By Mark Gross, Daniel Huybrechts, Dominic Joyce, Geir Ellingsrud, Loren Olson, Kristian Ranestad, Stein A. Stromme

This e-book is an multiplied model of lectures given at a summer season tuition on symplectic geometry in Nordfjordeid, Norway, in June 2001. The unifying characteristic of the ebook is an emphasis on Calabi-Yau manifolds. the 1st half discusses holonomy teams and calibrated submanifolds, targeting detailed Lagrangian submanifolds and the SYZ conjecture. the second one reviews Calabi-Yau manifolds and reflect symmetry, utilizing algebraic geometry. the ultimate half describes compact hyperkahler manifolds, that have a geometrical constitution very heavily with regards to Calabi-Yau manifolds.

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T. t. e. c : [0, TJ x (-e, e) -' N is a smooth map (e > 0) with c(t, 0) = c(t). We assume that all curves are geodesics. 8) := Then J(t) _ ac(t,s)Is =0 is a Jacobi field along c(t) = ca(t). 2) Chapter 2 Spaces of nonpositive curvature 34 Conversely, every Jacobi field along c(t) can be obtained by such a variation of c(t) through geodesics. t. 1). The second part is a consequence of the existence and smooth dependence on initial data for geodesics with prescribed initial value and initial direction.

1 (Mostow): Under the above assumptions (I'. 1I8)/SO(2). and if r is irreducible, and if there exists an isomorphism p:r - r' then the locally symrnetrw spaces r\G/K and r"\ /h' are isometric. e. conjugate subgroups of G. 1, we have normalized the metric of the symmetric space G/K. Of course. the symmetric structure is not lost if this metric is multiplied by a constant factor. In that. more general situation. Mostow's theorem says that two isomorphic lattices in symmetric spaces, satisfying the assumptions stated, are isometric tip to a scaling factor.

A lattice therefore is a free O(v) module of rank 2. Two lattices L1. L2 are called equivalent if there exists x E K' with L1 = A2We consider the set of equivalence classes of lattices as the set of vertices of a graph, with two vertices joined by an edge if and only if the corresponding classes have representatives L1, L2 with the following property: There exists an O(v) basis (e1,e2) for L1 for which (e1,lre2) is an O(v) basis for L2. e. a connected, nonempty graph without circuits (a circuit in it graph is a subgraph isomorphic to the graph with set of vertices Z/nZ and edges joining i and i + 1 for all i E Z/nZ, for some n E N).