g_\Omega )|^2 \le \frac{(A\cdot |\Lambda |)^{-1}}{\sqrt{2^n\det (\textrm{Im}\,\Omega )}}. \end{aligned}$$
Proof
(2.6) implies \((1)\Leftrightarrow (2)\). By (2.2), we know that (2) is equivalent to that \(C_{g_\Omega }^{\Lambda ^\circ }\) is interpolation (see 2.5) with
By Proposition 2.3, the above inequality is equivalent to that
Thus Theorem 2.4 gives \((2)\Leftrightarrow (3)\). \(\square \)
2.3 Proof of the Hörmander criterion (Theorem 1.7)
2.3.1 \(L^2\)-estimate for the \({\overline{\partial }}\)-equation
We shall use the following special case of Hörmander’s theorem (see [Dem, page 378, Theorem 6.5], see also Chapter 4 in [Hor90]):
Theorem 2.8
Fix a smooth (0, 1)-form v with \({\overline{\partial }}v=0\) on \({\mathbb {C}}^n\). Let \(\phi \) be a plurisubharmonic function such that \(\phi -\delta |z|^2\) is also plurisubhamonic on \({\mathbb {C}}^n\) for some positive constant \(\delta \). Then there is a smooth function a on \({\mathbb {C}}^n\) such that \({\overline{\partial }}u=v\) and
where \(|v|^2:=\sum |v_{{\bar{j}}}|^2\) for \(v=\sum v_{{\bar{j}}} d{\bar{z}}_j\).
Proof of the Hörmander criterion (Theorem 1.7)
Notice that the \(\beta \)-Seshadri constant does not depend on the choose of \(x\in {\mathbb {C}}^n/\Gamma \). Thus, if the Hörmander constant is bigger than one then there exist \(\gamma >1\) and an \(\omega _\textrm{euc}\)-psh function \(\psi \) on \({\mathbb {C}}^n/\Gamma \) such that \(\psi =\gamma T_\beta \) near \(0\in {\mathbb {C}}^n/\Gamma \) for some \(\beta \in {\mathcal {B}}\). Let
be the natural quotient mapping. Fix \(c=\{c_\lambda \}\) such that
Let us apply Theorem 2.8 to
where \(\chi \) is a smooth function on \({\mathbb {R}}\) that is equal to 1 near the origin and equals to 0 outside a smooth ball of radius r. Let us take r such that
Then we know that v is smooth, \({\overline{\partial }}v=0\) and
for some constant thay does not depend on the sequence \(c=\{c_\lambda \}\). Moreover, since \(\psi \) is \(\omega _\textrm{euc}\)-psh, we know that \(\phi (z)-(1-\gamma ^{-1})\pi |z|^2\) is plurisubharmonic. Thus Theorem 2.8 implies that there exists a smooth function u such that \({\overline{\partial }}u=v\) and
By a direct computation we know that \(e^{-T_\beta }\) is not integrable near \(0\in {\mathbb {C}}^n/\Gamma \), hence \(e^{-\phi }\) is not integrable near \(\Gamma \) and (2.7) implies that u vanishes at \(\Gamma \). Take
we know that F is holomorphic in \({\mathbb {C}}^n\),
for some constant \(C_1\) does not depend on c (notice that \(\psi \) is bounded from above) and \(F(\lambda )=c_\lambda \) for all \(\lambda \in \Gamma \). Thus \(\Gamma \) is a set of interpolation. The final statement is a direct consequence pf Proposition 1.6, which will be proved in Sect. 3.1. \(\square \)
In order to estimate the \(L^2\) norm of the extension F in the above proof, we shall introduce the following Ohsawa–Takegoshi type theorem [OT87] proved by Berndtsson and Lempert (see [BL16, Theorem 3.8], the main theorem in [GZ15] and [Blo13] for related results).
Theorem 2.9
Let \(\Gamma \) be a lattice in \({\mathbb {C}}^n\). Assume that there exists a non positive \(\Gamma \) invariant function \(\psi \) on \({\mathbb {C}}^n\) such that \(\psi (z)+\pi |z|^2\) is plurisubharmonic on \({\mathbb {C}}^n\), \(\psi \) is smooth outside \(\Gamma \) and \( \psi (z) - \gamma \log |z|^2\) is bounded near the origin for some constant \(\gamma >n\). Then for every sequence of complex numbers \(\{c_{\lambda }\}_{\lambda \in \Gamma }\) with \( \sum _{\lambda \in \Gamma } |c_\lambda |^2 e^{-\pi |\lambda |^2}=1, \) there exists \(F\in {\mathcal {F}}^2\) such that \(F(\lambda )=c_\lambda \) for all \(\lambda \in \Gamma \) and
Proof
Since \(\psi \) is \(\Gamma \) invariant, we know that \(\psi \) has isolated order \(\gamma \) log poles at \(\Gamma \), one may use Ohsawa–Takegoshi extension theorem to extend \(L^2\) functions from \(\Gamma \) to \({\mathbb {C}}^n\). Denote by F the extension with minimal \(L^2\) norm. By our assumption
Hence [BL16, Theorem 3.8] or the main theorem in [GZ15] implies
thus our theorem follows. \(\square \)
2.4 Transcendental lattices and jet interpolations
Let us first introduce the following definition for jet interpolations.
Definition 2.3
Let \(k\ge 0\) be an integer. Let \(\Gamma \) be a lattice in \({\mathbb {C}}^n\). Put
We say that \(\Gamma \) is a set of k-jet interpolation for \({\mathcal {F}}^2\) if there exists a constant \(C>0\) such that for every sequence of complex numbers \(\{c_{\lambda , \alpha }\}_{\lambda \in \Gamma , \alpha \in N_k }\) with \( \sum _{\lambda \in \Gamma , \alpha \in N_k} |c_{\lambda , \alpha }|^2 e^{-\pi |\lambda |^2}=1, \) there exists \(F\in {\mathcal {F}}^2\) with
and \(||F||^2 \le C\).
The proof of the Hörmander criterion above also implies the following result.
Theorem 2.10
Let \(k\ge 0\) be an integer. Let \(\Gamma \) be a transcendental lattice in \({\mathbb {C}}^n\). Assume that
then \(\Gamma \) is a set of k-jet interpolation for \({\mathcal {F}}^2\).
Proof
Since \(\Gamma \) is transcendental, by (3.2), we know that (2.8) implies that there exists a non positive \(\Gamma \) invariant function \(\psi \) on \({\mathbb {C}}^n\) such that \(\psi (z)+\pi |z|^2\) is plurisubharmonic on \({\mathbb {C}}^n\), \(\psi \) is smooth outside \(\Gamma \) and \( \psi (z) - \gamma \log |z|^2\) is bounded near the origin for some constant \(\gamma >n+k\). Thus the Hörmander \(L^2\) estimate with singular weight \(\psi \) (similar to the proof of the Hörmander criterion above) gives the above theorem. \(\square \)
Remark
If \(\Gamma \) is transcendental with \(|\Gamma |>\frac{n^n}{n!}\) then
thus we know that \(\sqrt{\frac{n+k}{n}} \,\Gamma \) is a set of k-jet interpolation for \({\mathcal {F}}^2\). In one dimensional case, we have the following theorem [GL09].
Theorem 2.11
Let \(\Gamma \) be a lattice in \({\mathbb {C}}\). Then the followings are equivalent:
-
(1)
\(\Gamma \) is a set of interpolation for \({\mathcal {F}}^2\);
-
(2)
\(\sqrt{k+1}\,\Gamma \) is a set of k-jet interpolation for \({\mathcal {F}}^2\) for some positive integer k;
-
(3)
\(|\Gamma |>1\).
For the higher-dimensional cases, we can prove the following result.
Theorem 2.12
Let \(\Gamma \) be a transcendental lattice in \({\mathbb {C}}^n\). If \(\sqrt{\frac{n+k}{n}} \,\Gamma \) is a set of k-jet interpolation for \({\mathcal {F}}^2\) for some non-negative integer k then \(|\Gamma | \ge \frac{(k+1)^n}{(n+k)^n}\frac{n^n}{n!}\).
Proof
Put \(\nabla ^\alpha F:=e^{\pi |z|^2}\partial ^{\alpha } (e^{-\pi |z|^2} F)\) and \(\Gamma _k:=\sqrt{\frac{n+k}{n}} \,\Gamma \). Assume that \(\sqrt{\frac{n+k}{n}} \,\Gamma \) is a set of k-jet interpolation for \({\mathcal {F}}^2\). Let us define
Then by the Balian-Low type theorem, we know that G is not identically zero on \({\mathbb {C}}^n\). We claim that G is \(\Gamma _k\) invariant. In fact, if we put
Then \(F \mapsto T_\lambda F\) is an isomorphism on \({\mathcal {F}}^2\) with \(||F||=||T_\lambda F||\), \(|T_\lambda F(0)|^2=|F(\lambda )|^2e^{-\pi |\lambda |^2}\) and
Hence G is \(\Gamma _k\) invariant. Put \(\psi =\log G\), we know that \(\psi \) is \(\Gamma \) invariant, \(\psi (z)+\pi |z|^2\) is plurisubharmonic on \({\mathbb {C}}^n\) and \(\psi (z)-(k+1)\log |z|^2\) is bounded above near \(z=0\). Since \(\Gamma \) is transcendental, we know that \(\Gamma _k\) is also transcendental, thus the \(\Gamma _k\) invariant analytic set \(\{\psi =-\infty \}\) is discrete. Hence (3.2) gives that
from which our theorem follows. \(\square \)
Remark
The above theorem is our motivation for the conjecture A after Theorem 1.1, moreover, notice that \(\lim _{k\rightarrow \infty } \frac{k+1}{n+k}=1\), the above theorem also suggests the following higher-dimensional analogue of Theorem 2.11.
Conjecture B
Let \(\Gamma \) be a transcendental lattice in \({\mathbb {C}}^n\). Then the followings are equivalent:
-
(1)
\(\Gamma \) is a set of interpolation for \({\mathcal {F}}^2\);
-
(2)
\(\sqrt{\frac{n+k}{n}} \,\Gamma \) is a set of k-jet interpolation for \({\mathcal {F}}^2\) for some positive integer k;
-
(3)
\(|\Gamma |>\frac{n^n}{n!}\).
Remark
From Proposition 1.4, we know that the above conjecture implies conjecture A.
3 Hörmander constants and Kähler embeddings
3.1 Hörmander constants and proof of Proposition 1.6
In Definitions 1.3 and 1.5 we have defined the Hörmander constants and the \(\beta \)-Seshadri constants for an n-dimensional compact Kähler manifold \((X, \omega )\). If all \(\beta _j=1/n\) and \(\omega \in c_1(L)\) for some ample line bundle L then we have
where \(\epsilon _x(\omega )\) denotes the Seshadri constant introduced by Demailly in [Dem92] (in fact, from (6.2) in [Dem92], we have \(n \epsilon _x(\omega ;\beta )=\gamma (L,x)\), but Theorem 6.4 in [Dem92] tells us that \(\gamma (L,x)\) is precisely the Seshadri constant used in algebraic geometry when L is ample). In general, the condition \(\sum \beta _j=1\) is used to make sure that
Remark
For transcendental \(\omega \) on a general compact Kähler manifold, we know that (see Theorem 3.2 below for the proof and generalizations) \( n \epsilon _x(\omega ;\beta ) \) is equal to the generalized Seshadri constant (also denoted by \(\epsilon _x(\omega )\)) defined by Tosatti in [Tos18, section 4.4]. In this general case, we shall prove the following result.
Proposition 3.1
Let \((X, \omega )\) be an n-dimensional compact Kähler manifold. Assume that X has no non-trivial analytic subvarieties, then
Proof
Note that for every \(\beta \in {\mathcal {B}}\), by [Dem, page 167, Corollary 7.4] (our definition of \(dd^c\) in Definition 1.5 is half of the one there) we have
which gives
Hence
On the other hand, we have the following identity proved by Tosatti in [Tos18, Theorem 4.6]
where the infimum runs over all positive-dimensional irreducible analytic subvarieties V containing x and \(\textrm{mult}_x V\) denotes the multiplicity of V at x. Hence if X has no non-trivial subvarieties then (put \(\beta _0= (1/n,\cdots , 1/n)\), use (3.1) and the remark above)
The above inequality and (3.4) together imply (3.2). \(\square \)
Proof of Proposition 1.6
Apply (3.2) to the case that \(X={\mathbb {C}}^n /\Gamma \) and \(\omega = dd^c(\pi |z|^2)\), we get immediately Proposition 1.6 (note that in this case
and \(\int _X \omega ^n = n! \,|\Gamma |\)). \(\square \)
3.2 Relation with the s-invariant
Our \(\beta \)-Seshadri constant is closely related to the s-invariant introduced by Cutkosky, Ein and Lazarsfeld in [CEL01].
Theorem 3.2
Let \((X, \omega )\) be an n-dimensional compact Kähler manifold. Assume that \(\omega \) lies in the first Chern class of a holomorphic line bundle L on X. Fix \(\beta =(\beta _1,\cdots , \beta _n)\in {\mathbb {R}}^n\) such that all \(\beta _j^{-1}\) are positive integers. Then
where \({\mathcal {I}}_\beta \) is the ideal of \({\mathcal {O}}_X\) generated by \(\{z_1^{1/\beta _1}, \cdots , z_n^{1/\beta _n}\}\) and
is the s-invariant of \({\mathcal {I}}_\beta \) with respect to L (see Definition 5.4.1 in [Laz04]), where \(\mu \) is the blowing-up of X along \({\mathcal {I}}_\beta \) with exceptional divisor E.
Proof
First let us prove \(\epsilon _x(\omega ;\beta )\ge \frac{1}{s_{L}({\mathcal {I}}_\beta )}\). Note that L is ample since \(\omega \) is positive, hence
By Example 5.4.10 in [Laz04], one may replace \(\mu \) by a desingularization \(f: Y\rightarrow X\) of \({\mathcal {I}}_\beta \) with exceptional divisor F, more precisely, we have
which implies that for every \(\gamma <\frac{1}{s_{L}({\mathcal {I}}_\beta )}\) there is a singular metric \(e^{-\phi }\) on \(f^*L\) with \(\gamma \)-log pole along F such that \(i\partial {\overline{\partial }}\phi >0\) on Y. Then the weight \(f_*\phi \) on L will have the \(\gamma T_\beta \)-singularity, from which we know that \(\epsilon _x(\omega ;\beta )\ge \gamma \). Hence \(\epsilon _x(\omega ;\beta )\ge \frac{1}{s_{L}({\mathcal {I}}_\beta )}\).
Now let us prove that \(\epsilon _x(\omega ;\beta )\le \frac{1}{s_{L}({\mathcal {I}}_\beta )}\). For every \(\gamma < \epsilon _x(\omega ;\beta )\), we can find a singular metric \(e^{-\psi }\) on L with \(\gamma T_\beta \)-singularity such that \(i\partial {\overline{\partial }}\psi >0\). Then \(e^{-f^*\psi }\) defines a singular metric on \(f^*L\) with \(\gamma \)-log pole along F such that \(i\partial {\overline{\partial }}(f^*\psi )>0\) on Y, from which we know that \(\epsilon _x(\omega ;\beta )\le \frac{1}{s_{L}({\mathcal {I}}_\beta )}\). \(\square \)
Remark
Since \({\mathcal {I}}_\beta \) are special monomial ideals, it is not hard to find the explicit desingularizations. Let us look at the simple example \(\beta =(1,\frac{1}{2})\). Then, in this case, we have
First, one may blow up the origin, so \(z_1=uv, z_2=u\) gives
then we can blow up the point \((u, v)=(0,0):={\tilde{0}}\), so \(v= ts, u=s\) gives
from which we know that \(F= 2\cdot |\{s=0\}|\).
3.3 Extremal property of the \(\beta \)-Seshadri constant
For s-invariant of a general monomial ideal
with isolated zero set \(\{x\}\), where P is the Newton polytope defined by
one may correspondingly define the P-Seshadri constant
where
Then the proof of Theorem 3.2 also implies
The reason why we only use \(\beta \)-Seshadri constants in this paper is that they have the following extremal property.
Theorem 3.3
\(\sup \{ \epsilon _x(\omega ;P): (1,\ldots , 1)\text { lies in the boundary of }P\}\) equals \(\sup _{\sum \beta _j=1}\epsilon _x(\omega ;\beta )\).
Proof
The proof follows from a very simple fact: for an arbitrary Newton polytope P such that \((1,\ldots , 1)\) lies in the boundary of P, one can always find \(\beta \in {\mathbb {R}}_{>0}^n\) with \(\sum \beta _j =1\) and
hence
gives the theorem. \(\square \)
Remark
The condition that \((1,\ldots , 1)\) lies in the boundary of P is equivalent to the following identity
see [How01, Gue12] for the proof and related results.
3.4 McDuff–Polterovich’s theorem
McDuff–Polterovich [MP94] proved that the Seshadri constant
is always no bigger than the following Gromov width of \((X, \omega )\) (see [LMS13, Theorem 1.1] and [EV16])
where means there exist a smooth injection \( f: B_r \hookrightarrow X \) such that \( f^*(\omega )=\omega _\textrm{euc}:=\frac{i}{2} \sum _{j=1}^n dz_j \wedge d{\bar{z}}_j.\) In fact, the proof in [MP94] also gives the following stronger result:
Theorem 3.4
Let \((X, \omega )\) be a compact Kähler manifold. Denote by \({\mathcal {K}}_\omega \) the space of Kähler metrics in the cohomology class \([\omega ]\). Then the Seshadri constant \(\epsilon _x(\omega )\) is equal to the following Kähler width
where
“" means that there exists an holomorphic injection \(f: B_r \hookrightarrow X\) such that \( f(0)=x\) and \(f^*({\tilde{\omega }})=\omega _\textrm{euc}\).
Proof
The proof here is different from McDuff–Polterovich’s approach in [MP94]. Our main idea is to use the following plurisubharmonic function on \({\mathbb {C}}^n\)
which satisfies
-
(L1)
\(\psi _r(z)-\pi r^2 \log |z|^2\) is bounded near 0.
-
(L2)
\(\psi _r(z) \le \pi |z|^2\) on \({\mathbb {C}}^n\) and \(\{z\in {\mathbb {C}}^n:\psi _r(z)<\pi |z|^2\}=B_r\).
-
(L3)
For every \(w\in B_r\) and \(0<\delta <1\) we have
$$\begin{aligned} \psi _r(f_\delta (w))= \psi _r(w) + \pi r^2 \log (\delta ^2), \end{aligned}$$where \( f_\delta (w):=(\delta w_1, \ldots , \delta w_n). \)
We shall use (L1)-(L3) to prove (P1), (P2) below which imply the theorem:
-
(P1)
If \(c_x(\omega ) > \pi r^2\) then \(\epsilon _x(\omega ) \ge \pi r^2\).
-
(P2)
If \(\epsilon _x(\omega ) > \pi r^2\) then \(c_x(\omega ) \ge \pi r^2\).
Proof of (P1): If \(c_x(\omega ) > \pi r^2 \) then then one may think of \(B_r\) as a Kähler subset of X. Let us define \({\tilde{\psi }}\) such that \({\tilde{\psi }}=\psi _r-\pi |z|^2\) on \(B_r\) and \({\tilde{\psi }}=0\) outside \(B_r\) in X. Then (L2) implies that \({\tilde{\psi }}\) is \(\omega \)-psh (note that \(\omega _\textrm{euc}=dd^c(\pi |z|^2)\)). Fix a small \(\varepsilon >0\) (assume that \(\varepsilon <\pi r^2\)), then by (L1) one may take a sufficiently big \(C>0\) such that
is equal to \({\tilde{\psi }}+C\) near the boundary of \(B_r\). Hence \(\psi \) extends to an \(\omega \)-psh function on X. Notice that \(\psi = (\pi r^2-\varepsilon )\log |z|^2 \) near \(z(x)=0\), hence \(\epsilon _x(\omega ) \ge \pi r^2-\varepsilon \) for all \(\varepsilon >0\) and (P1) holds.
Proof of (P2): To simplify the notation we take \(r=1\). If \(\epsilon _x(\omega ) > \pi \) then one may use the construction in (3.9) to produce an \(\omega \)-psh function \(\psi \) smooth on \(X\setminus \{x\}\) such that
and \(\omega +dd^c\psi >0\) on X, where \(\psi _1\) in defined in (3.8). For \(w\in B_1\) and \(0<\delta <1\) let us define
where \(\textrm{Max} \) denotes a regularized max function. By (L3) and (L2)
for \(w\in \overline{B_1}\), with identity holds if and only if \(w\in \partial B_1\); together with \((*)\), we know that for every \(0<\gamma <1\), one may take a small \(\delta \) such that \(dd^c \phi \) extends to a Kähler form \({\tilde{\omega }} \in {\mathcal {K}}_\omega \) with
Letting \(\gamma \rightarrow 1\) we finally get \(c_x(\omega )\ge \pi \). The proof is complete. \(\square \)
In order to generalize the above proof to general \(\beta \)-Seshadri constants, we need to construct the associated \(\beta \)-version of \(\psi _r\) in (3.8) (see Lemma 3.8 below). In the next subsection, we shall use the Legendre transform theory to decode the construction.
3.5 Iterated Legendre transform and proof of Theorem A
The main ingredient in our proof of Theorem A is the theory of iterated Legendre transform.
Definition 3.1
Let \(\phi \) be a smooth convex function on \({\mathbb {R}}^n\). We call
the Legendre transform of \(\phi \). Let \(A\subset {\mathbb {R}}^n\) be a closed set. We call
the iterated Legendre transform of \(\phi \) with respect to A.
Remark
Notice that
hence we always have
Lemma 3.5
Let \(\phi \) be a smooth convex function on \({\mathbb {R}}^n\). Then
if and only if \(\alpha =\nabla \phi (x)\), where
Proof
It follows from the fact that x is the maximum point of the following concave function
if and only if x is a critical point of \(\psi ^\alpha \). \(\square \)
Proposition 3.6
If \(\phi \) is smooth strictly convex and \(A \subset {\mathbb {R}}^n\) is closed then
Proof
By the above lemma, we have
with identity holds if and only if \(\alpha =\nabla \phi (x)\). Hence \(\nabla \phi (x)\) is the unique maximum point of the following function
The proof of Proposition 2.2 in [Wan18] implies that \(\rho ^x\) is smooth strictly concave on \(\nabla \phi ({\mathbb {R}}^n)\). Hence the supremum of \(\rho ^x\) on the complement of any small ball around \(\nabla \phi (x)\) must be strictly smaller than \(\phi (x)\). By our assumption, A is closed, thus \(\phi (x)=\phi _A (x)\) if and only if \(\nabla \phi (x) \in A\). \(\square \)
Definition 3.2
Let \(\phi \) be a smooth strictly convex function on \({\mathbb {R}}^n\) and A be a closed set in \({\mathbb {R}}^n\). We call
the Hele–Shaw domain of \((\phi , A)\).
The above proposition implies that
Our key observation is the following:
Theorem 3.7
Assume that \(\phi \) is smooth strictly convex and \(A \subset {\mathbb {R}}^n\) is closed. If \(x\in \Omega _A(\phi ) \) then
where \(\partial A\) denotes the boundary of A.
Proof
Since \(\nabla \phi (x)\) is the unique maximum point of the following concave function
and \(\nabla \phi (x) \notin A\), we know that for every point \(a\in A\),
is increasing on \(t\in [0,1]\). Take \({\hat{t}} \in [0,1]\) such that
we know that
Hence the theorem follows. \(\square \)
Lemma 3.8
With the notation in Theorem A, for each \(r>0\) there exists \(\psi _r \in \textrm{psh}({\mathbb {C}}^n)\) such that
-
(L1)
\(\psi _r-\pi r^2 T_\beta \) is bounded near 0.
-
(L2)
\(\psi _r(z) \le \pi |z|^2\) on \({\mathbb {C}}^n\) and \(\{z\in {\mathbb {C}}^n: \psi _r(z)<\pi |z|^2\}=B_r^\beta \).
-
(L3)
For every \(w\in B_r^\beta \) and \(0<\delta <1\) we have
$$\begin{aligned} \psi _r(f_\delta (w))= \psi _r(w) + \pi r^2 \log (\delta ^2), \end{aligned}$$where \( f_\delta (w):=(\delta ^{\beta _1}w_1, \ldots , \delta ^{\beta _n}w_n). \)
Proof
Let us consider
where
Then we have
and
where \(0\log 0:=0\). Hence
where
Proof of (L1): Notice that \(\psi _r-\pi r^2 T_\beta \) is bounded near 0 if and only if \(\phi _A(x)-\sup _{\alpha \in A_+} \alpha \cdot x\) is bounded on \((-\infty , 0]^n\). The above formula for \(\phi _A\) implies that
for all \(x\in (-\infty , 0]^n\). Hence (L1) follows.
Proof of (L2): Follows directly from (3.10).
Proof of (L3): Notice that (3.11) implies
from which (L3) follows. \(\square \)
Proof of Theorem A
Similar as the proof of Theorem 3.4 (replace \(B_r\) by \(B_r^\beta \)), we know that the lemma above gives Theorem A. \(\square \)
3.6 A partial converse of the Hörmander criterion
For general higher dimensional cases, we do not know whether the Hörmander criterion is an equivalent criterion or not. Based on Demailly–Păun’s generalized Nakai–Moishezon ampleness criterion [DP04], a theorem of Nakamaye [Nak96, Oht19], Lindholm’s result [Lin01] and the Balian-Low type theorem (see [Hei07, Theorem 10] and [AFK14, Theorem 1.5]), we obtain the following partial converse of the Hörmander criterion.
Theorem 3.9
Let \(\Gamma \) be a lattice in \({\mathbb {C}}^n\). Denote by \(\iota _\Gamma \) the Hörmander constant of \(({\mathbb {C}}^n/\Gamma , \omega _\textrm{euc})\).
-
(1)
If \(\Gamma \) is a set of interpolation for \({\mathcal {F}}^2\) and all irreducible analytic subvarieties of X are translates of complex tori then \(\iota _\Gamma >1/n\).
-
(2)
If \(\Gamma \) is a set of interpolation for \({\mathcal {F}}^2\) and the only positive dimensional irreducible analytic subvariety of X is X itself then
$$\begin{aligned} \iota _\Gamma =\frac{(n!\,|\Gamma |)^{1/n} }{n} > \frac{(n!)^{1/n} }{n}. \end{aligned}$$ -
(3)
Assume that \(\omega _\textrm{euc}\) is rational on \(\Gamma \) or the Picard number of X is \(n^2\). If \(\Gamma \) is a set of interpolation for \({\mathcal {F}}^2\) then \(\iota _\Gamma >1/(n\,e)\).
Proof
Denote by 0 the unit of the torus and write \(\omega :=\omega _\textrm{euc}\). The main idea is to use the following Demailly–Păun identity proved by Tosatti in [Tos18, Theorem 4.6]
where the infimum runs over all positive-dimensional irreducible analytic subvarieties V containing 0, and \(\textrm{mult}_0 V\) denotes the multiplicity of V at 0. Now let us prove Theorem (1), by our assumption, it suffices to show that for all complex subtorus
Choose a \({\mathbb {C}}\) linear subspace E of \({\mathbb {C}}^n\) such that \(V=E/(E\cap \Gamma )\). Notice that if \(\Gamma \) is a set of interpolation for \({\mathcal {F}}^2\), then \(E\cap \Gamma \) is a set of interpolation for \({\mathcal {F}}^2|_{E}\). Thus the Balian–Low type theorem implies that
which gives \(\epsilon _0(\omega )>1\), hence (1) follows. Now assume further that X has no non-trivial subvarieties, then (3.2) implies
thus (2) follows directly from the Balian–Low type theorem. To prove (3), we shall use the following inequality (see [Ito20, Lemma 3.2])
where the infimum runs over all positive-dimensional abelian subvarieties V containing 0. Notice that the right hand side of (3.12) is 1-homogeneous with respect to \(\omega \), we know that (3.12) also holds for all \(\omega =c\omega '\), where \(\omega '\) is integral on \(\Gamma \). In particular, it holds true if \(\omega \) is rational on \(\Gamma \). In case the Picard number of X is \(n^2\), we know that \(\omega \) can be approximated by rational \(\omega '\), hence (3.12) is true for all \(\omega \). By Balian–Low type theorem, we have
By Stirling’s approximation, we have
hence (3) follows. \(\square \)
3.7 Hörmander constants and densities of general discrete sets
Definition 3.3
Let S be a discrete set in \({\mathbb {C}}^n\). Let \(\psi \) be a non positive function such that \(\psi +\pi |z|^2\) is plurisubharmonic on \({\mathbb {C}}^n\). Let \(\gamma \) be positive number. We call \((\psi , \gamma )\) an S-admissible pair if \(\psi \) is smooth outside S, \(e^{-\psi /\gamma }\) is not integrable near every point in S and there exists a small constant \(\varepsilon _0>0\) such that
Assume that there exists an S-admissible pair, then we call
the Hörmander constant of S.
Since \(\psi \) equals to \(-\infty \) at S, (3.13) implies that
thus S is uniformly discrete. The proof of the Hörmander criterion also implies:
Theorem 3.10
Let S be a discrete set in \({\mathbb {C}}^n\). Assume that \(\iota (S)>1\). Then there exists a constant \(C>0\) such that for every sequence of complex numbers \(a=\{a_\lambda \}_{\lambda \in S}\) with
there exists \(F\in {\mathcal {F}}^2\) such that \(F(\lambda )=a_\lambda \) for all \(\lambda \in S\) and \(||F||^2 \le C\).
Definition 3.4
Let S be a discrete set in \({\mathbb {C}}^n\). We shall define the upper uniform density of S as
where \(n(z_0, r)\) denotes the number of points in
In case S is a lattice in \({\mathbb {C}}^n\), we know that \(D^{+}(S)^{-1}\) is equal to the Lebesgue measure of the torus \({\mathbb {C}}^n/S\). In the one-dimensional case, we also have the following general result.
Theorem 3.11
Let S be a uniformly discrete set in \({\mathbb {C}}\). Then \(D^{+}(S) \cdot \iota (S)=1\).
Proof
Since S is uniformly discrete, we have \(\iota (S) >0\), a change of variable argument gives
for every sufficiently small \(\varepsilon >0\). Thus Theorem 3.10 implies that \( (\iota (S)-\varepsilon )^{-1/2}S\) is \({\mathcal {F}}^2\) interpolating. Apply the main result in [OS98], we know that
Letting \(\varepsilon \) go to zero, we have
Now it suffices to show that
Assume that \(D^{+}(S)^{-1}=b\), then for every \(0<a<\gamma <b\),
for all sufficiently large r, which gives
for all \(z_0\in {\mathbb {C}}\). Apply the Berndtsson–Ortega construction in [BO95, page 113–114], the above inequality implies that \(\iota (S) \ge a\) for every \(a<b\). Hence \(\iota (S) \ge b=D^{+}(S)^{-1}\). The proof is complete. \(\square \)
Using results from [BO95, OS98], one may also generalize the above theorem to general weight function \(\phi \) with \(\phi _{z{\bar{z}}}\) bounded by two positive constants. For general higher-dimensional cases, by [Lin01, Theorem 2], we know that if S is \({\mathcal {F}}^2\) interpolating then \( D^{+}(S) \le 1\). However, in general, S may not be \({\mathcal {F}}^2\) interpolating even \(D^{+}(S)\) is small enough. Comparing with Theorem 3.10, this means that there exists S with very small upper uniform density whose Hörmander constant is also small.
3.8 Proof of Theorem 1.10
Notice that if \((({\mathbb {Z}}\oplus \frac{1}{2} {\mathbb {Z}})^2, e^{-\pi |t|^2})\) gives a frame in \(L^2({\mathbb {R}}^2)\) then \(({\mathbb {Z}}^2, e^{-\pi t^2})\) defines a frame in \(L^2({\mathbb {R}})\), which is not true by the Balian-Low type theorem. Now it suffices to prove (2) since (2) implies (1) by the remark after Corollary 1.9. Put
then we know that \((X,\omega )\) is of type (1, 4). Since the moduli space of polarized type \((d_1,d_2)\) (\(d_1, d_2\) are fixed positive integers) Abelian surfaces is equal to the Siegel upper half-space, to prove (2), by the Hörmander criterion, it suffices to show that the Seshadri constant of a generic polarized type (1, 4) Abelian surface is bigger than two. Since generically a polarized type (1, 4) Abelian surface has Picard number one, by Theorem 6.1 (b) in [Bau99], its Seshadri constant equals
where we use the fact that \(k=1, l=3\) is the primitive solution of the following Pell’s equation
The proof is complete.
4 Non transcendental examples
4.1 Gröchenig–Lyubarskii’s example
Let us look at the following lattice in \({\mathbb {R}}^2 \times {\mathbb {R}}^2 \) (see [GL20, page 3, (4)]):
Its symplectic dual is
Fix \(\Omega =i I_n\), where \(I_n\) denotes the identity matrix. With the notation in Proposition 1.4, we have
Let us estimate the Seshadri constant of
on the complex tori \(X:={\mathbb {C}}^2/\Gamma _{\Omega , \Lambda ^\circ }\). Notice that the Riemannian metric induced by \(\omega \) is precisely the euclidean metric \(|\cdot |\), hence
Thus the following ball
contains precisely one point in \(\Gamma _{\Omega , \Lambda ^\circ }\) and we can think of B as a Kähler ball in X, which gives (see Theorem 3.4) the following Seshadri constant inequality
However, from [GL20, page 3, (4)], we know that \((\Lambda , g_\Gamma )\) does not define a frame in \(L^2({\mathbb {R}}^2)\). Thus by Proposition 1.4, \(\Gamma _{\Omega , \Lambda ^\circ }\) is not a set of interpolation for \({\mathcal {F}}^2\). To summarize, we obtain:
Theorem 4.1
There exists a lattice in \({\mathbb {C}}^2\) whose Seshadri constant is bigger than one but it is not a set of interpolation for \({\mathcal {F}}^2\).
Remark
By the Hörmander criterion, we know every lattice in \({\mathbb {C}}^2\) with Seshadri constant bigger than two is a set of interpolation for \({\mathcal {F}}^2\). In the above example, one may further prove that
In fact the complex line \({\mathbb {C}}(0, 2/\sqrt{3})\) covers a subtorus, say
of X, thus [Tos18, Theorem 4.6] implies that
from which (4.2) follows.
4.2 Complex lattices
We call \(\Gamma \) a complex lattice if
One may verify the following:
Proposition 4.2
For a lattice \(\Gamma \) in \({\mathbb {C}}^n\), the followings are equivalent
-
(1)
\(\Gamma \) is a complex lattice;
-
(2)
\(\Gamma ={\mathbb {Z}}[i] \{\gamma _1,\ldots , \gamma _n\}\) for some \(\gamma _j\in {\mathbb {C}}^n\);
-
(3)
\(\Gamma =A {\mathbb {Z}}[i]^n\) for some \(A\in GL(n, {\mathbb {C}})\);
-
(4)
\(X:={\mathbb {C}}^n/\Gamma \) is biholomorphic to \({\mathbb {C}}^n/{\mathbb {Z}}[i]^n\).
Now we can prove a generalization of (4.2).
Theorem 4.3
Assume that \(\Gamma =A {\mathbb {Z}}[i]^n\) is a complex lattice. Then
where
Proof
Choose \(z=A^{-1}w\) as the new variable, one may assume that \(A=I_n\). Then
Denote by \(\psi _0(z)\) the Green function on \({\mathbb {C}}/{\mathbb {Z}}[i]\) satisfying
then we know that
satisfies \(2\pi \omega +i\partial {\overline{\partial }}\psi \ge 0\) and has order one log pole at \(\Gamma \). Thus we know that \( \epsilon _0(\omega )\ge e_{min}(A)\), together with Theorem 3.4, it gives the lower bound of the Seshadri constant that we need. To prove the upper bound, it suffices to choose a subtorus
then [Tos18, Theorem 4.6] gives
Take the infimum over all \(0\ne \gamma \in \Gamma \), the upper bound follows. \(\square \)
Remark
In case
we have
thus the above theorem gives
4.3 Seshadri sequence and Gröchenig’s result
In this section, we shall rephrase the main result of Gröchenig in [Gro11] in terms of the Seshadri constant. The main idea is to consider a sequence of extensions, more precisely, let
be an increasing sequence of complex Lie subgroups of X. We shall introduce the Seshadri constant \(\epsilon _j\), \(1\le j\le k\), for extension from \(X_{j-1}\) to \(X_{j}\). Let
be the covering map, where \(E_j\) is an \(n_j\) dimensional complex subspace of \({\mathbb {C}}^n\). Let
be the orthogonal decomposition with respect to the Euclidean metric \(\omega \). Then
define a lattice in \(F_j\). Put
(in general, \(X^\bot _{j-1}\) is not a subtorus of \(X_j\)). Denote by
the Seshadri constant at the origin of \(X^\bot _{j-1}\) with respect to \(\omega \).
Definition 4.1
We call (4.3) an admissible sequence of X if
X is said to be Seshadri admissible if it possesses an admissible sequence.
Theorem 4.4
Assume that X is Seshadri admissible, then \(\Gamma \) is a set of interpolation in \({\mathcal {F}}^2\).
Proof
Let \(\pi : {\mathbb {C}}^n \rightarrow X\) be the covering map. Put
It suffices to prove that each element f in \({\mathcal {F}}^2_{j-1}\) extends to an element F in \({\mathcal {F}}^2_j\) with \(||F||\le C_j ||f||\). Since \(\pi ^{-1}(X_j)\) is a disjoint union of translates of \(E_j\) and
the above extension problem reduces to the extension from \({\mathcal {F}}^2_{j-1}|_{\pi _j^{-1}(X_{j-1})}\) to \({\mathcal {F}}^2_j|_{E_j}\). Apply the Hörmander method, it suffices to construct an \(\omega \) plurisubharmonic function with order \((n_j-n_{j-1})\) log pole along \(\pi _j^{-1}(X_{j-1})\). Now the assumption \(\epsilon _j > n_j-n_{j-1}\) gives an \(\omega \) plurisubharmonic \(\psi _j\) with order \(n_j-n_{j-1}\) log pole at the origin of \(F_j\), the pull back of \(\psi _j\) along the natural projection
gives the function that we need. \(\square \)
Now we shall show how to use the above theorem to give a new proof of [Gro11, Theorem 9] on the Gabor frame property for \((\Lambda , g_\Omega )\). The setup for [Gro11, Theorem 9] is the following:
Based on Proposition 1.4, we shall prove a similar result with a weaker assumption, i.e. we shall only assume that \( \Gamma _{\Omega , \Lambda ^\circ }\) is a complex lattice. Let us write
where \(A\in GL(n, {\mathbb {C}})\). By the Iwasawa decomposition (see [Bum13, Proposition 26.1]), we have
where U is unitary and S is lower triangular with positive eigenvalues \(\lambda _j\) (U, S are uniquely determined by A, \(\lambda _j^{-1}\) is equal to \(\gamma _j\) in [Gro11, Theorem 9]). Since the Euclidean metric \(\omega \) is unitary invariant, one may assume that
Put
and
Then we have
and
Since \(n_j=j\), we have \(n_j-n_{j-1}=1\). Hence if
then X is Seshadri admissible. Thus Theorem 4.4 implies the following slight generalization of Gröchenig’s result (notice again that \(\gamma _j =\lambda _j^{-1}\)):
Theorem 4.5
(Theorem 9 in [Gro11]). Let \(\Gamma _{\Omega , \Lambda ^\circ }\) be a complex lattice. With the notation above, assume that \(\lambda _j>1\) for all \(1\le j\le n\). Then \(\Gamma _{\Omega , \Lambda ^\circ }\) is set of interpolation for \({\mathcal {F}}^2\) (and equivalently \((\Lambda , g_\Omega )\) defines a frame in \(L^2({\mathbb {R}}^n)\)).
5 Effective interpolation bounds
5.1 Interpolation bounds in terms of the Buser–Sarnak constant
We shall use Theorem 2.9 to prove the following:
Theorem 5.1
Fix a lattice \(\Gamma \) in \({\mathbb {C}}^n\). If
then every sequence of complex numbers \(a=\{a_\gamma \}\) with \(\sum _{\gamma \in \Gamma } |a_\gamma |^2 e^{-\pi |\gamma |^2}=1\) extends to a function in \({\mathcal {F}}^2\). Moreover,
where
Proof of the lower bound
Notice that (by an induction on n)
and
It suffices to show that
Notice that
The main observation is that the Taylor expansion
is now an orthogonal decomposition, i.e.
from which we know that
Now put \(t=\pi |\gamma |^2\), we have
which gives (5.2). \(\square \)
Proof of the upper bound
The main idea is to use Theorem 2.9. The definition of C implies that
is embedded ball in \(X:={\mathbb {C}}^n /\Gamma \). For
put (notice that \(\{\delta |z|^2 <1\} \subset B\))
Then we have
Denote by \(\psi \) the pull back to \({\mathbb {C}}^n\) of \( \pi \psi _\delta /\delta \). Apply Theorem 2.9 to \(\psi \), we get
Put
then \(n/C\le x<1\) and
Thus the upper bound follows from
The proof of Theorem B is now complete. \(\square \)
5.2 Interpolation bounds in terms of the Robin constant
In this subsection, we shall generalize Theorem 5.1 to the case that \(\epsilon _0(\omega )>n\). The main idea is to consider the following envelope with prescribed singularity
on the torus \(X={\mathbb {C}}^n/\Gamma \). Denote by \(\psi _a\) the pull back to \({\mathbb {C}}^n\) of \(\psi _a^\pi \).
Definition 5.1
We call \(\psi _a\) the a-envelope function on \({\mathbb {C}}^n\) associated to the lattice \(\Gamma \) and
the a-Robin constant of \(\Gamma \).
We have the following generalization of Theorems 5.1 and B.
Theorem 5.2
Assume that \(\epsilon :=\epsilon _0(\omega )>n\). Put
Then every sequence of complex numbers \(c=\{c_\gamma \}\) with \(\sum _{\gamma \in \Gamma } |c_\gamma |^2 e^{-\pi |\gamma |^2}=1\) extends to a function in \({\mathcal {F}}^2\). Moreover,
Proof
The proof of the lower bound is the same. For the upper bound, it suffices to apply Theorem 2.9 to \(\psi =\psi _a\). \(\square \)
Remark
In case \(n=1\), we know that
Moreover, \(\psi _\epsilon ^\pi \) is also well defined. In fact, we have the following
Proposition 5.3
\(\psi _\epsilon ^\pi \) is equal to the unique solution, say \(\psi \), of
on \(X={\mathbb {C}}/\Gamma \).
Proof
Since
from the Hodge theory, we know that up to a constant there exists a unique solution \(\psi \) such that
Thus if we assume further that \(\sup _X \psi =0\) then \(\psi \) is unique. Moreover, we know that \(\psi \) is smooth outside the origin and \(\psi -\epsilon \log |z|^2\) is smooth near the origin, thus
On the other hand, notice that \(\psi _\epsilon ^\pi -\psi \) is subharmonic, thus \(\psi _\epsilon ^\pi -\psi \) is equal to a constant, say A. Take \(z_0\) such that \(\psi (z_0)=0\), then
which gives
Hence \(\psi = \psi _\epsilon ^\pi \). \(\square \)
Remark
The above Proposition implies that \(\psi _\epsilon ^\pi \) is equal to the Arakelov Green function up to a non-zero constant (see [Fal84, page 393 and 417]).
5.3 Faltings’ identity for the Robin constant
Notice that
we know that \([\omega /\epsilon ]\) is the Chern class of the line bundle, \(L=[0]\), over X. Choose a metric h on L such that the Chern curvature satisfies
Denote by s the canonical section of [0], then we know that
which implies that
The pull back of the line bundle [0] to \({\mathbb {C}}\) is a trivial line bundle, thus one may identify the pull back to \({\mathbb {C}}\) of s with a holomorphic function, say \(f^s(z)\) on \({\mathbb {C}}\). In case \(\Gamma =\textrm{Span}_{{\mathbb {Z}}} \{1, \tau \}\), \(\textrm{Im}\, \tau =\epsilon \), (up to a constant) we have
(notice that \(\vartheta \left( \frac{1}{2}+\frac{\tau }{2}; \tau \right) =0\)) where
is known as the Jacobi theta function. The following formula for the Robin constant is based on the Faltings’ theta metric \(||\theta ||\) (see [Fal84, page 403, 413 and 416] or the function U below).
Proposition 5.4
Assume further that \(\Gamma =\textrm{Span}_{{\mathbb {Z}}} \{1, \tau \}\), \(\textrm{Im}\, \tau =\epsilon \). Denote by \(\psi _\epsilon \) the pull back to \({\mathbb {C}}\) of \(\psi _\epsilon ^\pi \), then
and the \(\epsilon \)-Robin constant of \(\Lambda \) defined by
satisfies
where \(\eta (\tau )\) is the Dedekind eta function defined by \( \eta (\tau ):=e^{\pi i\tau /12} \Pi _{n=1}^\infty (1-e^{2\pi i n \tau })\).
Proof
One may verify that U is \(\Gamma \)-invariant and
thus (5.5) follows. To prove (5.6), it suffices to show that
or equivalently
which follows from
(since both sides are cusp forms of degree 12 with the same leading term). \(\square \)
Remark
Notice that
and
Since \(\vartheta \) is an even function of z and depends only on \(e^{2\pi iz}\), we know that
is an even convex function of t. Moreover, since \(|\vartheta \left( z; \tau \right) e^{-\pi (\textrm{Im} \, z)^2/\epsilon }\big |\) is \(\Gamma \) invariant, we have
and
Notice that
with identity holds if \(\tau =i\epsilon \). By the Poisson summation formula, we have
take \(z=it\), we get
which gives
(for a higher-dimensional generalization of the above argument, see [Pa18, Lemma 8.2]). By (5.6), the above estimate gives the following lower bound for \(\rho _\epsilon /\epsilon \).
Theorem 5.5
The \(\epsilon \)-Robin constant of \(\Gamma =\textrm{Span}_{{\mathbb {Z}}} \{1, \tau \}\), \(\textrm{Im}\, \tau =\epsilon \) satisfies
with identity holds if \(\tau = i\epsilon \).
5.4 Proof of Theorem B
Proof of (1) and (2)
Notice that the disc of diameter \(\int _{0\ne \lambda \in \Gamma } |\lambda |\) is contained in a fundamental domain of \({\mathbb {C}}/\Gamma \). Hence \(|\Lambda |^{-1}=|\Gamma | \ge C\). The frame bounds estimate follows directly from Theorem 5.1 and Theorem 2.7. \(\square \)
Proof of (3)
For the lower bound, by Theorem 5.2 (let a go to \(\epsilon \)), it suffices to compute the \(\epsilon \)-Robin constant \(\rho \), where
is the Seshadri constant. Denote by \(\rho '\) the \(\textrm{Im}\, \tau /a\)-Robin constant of \({\mathbb {C}}/\langle 1, \tau /a\rangle \), then a change of variable argument gives
Thus
by Theorem 5.5. Apply Theorems 5.2 and 2.7, we get the lower bound. The upper bound follows directly from Theorems 2.7 and 5.1. \(\square \)
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Acknowledgements
We would like to thank A. Austad, E. Berge, U. Enstad, M. Faulhuber, L. Polterovich and E. Skrettingland for their feedback on earlier versions of the manuscript. Thanks are due to the referee for many helpful suggestions.
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Luef, F., Wang, X. Gaussian Gabor frames, Seshadri constants and generalized Buser–Sarnak invariants. Geom. Funct. Anal. 33, 778–823 (2023). https://doi.org/10.1007/s00039-023-00640-z
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DOI: https://doi.org/10.1007/s00039-023-00640-z
Keywords and phrases
- Gabor frame
- Bargmann transform
- Hörmander estimate
- Calabi Yau theorem
- Seshadri constant
- Ohsawa Takegoshi extension
- Berndtsson Lempert method