Differential Geometry

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 booklet is an extended model of lectures given at a summer season university on symplectic geometry in Nordfjordeid, Norway, in June 2001. The unifying function of the publication is an emphasis on Calabi-Yau manifolds. the 1st half discusses holonomy teams and calibrated submanifolds, targeting certain Lagrangian submanifolds and the SYZ conjecture. the second one reports Calabi-Yau manifolds and replicate symmetry, utilizing algebraic geometry. the ultimate half describes compact hyperkahler manifolds, that have a geometrical constitution very heavily concerning Calabi-Yau manifolds.

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Example text

N = 1. At this step we involve the following observation made by D. Cordero-Erausquin, M. Fradelizi and B. Maurey in their study and proof of the so-called B-conjecture, cf. [CE-F-M], Theorem 1. It is stated below as a lemma, where D(λ) is treated as a linear map. 2 (D. Cordero-Erausquin, M. Fradelizi and B. Maurey [CE-F-M]). For any symmetric convex body K in Rn , the function (t1 , . . , tn ) −→ γ(D(et1 , . . , etn )(K)) is log-concave on Rn . 1, introduce the function on Rn−1 v(t1 , . .

1), and the (Shannon) entropy power of X is N (X) = e2h(X)/n . We limit ourselves to random vectors X with h(X) < +∞; in this case, N (X) is a non-negative real number. Building on work of [32] and resolving a conjecture they made, [33] recently showed the following result. 1. Let X1 , . . , XM be independent Rn -valued random vectors, such that the entropy of each exists and is finite. Let β be a fractional partition using a collection G of subsets of [M ]. Then N (X1 + . . + XM ) ≥ βs N s∈G Xj .

Some applications of duality relations. , 1469, Springer, Berlin, 1991. [M-P] Milman, V. , and Pajor, A. Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. , 1376, Springer, Berlin, 1989. [P] Pisier, G. The volume of convex bodies and Banach space geometry. Cambridge Tracts in Mathematics, 94. Cambridge University Press, Cambridge, 1989, xvi+250 pp. C. The difference body of a convex body. Arch. Math. (Basel) 8 (1957), 220–233. C. Convex bodies associated with a given convex body.

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