Symplectic structure

In the case of k = C and Kx = 0, has a further nice structure. Suppose X has a holomorphic symplectic form u, i.e. to is an element in H°(X,Q x) which is nondegenerate at every point x G X. Usually, the moduli space inherits a nice property of the base space, and in the case of X, this is the case as the next theorem shows. 

Theorem 1.10 (Fujiki (n = 2) [21], Beauville (n 2) [6]). Suppose X has a holomorphic symplectic form uj. Then X also has a holomorphic symplectic form. 

Let S X be the subset of SnX consisting of Yhvi[xi] (xi distinct) with 2. Its inverse image by the Hilbert-Chow morphism 7r X — SnX (resp. the quotient map Xn — SnX) is denoted by X (resp. X ). Let us denote by A C Xn the big diagonal consisting of elements (aq. xn) with x = Xj for some i j. Then A fl X is smooth of codimension 2 in X . Moreover, generalizing (1.4), we have the following commutative diagram 

The holomorphic symplectic form uonX induces one on X , which we still denote by ui. Its pull-back rfuj is invariant under the action of n, hence defines a holomorphic 2-form cp on X with p cp = r ui. Then we have 

Therefore we have di ipn = 0, hence p is a holomorphic symplectic form on xj. Now, Xl l X is of codimension 2 in X, hence p extends to the whole X as a holomorphic form by the Hartogs theorem. We still have div pn = 0 in X, hence p is non-degenerate.  

On the other hand, the left hand side is equal to 

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A mapping is said to be symplectic or canonical if it preserves the differential form dp A dq which defines the symplectic structure in the phase space. Differential forms provide a geometric interpretation of symplectic-ness in terms of conservation of areas which follows from Liouville s theorem [14]. In one-degree-of-freedom example symplecticness is the preservation of oriented area. An example is the harmonic oscillator where the t-flow is just a rigid rotation and the area is preserved. The area-preserving character of the solution operator holds only for Hamiltonian systems. In more then one-degree-of-freedom examples the preservation of area is symplecticness rather than preservation of volume [5]. 

We focus on so-called symplectic methods [18] for the following reason It has been shown that the preservation of the symplectic structure of phase space under a numerical integration scheme implies a number of very desirable properties. Namely,... 

Equations (4) are called quasi-Hamiltonian because, even though they employ generalized velocities, they describe the motion in the space of canonical variables. Accordingly, numerical trajectories computed with appropriate integrators will conserve the symplectic structure. Eor example, an implicit leapfrog integrator can be expressed as... 

In Chapter 3, we shall explain the hyper-Kahler structure, which is closely related with the holomorphic symplectic structure. Actually, a hyper-Kahler manifold has a holomorphic symplectic structure as we shall explain in Chapter 3. The converse is also true if X is compact. 

The holomorphic symplectic structure on restricts to that on X. In particular,... 

The Hilbert scheme of points on the cotangent bundle of a Riemann surface has a natural holomorphic symplectic structure together with a natural C -action. In this case, the unstable manifold is very important since it becomes a Lagrangian submanifold. The same kind of situation appears in many cases, for example when one studies the moduli space of Higgs bundle or the quiver varieties [62], and it is worth explaining this point before studying the specific example. 

This gives an isomorphism between dense open sets of T EhJ and A/s, at least locally. Moreover, it is shown in [40] that this isomorphism preserves symplectic structures. 

S. Mukai, Symplectic structure of the moduli space of sheaves on an abelian or K3 surface. Invent. Math. 77 (1984), 101-116. 

The symplectic form on the moduli space M.r,c, c2 is described as follows. Let be a stable sheaf on X, then the tangent space of MriCuC2 at is given by TsM.r,cllC2 = Ext1 ( , ), and the symplectic structure is defined by... 

Proposition 1.12. Let X be a compact Kohler manifold which admits a holomorphic symplectic structure. Then X has a hyper-Kahler metric. 

The holomorphic symplectic structure on (C2) restricts to that on A. In particular, it implies that the canonical bundle is trivial, i.e. Kx = Ox- Hence the resolution is minimal. ... 

The proof is almost the same as that of the statement (2). Other components have symplectic structures, but are contained in the isotropic subvariety 7r 1(iV [0]). This is possible only when components are 0-dimensional. ... 

Moreover, it is shown in [40] that this isomorphism preserves symplectic structures. 

With the connection of PDEs, and especially soliton forms, to group symmetries established, one can conclude that if the Maxwell equation of motion that includes electric and magnetic conductivity is in soliton (SGE) form, the group symmetry of the Maxwell field is SU(2). Furthermore, because solitons define Hamiltonian flows, their energy conservation is due to their symplectic structure. 

Based on the paper Chiou et al the closed Newton-Cotes differential schemes can be presented as multilayer symplectic structures. 

We note here than in ref. 3 Chiou et al. have re-written Open Newton-Cotes differential schemes as multilayer symplectic structures based on (44). 

As in the previous paragraph we base our study on ref. 27 where Zhu et al. have proved the symplectic structure of the well-known second-order differential scheme (SOD),... 

One way of quantifying the sensitive dependence on initial conditions in a nonlinear dynamics model is via Lyapunov exponents. Usually this is done by introducing the variational equations which describe the time-dependent variation of perturbations of a solution of a dynamical system. Besides giving us a means to verify the chaotic nature of given system, the variational equations prove useful in describing the symplectic structure (taken up in the next chapter) which is essential to the design of effective numerical methods for molecular dynamics. 

In what follows, we discuss the development of constrained integration methods, with a focus on the symplectic structure. Although we do this in a different way here, our presentation is based on the article [229]. 

A class of methods that do provide the necessary features can be found in the work of Feng Kang [133], referred to as J-splitting by McLachlan and Quispel [261]. Let J be the skew-symmetric canonical symplectic structure matrix. The idea is to consider a splitting of J into a finite number K of skew-symmetric matrices 7 , i=, K. This induces a splitting of the Hamiltonian vector field into K vector... 

As an example of such a y-splitting, we may take AT = Ne with each 7, to be defined by the components of the canonical symplectic structure matfix having to do with the ith position and momentum pair, thus the equations of motion for the /th vector field become... 

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