Linear symplectic geometry

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Symplectic Geometry ◽  
Vector Bundles ◽  
Maslov Index ◽  
Vector Spaces ◽  
Complex Structures ◽  
Homotopy Equivalence ◽  
Linear Complex ◽  
Symplectic Forms ◽  
Lagrangian Subspaces ◽  
First Chern Class

The second chapter introduces the basic concepts of symplectic topology in the linear algebra setting, such as symplectic vector spaces, the linear symplectic group, Lagrangian subspaces, and the Maslov index. In the section on linear complex structures particular emphasis is placed on the homotopy equivalence between the space of symplectic forms and the space of linear complex structures. The chapter includes sections on symplectic vector bundles and the first Chern class.

Constructing symplectic manifolds

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Final Section ◽  
Symplectic Manifolds ◽  
Homotopy Equivalence ◽  
Codimension Two ◽  
Open Manifolds ◽  
Connected Sums ◽  
Symplectic Forms ◽  
Ambient Manifold ◽  
Blowing Up ◽  
Four Manifolds

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

scholarly journals ON THE STABILITY OF SYZYGY BUNDLES

2011 ◽  
Vol 22 (04) ◽  
pp. 515-534 ◽  
Author(s):  
IUSTIN COANDĂ
Keyword(s):  
Projective Space ◽  
Complete Intersection ◽  
Vector Bundles ◽  
Elementary Proof ◽  
Base Point ◽  
Vector Spaces ◽  
Similar Argument ◽  
Vanishing Theorem ◽  
The Stability ◽  
Koszul Cohomology

We are concerned with the problem of the stability of the syzygy bundles associated to base-point-free vector spaces of forms of the same degree d on the projective space of dimension n. We deduce directly, from M. Green's vanishing theorem for Koszul cohomology, that any such bundle is stable if its rank is sufficiently high. With a similar argument, we prove the semistability of a certain syzygy bundle on a general complete intersection of hypersurfaces of degree d in the projective space. This answers a question of H. Flenner [Comment. Math. Helv.59 (1984) 635–650]. We then give an elementary proof of H. Brenner's criterion of stability for monomial syzygy bundles, avoiding the use of Klyachko's results on toric vector bundles. We finally prove the existence of stable syzygy bundles defined by monomials of the same degree d, of any possible rank, for n at least 3. This extends the similar result proved, for n = 2, by L. Costa, P. Macias Marques and R. M. Miro-Roig [J. Pure Appl. Algebra214 (2010) 1241–1262]. The extension to the case n at least 3 has been also, independently, obtained by P. Macias Marques in his thesis [arXiv:0909.4646/math.AG (2009)].

scholarly journals Two-Categorical Bundles and their Classifying Spaces

2012 ◽  
Vol 10 (2) ◽  
pp. 299-369 ◽  
Author(s):  
Nils A. Baas ◽  
Marcel Bökstedt ◽  
Tore August Kro
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Vector Spaces ◽  
Bundle Theory ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Principal Bundles

AbstractFor a 2-category 2C we associate a notion of a principal 2C-bundle. For the 2-category of 2-vector spaces, in the sense of M.M. Kapranov and V.A. Voevodsky, this gives the 2-vector bundles of N.A. Baas, B.I. Dundas and J. Rognes. Our main result says that the geometric nerve of a good 2-category is a classifying space for the associated principal 2-bundles. In the process of proving this we develop powerful machinery which may be useful in further studies of 2-categorical topology. As a corollary we get a new proof of the classification of principal bundles. Another 2-category of 2-vector spaces has been proposed by J.C. Baez and A.S. Crans. A calculation using our main theorem shows that in this case the theory of principal 2-bundles splits, up to concordance, as two copies of ordinary vector bundle theory. When 2C is a cobordism type 2-category we get a new notion of cobordism-bundles which turns out to be classified by the Madsen–Weiss spaces.

STRING EQUATIONS OF MOTION FROM VANISHING CURVATURE

1991 ◽  
Vol 06 (08) ◽  
pp. 1319-1333 ◽  
Author(s):  
MARK J. BOWICK ◽  
KONG-QING YANG
Keyword(s):  
Vector Bundles ◽  
Adiabatic Approximation ◽  
Equations Of Motion ◽  
Bosonic String ◽  
Complex Structures ◽  
Closed Bosonic String

The equations of motion for the massless modes of the closed bosonic string are obtained in the adiabatic approximation from the requirement of the vanishing of the curvature of appropriate vector bundles over the space of complex structures Diff S1/S1. This vanishing is required for physical states to be independent of string parametrization.

SUPER KRICHEVER FUNCTOR

1991 ◽  
Vol 02 (06) ◽  
pp. 741-760 ◽  
Author(s):  
MOTOHICO MULASE ◽  
JEFFREY M. RABIN
Keyword(s):  
Line Bundle ◽  
Vector Bundles ◽  
Vector Spaces ◽  
Contravariant Functor ◽  
Infinite Dimensional ◽  
Arbitrary Rank ◽  
Geometric Data

A supersymmetric generalization of the Krichever map is proposed. This map assigns injectively a point of an infinite dimensional super Grassmannian to a set of geometric data consisting of an arbitrary algebraic super manifold of dimension 1|1 defined over a field of any characteristic and a line bundle on it. The naturality of this map comes from the fact that it is obtained as the restriction of a contravariant functor to certain special objects. This functor gives an antiequivalence between the category of infinite dimensional super vector spaces satisfying the Fredholm condition together with their stabilizers, and the category of algebraic super curves and certain sheaves on them including all even vector bundles of arbitrary rank.

On complex structures in 8-dimensional vector bundles

1998 ◽  
Vol 95 (1) ◽  
pp. 323-330 ◽  
Author(s):  
Martin Čadek ◽  
Jiří Vanžura
Keyword(s):  
Vector Bundles ◽  
Complex Structures ◽  
Dimensional Vector

Integrable Systems Associated to a Hopf Surface

2003 ◽  
Vol 55 (3) ◽  
pp. 609-635 ◽  
Author(s):  
Ruxandra Moraru
Keyword(s):  
Elliptic Curve ◽  
Moduli Spaces ◽  
Symplectic Geometry ◽  
Vector Bundles ◽  
Integrable Hamiltonian System ◽  
Complex Surface ◽  
Point Of View ◽  
Infinite Cyclic Group ◽  
Completely Integrable Hamiltonian System ◽  
Hopf Surface

AbstractA Hopf surface is the quotient of the complex surface by an infinite cyclic group of dilations of . In this paper, we study the moduli spaces of stable -bundles on a Hopf surface , from the point of view of symplectic geometry. An important point is that the surface is an elliptic fibration, which implies that a vector bundle on can be considered as a family of vector bundles over an elliptic curve. We define a map that associates to every bundle on a divisor, called the graph of the bundle, which encodes the isomorphism class of the bundle over each elliptic curve. We then prove that the map G is an algebraically completely integrable Hamiltonian system, with respect to a given Poisson structure on . We also give an explicit description of the fibres of the integrable system. This example is interesting for several reasons; in particular, since the Hopf surface is not Kähler, it is an elliptic fibration that does not admit a section.

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