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**Extra resources for Algebraic Geometry Santa Cruz 1995, Part 2: Summer Research Institute on Algebraic Geometry, July 9-29, 1995, University of California, Santa Cruz**

**Sample text**

It is however possible that the break-down phenomenon is unstable and thus impossible to detect by numerical calculations. The regularity of solutions for small initial conditions has been verified in the case of supercritical Wave Maps 16 and in the case of the Einstein vacuum equations. In these Lectures I will present our attempts to attack the Conjecture, or rather the regularity part of it, from the local point of view the "well posedness" of the initial value problem. P. is well posed in the energy norm 17 then global regularity is automatically implied by Energy conservation.

Since ind D = 2. we see that the Kuranishi model of a neighborhood of (8,0) in Me is given by the quotient by 51 of the zero set of the 5 1 -equivariant map cp : ker D = c2+r ---+ coker DEB H! (X; R) = C r EB R2. After taking the quotient by 51 (which acts trivially on the R summands), we may think of cp as a section of the bundle EEl rEB lR,2 r yo T1 ccpl+r \ {c} 24 Ronald Fintushel where c cpI+r denotes the cone, with cone point c, and ~2 is a trivial real 2plane bundle. The local model of Me is the zero set of cpo Again, let Me denote the complement in Me of its intersection with the open cone neighborhood of (8,0).

Let M~ denote the complement in Me of the open cone neighhorhood of (8,0). This is a compact (2k - I)-manifold with boundary Cpk-1. Since a characteristic line bundle has at most one reducible solution to the Seiberg-Witten 22 Ronald Fintushel equations, the restriction of the basepoint map is a fibration M~ ---t M~ and this restricts over the boundary as the canonical 8 1-fibration 8 2k - 1 ---t Cpk-1. e. the restriction of the class (3 of Lecture 2 to 8M~ is just h. So we get the contradiction in case k claimed.