Download Calabi-Yau Manifolds and Related Geometries by Mark Gross, Daniel Huybrechts, Dominic Joyce, Geir PDF

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).

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