Symplectic form

As we mentioned at the beginning, Xl" inherits structures from X. First of all, it is a scheme. It is projective if X is projective. These follows from Grothendieck s construction of Hilbert schemes. A nontrivial example is a result by Beauville [6] Xl" has a holomorphic symplectic form when X has one. When X is projective, X has a holomorphic symplectic form only when X is a X3 surface or an abelian surface by the classification theory. We also have interesting noncompact examples X = or X = r S where E is a Riemann surface. These examples are particularly nice because of the existence of a C -action, which naturally induces an action on Xl" . (See Chapter 7.)... 

In the case of k = C and Kx = 0, Xl" has a further nice structure. Suppose X has a holomorphic symplectic form u, i.e. u is an element in H X, which is nondegenerate at every point x [Pg.8]

The holomorphic symplectic form a on X induces one on X, which we still denote by cj. Its pull-back rj uj is invariant under the action of 0,, hence defines a holomorphic 2-form

[Pg.8]

In the last part, we use the fact that X is a K3 surface or an abelian surface. For the proof of the non-degeneracy and the closedness, we refer to [56]. In Chapter 3, we shall show that the framed moduli space of torsion free sheaves on has a holomorphic symplectic form. 

Proof. Since X has a holomorphic symplectic form u), we can identify TX and T X by using uj. This implies cfyX) = 0. By the Calabi conjecture proved by Yau [81], there exists a Ricci-flat Kahler metric on X. Using the Ricci-fiatness, the Bochner-Weitzenbock formula gives... 

Exercise 1.13. Compare Beauville s symplectic form and Mukai s symplectic form on the open set pY), where X is the open stratum of S X and tt is the... 

This means uJc is of type (2,0). It is clear that dujc = 0 and ujc is non-degenerate. Hence Uc is a holomorphic symplectic form. 

Now we fix a Riemannian metric g which is invariant under the T-action. The symplectic form UJ together with the Riemannian metric g gives an almost complex structure I defined by uj v,x) = g Iv,w). With this almost complex structure, we regard the tangent space Tj-X as a complex vector space. Let X = be the decomposition into the... 

Let be a Kahler manifold with a holomorphic symplectic form cjc- Suppose there exists a C -action on X with the property that tplujc = tuJc for t G C, where we denote the C -action on X hy il)t X X. Let C, be a connected component of the fixed point set of the C -action, and consider the subset defined by... 

Now we shall study the Hilbert scheme of points on the cotangent bundle of a Riemann surface. Let E be a Riemann surface and T E its cotangent bundle. There exists a natural holomorphic symplectic form uc on T E. The multiplication by a complex number on each fiber gives a natural C -action on T E, and with respect to this action we have "(p uJc = tuc for t E C, where we denote the action of t by T E T E. As explained in Theorem 1.10, the Hilbert scheme inherits a holomorphic symplectic form and... 

Remark 7.8. In order to compute the Poincare polynomial, we do not actually need the holomorphic symplectic form on T E. The fact that it is locally isomorphic to T C is enough. Hence the above argument holds and shows Gottsche s formula also for the case of the total space of a holomorphic line bundle over E, not necessarily T E. The only difference is that the unstable manifold W becomes Lagrangian in the case of T E. 

Now we shall explain how much T E[" 1 looks like A/s- In both cases, there exists a holomorphic symplectic form ujc and a C -action which satisfies 4>luJc = for t E C. The holomorphic symplectic form on As is given as follows. For (, ) G A/s, the tangent space of As at E, ) is given by... 

By the Serre duality, there exists a natural non-degenerate pairing between ii (E, End(i/)) and ii°(E, End( /) 0 ids), and this defines a holomorphic symplectic form on A/s- The... 

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. 

Theorem 1.11 (Mukai [56]). LetX be a K3 surface or an abelian surface and M.rfiuC2 be the moduli space of stable sheaves on X with fixed rank r and Chem classes Ci,C2, then Mr,Cl,c2 has a holomorphic symplectic form. 

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... 

Theorem 1.16. The subvariety 7r 1(n[0]) is isotropic with respect to the holomorphic symplectic form on (C2) / 2- In particular, dim7r-1(n[0]) < n — 1. Moreover, there exists at least one n — 1-dimensional component. 

It would be valuable to have a general method for deriving volume conserving methods. It turns out that volume conservation is, itself, most readily obtained as a consequence of a more fundamental property of Hamiltonian flows, the conservation of the symplectic form. 

Describing the Hamiltonian structure for a constrained system is a little complicated to do formally. The simplifying concept that we exploit is that the symplectic 2-form in the ambient space can be projected to the co-tangent bundle to define an associated symplectic form on the manifold. 

Thus we see that the projected version of the symplectic form in the ambient space is conserved by the differential equation system (4.7)-(4.9). This is what is meant by saying that the constrained system is analogous to a Hamiltonian system. We may also think of its flow map as being a symplectic map of the co-tangent bundle. 

In particular, the dimension of each orbit of coadjoint representation is even. It is easily seen that the symplectic form oj is invariant under the coadjoint action that is, Ad a = a , h G 0. It also turns out that du = 0. 

An irreducible even-dimensional complex torus (see Example 1 below) may serve as an example of a manifold integrated additively, but not integrated in the weak sense. However, any symplectic form on a torus in some linear coordinates is written in the canonical form as... 

Theorem 3.4.2 (MARKUSHEVICH). Let S be a K3-type surface, and let ws be a symplectic form. The following assertions hold true ... 

The symplectic form

form prjws d-pr ws on 5 X 5. Consider the mapping... 

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