Differential isomorphism and equivalence of algebraic varieties

Berest, Yuri; Wilson, George

doi:10.1017/cbo9780511526398.007

Cited by 14 publications

(40 citation statements)

References 45 publications

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“…By [10,Prop. 5.2], the map W → (X, L) is a bijection from Gr ad to the set of isomorphism classes of pairs of a framed affine curve and line bundle on it.…”

Section: Wilson's Adelic Grassmannianmentioning

confidence: 99%

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Reflective prolate-spheroidal operators and the KP/KdV equations

Casper

Grünbaum

Yakimov

et al. 2019

Proc. Natl. Acad. Sci. U.S.A.

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Beginning with the work of Landau, Pollak and Slepian in the 1960s on timeband limiting, commuting pairs of integral and differential operators have played a key role in signal processing, random matrix theory, and integrable systems. Previously, such pairs were constructed by ad hoc methods, which essentially worked because a commuting operator of low order could be found by a direct calculation.We describe a general approach to these problems that proves that every point W of Wilson's infinite dimensional adelic Grassmannian Gr ad gives rise to an integral operator TW , acting on L 2 (Γ) for a contour Γ ⊂ C, which reflects a differential operator with rational coefficients R(z, ∂z) in the sense that R(−z, −∂z) • TW = TW • R(w, ∂w) on a dense subset of L 2 (Γ). By using analytic methods and methods from integrable systems, we show that the reflected differential operator can be constructed from the Fourier algebra of the associated bispectral function ψW (x, z). The exact size of this algebra with respect to a bifiltration is in turn determined using algebro-geometric methods. Intrinsic properties of four involutions of the adelic Grassmannian naturally lead us to consider the reflecting property above in place of plain commutativity. Furthermore, we prove that the time-band limited operators of the generalized Laplace transforms with kernels given by the rank one bispectral functions ψW (x, −z) always reflect a differential operator. A 90 • rotation argument is used to prove that the time-band limited operators of the generalized Fourier transforms with kernels ψW (x, iz) admit a commuting differential operator. These methods produce vast collections of integral operators with prolate-spheroidal properties, associated to the wave functions of all rational solutions of the KP hierarchy vanishing at infinity, introduced by Krichever in the late 1970s.that is, the eigenfunctions of the integral operatorThis is a numerically extremely unstable problem. Landau, Pollak and Slepian [43,32] bypassed this issue based on the property thatcommutes with EE * . The differential operator R(z, ∂ z ) is the "radial part" of the Laplacian in prolate-spheroidal coordinates, and (the joint) eigenfunctions of it and the integral operator EE * are known as the prolate-spheroidal functions, which can be computed numerically in very stable ways. The commuting pair can be traced back to Bateman [6,

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“…By [10,Prop. 5.2], the map W → (X, L) is a bijection from Gr ad to the set of isomorphism classes of pairs of a framed affine curve and line bundle on it.…”

Section: Wilson's Adelic Grassmannianmentioning

confidence: 99%

“…Note in particular that W n = z −n A Wn so that L ∼ = O Xn for X n = Spec(A Wn ). The curve X n is one of the special extremal rational, unicursal curves considered in [10,29]. In particular X n is the simplest unicursal curve having differential genus n, as described below.…”

Section: Wilson's Adelic Grassmannianmentioning

confidence: 99%

Reflective prolate-spheroidal operators and the KP/KdV equations

Casper

Grünbaum

Yakimov

et al. 2019

Proc. Natl. Acad. Sci. U.S.A.

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“…The reader may also wish to see [BW3] for a recent review of a problem closely related to the subject of the present paper, the problem of classifying affine varieties up to differential isomorphism, where cuspidal quotients of affine curves play an important role.…”

Section: A Quasi-inverse Functor Is Given By the Derham Functormentioning

confidence: 99%

Cusps and 𝒟-modules

Ben‐Zvi

Nevins

2003

J. Amer. Math. Soc.

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We study interactions between the categories of D \mathcal {D} -modules on smooth and singular varieties. For a large class of singular varieties Y Y , we use an extension of the Grothendieck-Sato formula to show that D Y \mathcal {D}_Y -modules are equivalent to stratifications on Y Y , and as a consequence are unaffected by a class of homeomorphisms, the cuspidal quotients. In particular, when Y Y has a smooth bijective normalization X X , we obtain a Morita equivalence of D Y \mathcal {D}_Y and D X \mathcal {D}_X and a Kashiwara theorem for D Y \mathcal {D}_Y , thereby solving conjectures of Hart-Smith and Berest-Etingof-Ginzburg (generalizing results for complex curves and surfaces and rational Cherednik algebras). We also use this equivalence to enlarge the category of induced D \mathcal {D} -modules on a smooth variety X X by collecting induced D X \mathcal {D}_X -modules on varying cuspidal quotients. The resulting cusp-induced D X \mathcal {D}_X -modules possess both the good properties of induced D \mathcal {D} -modules (in particular, a Riemann-Hilbert description) and, when X X is a curve, a simple characterization as the generically torsion-free D X \mathcal {D}_X -modules.

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“…Let us note that the problem of studying isomorphisms of algebras of differential operators is also actively studied in different contexts, mostly for singular affine varieties; see [2] for a recent review.…”

Section: Introductionmentioning

confidence: 99%

Differential operators on toric varieties and Fourier transform

Felder

Rossi

2008

Sel. math., New ser.

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We show that Fourier transforms on the Weyl algebras have a geometric counterpart in the framework of toric varieties, namely they induce isomorphisms between twisted rings of differential operators on regular toric varieties, whose fans are related to each other by reflections of one-dimensional cones. The simplest class of examples is provided by the toric varieties related by such reflections to projective spaces. It includes the blow-up at a point of the affine space and resolution of singularities of varieties appearing in the study of the minimal orbit of sln+1. (2000). Primary 13S32; Secondary 14M25. Mathematics Subject Classification

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Differential isomorphism and equivalence of algebraic varieties

Cited by 14 publications

References 45 publications

Reflective prolate-spheroidal operators and the KP/KdV equations

Reflective prolate-spheroidal operators and the KP/KdV equations

Cusps and 𝒟-modules

Differential operators on toric varieties and Fourier transform

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