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Symplectic vector space
About: Symplectic vector space is a research topic. Over the lifetime, 2048 publications have been published within this topic receiving 53456 citations.
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TL;DR: In this paper, a new elementary geometric proof exploiting the positive curvature of complex projective space is presented for a basic lemma in the theory of lagrangian pairs in hermitian vector space.
11 citations
TL;DR: In this article, the authors consider manifolds satisfying a weakly Lefschetz property, i.e., the s-Lefschetz property, and construct compact manifolds which are s-lefscherez but not (s + 1)-Lefschez.
Abstract: For a symplectic manifold, the harmonic cohomology of symplectic divisors (introduced by Donaldson, 1996) and of the more general symplectic zero loci (introduced by Auroux, 1997) are compared with that of its ambient space. We also study symplectic manifolds satisfying a weakly Lefschetz property, that is, the s-Lefschetz property. In particular, we consider the symplectic blow-ups CP m of the complex projective space CP m along weakly Lefschetz symplectic submanifolds M ⊂ CP m . As an application we construct, for each even integer s > 2, compact symplectic manifolds which are s-Lefschetz but not (s + 1)-Lefschetz.
11 citations
TL;DR: In this paper, the structure of a Lie-algebra L over a field F is determined by the bilinear formon L that is derived from a matrix representation of L with finite degree d(Δ) by forming the trace of the matrix products.
Abstract: To what extent is the structure of a Lie-algebra L over a field F determined by the bilinear formon L that is derived from a matrix representationof L with finite degree d(Δ) by forming the trace of the matrix productsSuch a bilinear form is a function with two arguments in L, values in F and the properties:
11 citations
TL;DR: The symplectic structure, which will be destroyed in the numerical process due to roundoff errors when working with a symplectic (butterfly) matrix, will be forced by working just with the parameters.
Abstract: The SR algorithm is a structure-preserving algorithm for computing the spectrum of symplectic matrices. Any symplectic matrix can be reduced to symplectic butterfly form. A symplectic matrix B in butterfly form is uniquely determined by 4n - 1 parameters. Using these 4n - 1 parameters, we show how one step of the symplectic SR algorithm for B can be carried out in O(n) arithmetic operations compared to O(n 3 ) arithmetic operations when working on the actual symplectic matrix. Moreover, the symplectic structure, which will be destroyed in the numerical process due to roundoff errors when working with a symplectic (butterfly) matrix, will be forced by working just with the parameters.
11 citations
12 Dec 2018
TL;DR: In this article, the singular moduli spaces of Higgs bundles of degree 0 and rank n on a compact Riemann surface of genus g are proved to be symplectic singularities, in the sense of Beauville [Bea00] and admit a projective projective resolution if and only if g = 1$ or n = 2,2.
Abstract: In this paper, we study the algebraic symplectic geometry of the singular moduli spaces of Higgs bundles of degree $0$ and rank $n$ on a compact Riemann surface $X$ of genus $g$. In particular, we prove that such moduli spaces are symplectic singularities, in the sense of Beauville [Bea00], and admit a projective symplectic resolution if and only if $g=1$ or $(g, n)=(2,2)$. These results are an application of a recent paper by Bellamy and Schedler [BS16] via the so-called Isosingularity Theorem.
10 citations