Symplectic Manifolds

KAn-i for A an abelian surface, using the Weil conjectures. Here KAn-i is a symplectic manifold, defined as the kernel of the map —> A given by composing... 

This is a special case of the moment map which is defined for an action on a general symplectic manifold. Consider a function G —s- R>o given by... 

Kahler structures are easy to construct and flexible. For example, any complex submanifold of a Kahler manifold is again Kahler, and a Kahler metric is locally given by a Kahler potential, i.e. uj = / ddu for a strictly pseudo convex function u. However, hyper-Kahler structures are neither easy to construct nor flexible (even locally). A hypercomplex submanifold of a hyper-Kahler manifold must be totally geodesic, and there is no good notion of hyper-Kahler potential. The following quotient construction, which was introduced by Hitchin et al.[39] as an analogue of Marsden-Weinstein quotients for symplectic manifolds, is one of the most powerful tool for constructing new hyper-Kahler manifolds. 

For a later purpose (Chapter 7), we shall explain the perfectness of the Morse function given by the moment map of a torus action on a general symplectic manifold. However, when the fixed points of a torus action are all isolated, such as the case of the... 

Let X, cj) be a compact symplectic manifold and T a compact torus. We suppose that there exists a T-action on X preserving uj with the corresponding moment map /r X t. As explained in Chapter 3, /j, X t is called a moment map if it satisfies... 

Notice that our argument also gives the proof of the perfectness of the Morse function in the case of a noncompact symplectic manifold if the appropriate conditions on / are satisfied. For example, the condition that / ((—oo, c]) is compact for all c G IR is sufficient, and this is the case for as will be shown later. 

The unstable manifold W/ becomes important in Chapter 7 when we study a holomorphic symplectic manifold. 

In Chapter 5, we have studied Morse theory on a symplectic manifold X given by an action of a compact torus T. As noted there, when X is a Kahler manifold, the gradient flow is given by the associated holomorphic action of the complexification T of T. Hence, the stable and the unstable manifolds can be expressed purely in terms of the group action. 

Moduli of vector bundles on KS surfaces, and symplectic manifolds, Sugaku Exposition 1... 

Q j simultaneously at all the points of a certain (maybe small) open neighbourhood of any point on a symplectic manifold. This fact constitutes the content of the known Darhoux theorem which has no analogue (in the indicated sense) in Riemannian geometry. 

Definition 1.2.7 Local coordinates Pi,..MPn)7i) j n oii a symplectic manifold are called symplectic if in these coordinates the form o) is written in the canonical form. 

Hamiltonian and Locally Hamiltonian Vector Fields and the Poisson Bracket We will denote by V[M) an infinite-dimensional linear space of all smooth vector fields on a manifold M. If is a symplectic manifold, then in V M) there exists a... 

Definition 1.2.9 A smooth vector field u on a symplectic manifold M is called locally Hamiltonian if for any point x G Af, there exists such an open neighbourhood U[x) of the point x and such a smooth function Fu defined on this neighbourhood that V — sgrad Fez, that is, the field v is Hamiltonian in the neighbourhood U with the local Hamiltonian Fu ... 

It is readily seen that this field is locally Hamiltonian on a symplectic manifold R 0 supplied with a 2-form dx A dy. Indeed, as the local Hamiltonian Fu it suffices to take the function of the polar angle

[Pg.20]

However, this field is not a globally Hamiltonian one because the polar angle (x, y) is a multivalued function on a plane without a point, and the above-mentioned local Hamiltonians cannot be "sewed into one smooth single-valued function. This results from the fact that the symplectic manifold R 0 is not simply-connected. 

Theorem 1.2.2. A smooth vector held v on a symplectic manifold M with a symplectic structure locally Hamiltonian if and only if it preserves its symplectic structure, that is, if the derivative of the form u) in the direction of the held v is sero or, in other words, if g oj = cj at all L... 

PROOF By virtue of the Darboux theorem (see Theorem 1.2.1) it suffices to prove this assertion only for the case where M is the simplest symplectic manifold with the canonical structure (jj = Y dpi A dqi. Let the form (jj be preserved by a one-parameter group gt, that is, = 0, where 7(f) are integral trajectories... 

In the case of a two-dimensional Riemannian manifold Af with a Riemannian metric gij and with the form of the Riemannian area w = y/det(gij)dx A dy as a symplectic structure (see above), the condition that the group (3 of diffeomorphisms gt preserve the form cj is equivalent to the condition that the domain areas be preserved on the surface when these domains are shifted by the diffeomorphisms gt. Thus, the shifts along integral trajectories of a Hamiltonian field on a two-dimensional symplectic manifold preserve the domain areas. 

In the case of a two-dimensional symplectic manifold, the condition of the locally Hamiltonian character of the field admits another vivid geometrical interpretation. Let gij be a Riemannian metric on and let u) = y/det gij)dx A dy be the form of the Riemannian area. By virtue of the Darboux theorem, one can always choose local coordinates p and q such that the form oj be written in the canonical form dp A dq. Here p and q are certain functions of x and y (and vice versa). Let t be a locally Hamiltonian field t = (i (a , y),Q(x,y)), where P and Q are coordinates of the field in the local system of coordinates p and q. Let us interpret the field v as a velocity field of the flow of liquid of constant density (equal to unity) on the surface M. Let us investigate the variation of the mass of the liquid bounded by an infinitesimal rectangle on the surface when it is shifted along integral trajectories of the field v. It is clear that the mass of this liquid is equal to the area of the rectangle. Therefore, the mass of the liquid contained in a bounded (sufficiently small) domain on is equal to the area of the domain. 

Thus, in the case of a two-dimensional symplectic manifold, the locally Hamiltonian vector fields are exactly the flows of incompressible liquid, that is, the vector fields with zero divergence. In other words, the condition for the local Hamiltonian properties of the field v in the two-dimensional case is equivalent to the condition div(v) = 0. 

We have given an example of a locally Hamiltonian vector field on R 0 which is not a Hamiltonian field. The symplectic manifold was not compact here. But such kind of examples may be constructed on compact closed (i.e., without a boundary) manifolds as well. Take, for instance, an ordinary flat torus supplied with a Euclidean metric gij = Si, Then the 2-form of the area on this torus will be written in Cartesian coordinates x, y as follows dx A dy, i.e., this 2-form is canonical. On the torus, consider a vector field v = (1,0) given by a uniform liquid motion along its parallels (Fig. 7). 

FVom the proof of Theorem 1.2.2, we see that the local Hamiltonian properties of the field v are equivalent to the closedness of the differential 1-form a = 5 —Yidpi + Xidgi on the manifold Af. For a field to be globally Hamiltonian, it is sufficient that this form be exact. For instance, this will always be the case on (the Poincari /emma). If, however, a symplectic manifold is not simply-connected, then closed but not exact 1-forms may exist on it. This will be so if a group of one-dimensional cohomologies (Af, R) is nonzero. In both our examples, we deal with a nonzero group Ar (Af, R), namely... 

Proposition 1.2.2. Let a symplectic manifold M have a zero Brst group of real cohomologies JI M,R) (for instance, this will always be the case with a simply-connected manifold). Then any locally Hamiltonian vector held on the manifold will be at the same time globally Hamiltonian. 

Now let us proceed to describe an important operation of calculating a Poisson bracket of two functions on a symplectic manifold. 

Proposition 1.2.4. Such a linear inBnite dimensional space C M) of smooth functions f on a symplectic manifold M, which is endowed with the operation of calculating the Poisson bracket oatiiralfy transforms into... 

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