Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0706.3696

The Mellin transform and spectral properties of toric varieties

Victor Guillemin·Zuoqin Wang
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this article we apply results of \cite{W} on the twisted Mellin transform to problems in toric geometry. In particular we use these results to describe the asymptotics of probability densities associated with the monomial eigenstates, $z^k$, $k \in \ZZ^d$, in Bargmann space and prove an "upstairs" version of the spectral density theorem of \cite{BGU}. We also obtain for the $z^k$'s, "upstairs" versions of the results of \cite{STZ} on distribution laws for eigenstates on toric varieties.

Explore related subjects

math.SG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Victor Guillemin, Zuoqin Wang. 2007-06-27. The Mellin transform and spectral properties of toric varieties. https://arxiv.org/abs/0706.3696

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The contrast between real and $p$-adic integrability

A fundamental question in $p$-adic symplectic geometry and $p$-adic Hamiltonian mechanics is to what extent we can learn new information about integrable systems with coefficients in the rational numbers $\mathbb{Q}$, if we view them as $p$-adic systems instead of real ones. We combine geometric analytic estimates with Galois theory in order to show that at a local level, these real and $p$-adic viewpoints give completely different information. Then we focus on understanding this local behavior for the $p$-adic case by proving effective methods to compute local normal forms. Our results are motivated by a program outlined by Pelayo-Voevodsky-Warren a decade ago, to develop a $p$-adic theory of integrable systems and symplectic geometry.

math.SG↗
arXiv2026

A Smooth Hamiltonian Diffeomorphism with Two Fixed Points on $S^2\times S^2$

We give a construction of a Hamiltonian diffeomorphism of the equal-area symplectic product $S^2\times S^2$ with two fixed points. The argument starts with a four-fixed-point map, passes to rescaled cotangent coordinates near the antidiagonal, and reduces the local fixed-point equations to two planar saddle problems. A relative modification merges each pair of saddles, and mixed generating functions are used to transfer the modification to the original map. We include detailed explanations of the normal form along a level arc, the Moser equation, uniform existence of mixed generating functions, the order of parameter choices, and the global graph argument. This construction gives a counterexample to the degenerate version of Arnold's conjecture proposed by Arnold\cite{Arnold86} when the manifold is standard $S^2\times S^2$. This work is done with ChatGPT. The author gave the idea to study the limit behavior of $Φ_ρ$. GPT suggested a key idea of using second generating function to construct this Hamiltonian. Also, most of the computation is done by GPT.

math.SG↗
arXiv2026

Cluster structures from Legendrian double twist knots: a comparative approach

We undertake a systematic comparison of cluster algebras associated to Legendrian knots of the same smooth knot type. We describe a cluster structure on the m-graded augmentation variety of Legendrian representatives of a family of double twist knots that includes all orientably Lagrangian fillable twist knots. For different Legendrian representatives of the same smooth knot type, we show that these cluster structures are equivalent. Along the way, we characterize the representatives that admit Maslov-m exact Lagrangian fillings, construct a Catalan number of such fillings for each representative, and show that any Maslov-0 fillable representative is Legendrian isotopic to the (-1)-framed closure of a positive braid. Finally, we give two Legendrian representatives of a cabled twist knot that yield equivalent cluster structures of infinite type.

math.SG↗