We prove that for @math there exists a compact subset @math of the closed ball in @math of radius @math , such that @math has Hausdorff dimension @math and does not symplectically embed into the standard open symplectic cylinder. The second main result is a lower bound on the @math -th regular coisotropic capacity, which is sharp up to a factor of 3. For an open subset of a geometrically bounded, aspherical symplectic manifold, this capacity is a lower bound on its displacement energy. The proofs of the results involve a certain Lagrangian submanifold of linear space, which was considered by M. Audin and L. Polterovich.
As a consequence of Theorem and [Theorem 4] SZSmall , the value of the capacity @math on the ball @math lies between @math and @math . In the case @math this value can be exactly calculated, if we modify the definition of @math by restricting to Lagrangian submanifolds. Namely, the so obtained capacity @math satisfies [A_ ^+(B^4)= . ] To see this, we denote by @math the standard torus in @math . For every @math the rescaled torus @math is a Lagrangian submanifold of @math , with minimal area @math . It follows that @math . To see the opposite inequality, note that every orientable closed connected Lagrangian submanifold @math is diffeomorphic to the torus @math , since its Euler characteristic vanishes. For such an @math , K. Cieliebak and K. Mohnke proved @cite_3 that @math . The statement follows.
@cite_3
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Let @math be a smooth scheme over an algebraically closed field @math of characteristic zero and @math a regular function, and write @math Crit @math , as a closed subscheme of @math . The motivic vanishing cycle @math is an element of the @math -equivariant motivic Grothendieck ring @math defined by Denef and Loeser math.AG 0006050 and Looijenga math.AG 0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435. We prove three main results: (a) @math depends only on the third-order thickenings @math of @math . (b) If @math is another smooth scheme, @math is regular, @math Crit @math , and @math is an embedding with @math and @math an isomorphism, then @math equals @math "twisted" by a motive associated to a principal @math -bundle defined using @math , where now we work in a quotient ring @math of @math . (c) If @math is an "oriented algebraic d-critical locus" in the sense of Joyce arXiv:1304.4508, there is a natural motive @math , such that if @math is locally modelled on Crit @math , then @math is locally modelled on @math . Using results from arXiv:1305.6302, these imply the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with "orientation data", as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory arXiv:0811.2435, and on intersections of oriented Lagrangians in an algebraic symplectic manifold. This paper is an analogue for motives of results on perverse sheaves of vanishing cycles proved in arXiv:1211.3259. We extend this paper to Artin stacks in arXiv:1312.0090. Let @math be an induced subgraph of the torus @math . We show that when @math is even and @math divides some power of @math , then for sufficiently large @math the torus @math has a perfect vertex-packing with induced copies of @math . On the other hand, disproving a conjecture of Gruslys, we show that when @math is odd and not a prime power, then there exists @math such that @math divides some power of @math , but there is no @math such that @math has a perfect vertex-packing with copies of @math . We also disprove a conjecture of Gruslys, Leader and Tan by exhibiting a subgraph @math of the @math -dimensional hypercube @math , such that there is no @math for which @math has a perfect edge-packing with copies of @math . The full' edge isoperimetric inequality for the discrete cube (due to Harper, Bernstein, Lindsay and Hart) specifies the minimum size of the edge boundary @math of a set @math , as a function of @math . A weaker (but more widely-used) lower bound is @math , where equality holds iff @math is a subcube. In 2011, the first author obtained a sharp stability' version of the latter result, proving that if @math , then there exists a subcube @math such that @math . The weak' version of the edge isoperimetric inequality has the following well-known generalization for the @math -biased' measure @math on the discrete cube: if @math , or if @math and @math is monotone increasing, then @math . In this paper, we prove a sharp stability version of the latter result, which generalizes the aforementioned result of the first author. Namely, we prove that if @math , then there exists a subcube @math such that @math , where @math . This result is a central component in recent work of the authors proving sharp stability versions of a number of Erd o s-Ko-Rado type theorems in extremal combinatorics, including the seminal complete intersection theorem' of Ahlswede and Khachatrian. In addition, we prove a biased-measure analogue of the full' edge isoperimetric inequality, for monotone increasing sets, and we observe that such an analogue does not hold for arbitrary sets, hence answering a question of Kalai. We use this result to give a new proof of the full' edge isoperimetric inequality, one relying on the Kruskal-Katona theorem. Let @math be a geometrically bounded symplectic manifold, @math be a closed, regular coisotropic submanifold, and @math be a Hamiltonian diffeomorphism. The main result of this article is that the number of leafwise fixed points of @math is bounded below by the sum of the Betti numbers of @math , provided that the Hofer distance between @math and the identity is small enough and the pair @math is non-degenerate. As an application, I prove a presymplectic non-embedding result. A version of the Arnold-Givental conjecture for coisotropic submanifolds is also discussed.