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We prove that every strong symplectic filling of a planar contact manifold admits a symplectic Lefschetz fibration over the disk, and every strong filling of similarly admits a Lefschetz fibration over the annulus. It follows that strongly fillable planar contact structures are also Stein fillable, and all strong fillings of are equivalent up to symplectic deformation and blowup. These constructions result from a compactness theorem for punctured -holomorphic curves that foliate a convex symplectic manifold. We use it also to show that the compactly supported symplectomorphism group on is contractible, and to define an obstruction to strong fillability that yields a non-gauge-theoretic proof of Gay's recent nonfillability result [G] for contact manifolds with positive Giroux torsion
We study quotients of the Weyl algebra by left ideals whose generators consist of an arbitrary -graded binomial ideal in along with Euler operators defined by the grading and a parameter . We determine the parameters for which these -modules (i) are holonomic (equivalently, regular holonomic, when is standard-graded), (ii) decompose as direct sums indexed by the primary components of , and (iii) have holonomic rank greater than the rank for generic . In each of these three cases, the parameters in question are precisely those outside of a certain explicitly described affine subspace arrangement in . In the special case of Horn hypergeometric -modules, when is a lattice-basis ideal, we furthermore compute the generic holonomic rank combinatorially and write down a basis of solutions in terms of associated -hypergeometric functions. This study relies fundamentally on the explicit lattice-point description of the primary components of an arbitrary binomial ideal in characteristic zero, which we derive in our companion article [DMM]
We explore some geometric and analytic consequences of a curvature condition introduced by Ma, Trudinger, and Wang in relation to the smoothness of optimal transport in curved geometry. We discuss a conjecture according to which a strict version of the Ma-Trudinger-Wang condition is sufficient to prove regularity of optimal transport on a Riemannian manifold. We prove this conjecture under a somewhat restrictive additional assumption of nonfocality; at the same time, we establish the striking geometric property that the tangent cut locus is the boundary of a convex set. Partial extensions are presented to the case when there is no pure focalization on the tangent cut locus
We investigate -regularity of minimizers to in two dimensions for certain classes of nonsmooth convex functionals . In particular, our results apply to the surface tensions that appear in recent works on random surfaces and random tilings of Kenyon, Okounkov, and others
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