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V. Guillemin

Publications and source records attributed to V. Guillemin.

9 recordsLinked to original sources
arXiv2011

Symplectic Origami

An origami manifold is a manifold equipped with a closed 2-form which is symplectic except on a hypersurface where it is like the pullback of a symplectic form by a folding map and its kernel fibrates with oriented circle fibers over a compact base. We can move back and forth between origami and symplectic manifolds using cutting (unfolding) and radial blow-up (folding), modulo compatibility conditions. We prove an origami convexity theorem for hamiltonian torus actions, classify toric origami manifolds by polyhedral objects resembling paper origami and discuss examples. We also prove a cobordism result and compute the cohomology of a special class of origami manifolds.

math.SG↗
arXiv2007

The spectral density function of a toric variety

For a Kahler manifold (X, ω) with a holomorphic line bundle L and metric h such that the Chern form of L is ω, the spectral measures are the measures μ_N = \sum |s_{N,i}|^2 ν, where \{s_{N,i}\}_i is an L^2-orthonormal basis for H^0(X, L^{\otimes N}), and νis Liouville measure. We study the asymptotics in N of μ_N for (X, L) a Hamiltonian toric manifold, and give a precise expansion in terms of powers 1/N^j and data on the moment polytope Δof the Hamiltonian torus K acting on X. In addition, for an infinitesimal character k of K and the unique unit eigensection s_{Nk} for the character Nk of the torus action on H^0(X, L^N), we give a similar expansion for the measures μ_{Nk} = |s_{Nk}|^2 ν. A final remark shows that the eigenbasis \{s_{k}, k \in Δ\cap \mathbb{Z}^{\dim K} \} is a Bohr-Sommerfeld basis in the sense of Tyurin. Some of the present results are related to work of Shiffman, Tate and Zelditch. The present paper uses no microlocal analysis, but rather an Euler-Maclaurin formula for Delzant polytopes.

math.SP↗
arXiv2006

"Bottom of the well" semi-classical trace invariants

Let $\hat H$ be an h-admissible pseudodifferential operator whose principal symbol, $H$, has a unique non-degenerate global minimum. We give a simple proof that the semi-classical asymptotics of the eigenvalues of $\hat H$ corresponding to the "bottom of the well" determine the Birkhoff normal form of $H$ at the minimum. We treat both the resonant and the non-resonant cases.

math.SP↗
arXiv2005

Some inverse spectral results for semi-classical Schrödinger operators

We consider a semi-classical Schrödinger operator, -h^2Δ+ V(x). Assuming that the potential admits a unique global minimum and that the eigenvalues of the Hessian are linearly independent over the rationals, we show that the low-lying eigenvalues of the operator determine the Taylor series of the potential at the minimum.

math.SP↗
arXiv2005

Toric symplectic singular spaces I: isolated singularities

We generalize a theorem of Delzant classifying compact connected symplectic manifolds with completely integrable torus actions to certain singular symplectic spaces. The assumption on singularities is that if they are not finite quotient then they are isolated.

math.SG↗
arXiv2003

Potential functions and actions of tori on Kaehler manifolds

Let M be a Kaehler manifold with a free, holomorphic and Hamiltonian action of the standard n-torus T. We give a simple, explicit and canonical formula for the Kaehler potential on the Kaehler reduction of M. As a consequence we can derive improvements of several classical results known for more general Hamiltonian reductions. Among these are a forms-level proof of the Duistermaat-Heckman theorem; an elementary proof of Atiyah's proof of the convexity of the moment image of a complexified T-orbit; another formula due to Biquard-Gauduchon for the Kaehler potential; and a formula in terms of moment data for the Kaehler metric on a toric variety, due originally to the second author.

math.SG↗
arXiv2002

Kaehler cuts

A symplectic cut of a manifold M with a Hamiltonian circle action is a symplectic quotient of M x C. If M is Kaehler then, since C is Kaehler, the cut space is Kaehler as well. The symplectic structure on the cut is well understood. In this paper we describe the complex structure (and hence the metric) on the cut. We then generalize the construction to the case where M has a torus action and C is replaced by a toric Kaehler manifold.

math.DG↗