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

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

ISBN-10: 3540440593

ISBN-13: 9783540440598

This booklet is an multiplied model of lectures given at a summer time university 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 specific Lagrangian submanifolds and the SYZ conjecture. the second one stories Calabi-Yau manifolds and replicate symmetry, utilizing algebraic geometry. the ultimate half describes compact hyperkahler manifolds, that have a geometrical constitution very heavily on the topic of Calabi-Yau manifolds.

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Additional info for Calabi-Yau Manifolds and Related Geometries

Example text

Hence for each natural number n > 0 each connected component of the open set U , = =, W, is also contained in a contractible open subspace of X. Trivially { U,ln E N} is a countable open covering of X , and it remains only to show that it is locally finite. Since { 51j E J } is locally finite, for each x E X there is an n > 0 and an open neighborhood of x which meets at most n of the sets 5;consequently that neighborhood of x does not intersect U , for any m > n, which completes the proof. a.

For any natural number n > 0 let R" be the standard real n-dimensional vector space, in the metric topology arising from the euclidean norm R" II II R, Il(rl.. . , x,,)Il = . . + x,'. The closed n-disk D" and (n - I)-sphere S"- c D" consist of those points x E R" satisfying llxll 1 and llxll = 1, respectively, in the relative topology of R"; in particular D' is a closed interval and So consists of two points. An (open) n-cell is any space homeomorphic to the interior D" - S"- I of D". Let iD:il be any family of homeomorphic copies of D" and [ S ; - l ; lthe corresponding family of homeomorphic copies of S"- c D", with disjoint D: and uolS:- I , respectively.

L ,. , i max rjl, , j,,,la(Jl,. . ,j,)I. , . , i,, I i, < . . < i,) is a mutually disjoint family, F l , , , i , < . . < i,) is a mutually disjoint family, and the two families collectively cover G"(R"'); it remains to show that each Ui,. . i,,,, each y , ,. , ,,,, and each nonvoid intersection U i l . , . i,,, n . j,, is of 0th type. Each point of U i l , , . i,,, can be regarded as a onedimensional subspace of E R ~ spanned ' by an element of the form (xi, y , ) A . . A (xi,,, y,), where each of y , , .

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Calabi-Yau Manifolds and Related Geometries by Mark Gross, Daniel Huybrechts, Dominic Joyce, Geir Ellingsrud, Loren Olson, Kristian Ranestad, Stein A. Stromme

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