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We prove three fundamental properties of counting holomorphic cylinders in log Calabi–Yau surfaces: positivity, integrality and the gluing formula. Positivity and integrality assert that the numbers of cylinders, defined via virtual techniques, are in fact nonnegative integers. The gluing formula roughly says that cylinders can be glued together to form longer cylinders, and the number of longer cylinders equals the product of the numbers of shorter cylinders. Our approach uses Berkovich geometry, tropical geometry, deformation theory and the ideas in the proof of associativity relations of Gromov–Witten invariants by Maxim Kontsevich. These three properties provide evidence for a conjectural relation between counting cylinders and the broken lines of Gross, Hacking and Keel.
Tautological classes, or generalised Miller–Morita–Mumford classes, are basic characteristic classes of smooth fibre bundles, and have recently been used to describe the rational cohomology of classifying spaces of diffeomorphism groups for several types of manifolds. We show that rationally tautological classes depend only on the underlying topological block bundle, and use this to prove the vanishing of tautological classes for many bundles with fibre an aspherical manifold.
Let be a family of unimodal maps with topological entropies , and be their natural extensions, where . Subject to some regularity conditions, which are satisfied by tent maps and quadratic maps, we give a complete description of the prime ends of the Barge–Martin embeddings of into the sphere. We also construct a family of sphere homeomorphisms with the property that each is a factor of , by a semiconjugacy for which all fibers except one contain at most three points, and for which the exceptional fiber carries no topological entropy; that is, unimodal natural extensions are virtually sphere homeomorphisms. In the case where is the tent family, we show that is a generalized pseudo-Anosov map for the dense set of parameters for which is postcritically finite, so that is the completion of the unimodal generalized pseudo-Anosov family introduced by de Carvalho and Hall (Geom. Topol. 8 (2004) 1127–1188).
An elliptic orbifold is the quotient of an elliptic curve by a finite group. In 2001, Eskin and Okounkov proved that generating functions for the number of branched covers of an elliptic curve with specified ramification are quasimodular forms for . In 2006, they generalized this theorem to branched covers of the quotient of an elliptic curve by , proving quasimodularity for . We generalize their work to the quotient of an elliptic curve by for , , , proving quasimodularity for , and extend their work in the case .
It follows that certain generating functions of hexagon, square and triangle tilings of compact surfaces are quasimodular forms. These tilings enumerate lattice points in moduli spaces of flat surfaces. We analyze the asymptotics as the number of tiles goes to infinity, providing an algorithm to compute the Masur–Veech volumes of strata of cubic, quartic, and sextic differentials. We conclude a generalization of the Kontsevich–Zorich conjecture: these volumes are polynomial in .
We define an infinite family of linearly independent, integer-valued smooth concordance homomorphisms. Our homomorphisms are explicitly computable and rely on local equivalence classes of knot Floer complexes over the ring . We compare our invariants to other concordance homomorphisms coming from knot Floer homology, and discuss applications to topologically slice knots, concordance genus and concordance unknotting number.
We present an analytic construction of complete noncompact –dimensional Ricci-flat manifolds with holonomy . The construction relies on the study of the adiabatic limit of metrics with holonomy on principal Seifert circle bundles over asymptotically conical –orbifolds. The metrics we produce have an asymptotic geometry, so-called ALC geometry, that generalises to higher dimensions the geometry of –dimensional ALF hyperkähler metrics.
We apply our construction to asymptotically conical –metrics arising from self-dual Einstein –orbifolds with positive scalar curvature. As illustrative examples of the power of our construction, we produce complete noncompact –manifolds with arbitrarily large second Betti number and infinitely many distinct families of ALC –metrics on the same smooth –manifold.
We show that a smooth complex projective threefold admits a holomorphic one-form without zeros if and only if the underlying real –manifold is a –fibre bundle over the circle, and we give a complete classification of all threefolds with that property. Our results prove a conjecture of Kotschick in dimension three.
We establish topological regularity and stability of –dimensional spaces (up to a small singular set), also called noncollapsed in the literature. We also introduce the notion of a boundary of such spaces and study its properties, including its behavior under Gromov–Hausdorff convergence.
We introduce Riemannian metrics of positive scalar curvature on manifolds with Baas–Sullivan singularities, prove a corresponding homology invariance principle and discuss admissible products.
Using this theory we construct positive scalar curvature metrics on closed smooth manifolds of dimension at least five which have odd order abelian fundamental groups, are nonspin and atoral. This solves the Gromov–Lawson–Rosenberg conjecture for a new class of manifolds with finite fundamental groups.
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