Riemann surface

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.)... 

One of the advantages of the hyper-Kahler structure is that one can identify two apparently different complex manifolds with one hyper-Kahler manifold. Namely, a hyper-Kahler manifold X, g, I, J, K) gives two complex manifolds (X,/) and (X, J), which are not isomorphic in general. For example, on a compact Riemann surface, the moduli space of Higgs bundles and the moduli space of flat PGLr(C)-bundles come from one hyper-Kahler manifold, namely moduli space of 2D-self-duality equation (see [36] for detail.)... 

Hilbert scheme on the cotangent bundle of a Riemann surface... 

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. 

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... 

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

Let E be a compact Riemann surface. The moduli space As of Higgs bundle over E is dehned by... 

M.F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. London A 308 (1982), 524-615. 

N.J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59-126. 

The Hilbert scheme T is analogous in many ways with the moduli space of Higgs bundle over a Riemann surface introduced by Hitchin [36]. 

Vol. 1288 Yu. L. Rodin, Generalized Analytic Functions on Riemann Surfaces. V, 128 pages, 1987. 

A Riemann surface is a 2-dimensional compact differentiable surface, together with an infinitesimal element of length (see textbooks on differential and Riemannian geometry, for example, [Nak90]). The curvature K(x) at a point x is the coefficient a in the expansion ... 

Given a map M, its circle-packing representation (see [Moh97]) is a set of disks on a Riemann surface E of constant curvature, one disk D(v, rv) for each vertex v of M, such that the following conditions are fulfilled ... 

A map M is called reduced (see [Moh97, Section 3]) if its universal cover is 3-connected and is a cell-complex. It is shown in [Moh97, Corollary 5.4] that reduced maps admit unique primal-dual circle packing representations on a Riemann surface of the same genus moreover, a polynomial time algorithm allows one to find the coordinates of those points relatively easily. This means that the combinatorics of the map determines the structure of the Riemann surface. 

Jos06] J. Jost, Compact Riemann Surfaces An Introduction to Contemporary Mathematics, 3rd edition. Universitext, Springer Verlag, 2006. 

D. Sokolovski, S.K. Sen, V. Aquilanti, S. Cavalli, D. De Fazio, Interacting resonances in the F+H2 reaction revisited Complex terms, Riemann surfaces, and angular distributions, J. Chem. Phys. 126 (2007) 084305. 

The main idea is as follows. Let us consider the plane in which our chain is placed as a complex one, z = x + iy. (z = z(x, >)) and let us find the conformal transformation, z = z( ), of the plane z with the obstacle to the Riemann surface, = + b], which does not contain an obstacle (such a transformation means the transfer to the covering space). Due to the conformal invariance of Brownian motion1, in the covering space a random process will be obtained corresponding to the initial one on the plane z but without any topological constraints. 

Fig. 2. Conformal transformation of z-plane with one point removed to the covering Riemann surface, f, without any peculiar points...

CONFORMAL MAPPING ON RIEMANN SURFACES, Harvey Cohn. Lucid, insightful book presents ideal coverage of subject. 334 exercises make book perfect for self-study. 55 figures. 352pp. 5X x 8X. 64025-6 Pa. 8.95... 

A minimal surface can be represented (locally) by a set of three integrals. They represent the inverse of a mapping from the minimal surface to a Riemann surface. The mapping is a composite one first the minimal surface is mapped onto the unit sphere (the Gauss map), then the sphere itself is mapped onto the complex plane by stereographic projection. Under these operations, the minimal surface is transformed into a multi-sheeted covering of the complex plane. Any point on the minimal surface (except flat points), characterised by cartesian coordinates (x,y,z) is described by the complex number (o, which... 

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