Hyperplane section

Abstract: We derive a formula connecting the derivatives of parallel section functions of an origin-symmetric star body in R n with the Fourier transform of powers of the radial function of the body. A parallel section function (or (ni 1)dimensional X-ray) gives the ((ni 1)-dimensional) volumes of all hyperplane sections of the body orthogonal to a given direction. This formula provides a new characterization of intersection bodies in R n and leads to a unifled analytic solution to the Busemann-Petty problem: Suppose that K and L are two origin-symmetric convex bodies inR n such that the ((ni 1)-dimensional) volume of each central hyperplane section of K is smaller than the volume of the corresponding section of L; is the (n-dimensional) volume of K smaller than the volume of L? In conjunction with earlier established connections between the Busemann-Petty problem, intersection bodies, and positive deflnite distributions, our formula shows that the answer to the problem depends on the behavior of the (ni 2)-nd derivative of the parallel section functions. The a‐rmative answer to the Busemann-Petty problem for n• 4 and the negative answer for n‚ 5 now follow from the fact that convexity controls the second derivatives, but does not control the derivatives of higher orders.

Abstract: The modified elliptic genus for an M5-brane wrapped on a four-cycle of a Calabi-Yau threefold encodes the degeneracies of an infinite set of BPS states in four dimensions. By holomorphy and modular invariance, it can be determined completely from the knowledge of a finite set of such BPS states. We show the feasibility of such a computation and determine the exact modified elliptic genus for an M5-brane wrapping a hyperplane section of the quintic threefold.

Abstract: This is an informal (and mostly conjectural) discussion of some aspects of Fukaya categories. We start by looking at exact symplectic manifolds which are obtained from a closed Calabi-Yau by removing a hyperplane section. We look at the possible geometric significance of Hochschild cohomology in this situation, and how one can try to get from the Fukaya category of the exact manifold to that of the closed Calabi-Yau. Also included is a brief discussion of the role of Lefschetz pencils, and a bit of general deformation theory. To appear in the Proceedings of the Beijing ICM.

Abstract: One of the themes in algebraic geometry is the study of the relation between the ``topology'' of a smooth projective variety and a (``general'') hyperplane section. Recent results of Nori produce cohomological evidence for a conjecture that a general hypersurface of sufficently large degree should have no ``interesting'' cycles. We compute precise bounds for these results and show by example that there are indeed interesting cycles for degrees that are not high enough. In a different direction Esnault, Nori and Srinivas have shown connectivity for intersections of small multidegree. We show analogous cycle-theoretic connectivity results.