Hilbert scheme on the cotangent bundle of a Riemann surface

Hilbert scheme on the cotangent bundle of a Riemann surface 

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. 

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. 

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 

This is the unstable manifold W for the appropriate choice of G t in Chapter 5. 

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HILBERT SCHEME ON THE COTANGENT BUNDLE OF A RIEMANN SURFACE... 

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... 

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