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We show that any orientable Seifert 3-manifold is diffeomorphic to a connected component of the set of real points of a uniruled real algebraic variety, and prove a conjecture of Janos Kollar.
Consider a hyperbolic surface X of infinite area. The Liouville map L assigns to any quasiconformal deformation of X a measure on the space G(X) of geodesics of the universal covering X of X. We show that the Liouville map L is a homeomorphism from the Teichmuller space T(X) onto its image, and that the image L(T(X)) is closed and unbounded. The set of asymptotic rays to L(T(X)) consists of all bounded...
We study an integration theory in circle equivariant cohomology in order to prove a theorem relating the cohomology ring of a hyperkahler quotient to the cohomology ring of the quotient by a maximal abelian subgroup, analogous to a theorem of Martin for symplectic quotients. We discuss applications of this theorem to quiver varieties, and compute as an example the ordinary and equivariant cohomology...
We describe explicitly the algebras of degree zero operations in connective and periodic p-local complex K-theory. Operations are written uniquely in terms of certain infinite linear combinations of Adams operations, and we give formulas for the product and coproduct structure maps. It is shown that these rings of operations are not Noetherian. Versions of the results are provided for the Adams summand...
The A-polynomial of a manifold whose boundary consists of a single torus is generalised to an eigenvalue variety of a manifold whose boundary consists of a finite number of tori, and the set of strongly detected boundary curves is determined by Bergman's logarithmic limit set, which describes the exponential behaviour of the eigenvalue variety at infinity. This enables one to read off the detected...
For any 3-manifold M 3 and any nonnegative integer g, we give here examples of metrics on M each of which has a sequence of embedded minimal surfaces of genus g and without Morse index bounds. On any spherical space form we construct such a metric with positive scalar curvature. More generally, we construct such a metric with Scal>0 (and such surfaces) on any 3-manifold which carries a...
Let Y be a finite full subcomplex of a simplicial complex X. For any subdivision X' of X keeping Y invariant, and for ε small enough relatively to X', we define the ε-barycentric derived neighbourhood V ε (X',Y) of Y in X'. Theorem: for small enough ε, and for any simplex K of Y, the transverse stars of K in V ε (X,Y) and V ε (X',Y) have the same support. As a consequence,...
We develop the idea of self-indexing and the technology of gradient-like vector fields in the setting of Morse theory on a complex algebraic stratification. Our main result is the local existence, near a Morse critical point, of gradient-like vector fields satisfying certain “stratified dimension bounds up to fuzz” for the ascending and descending sets. As a global consequence of this, we derive the...
We study a homotopy invariant of phantom maps called the Gray index. In particular, it is conjectured that the Gray index of an essential phantom map between finite-type spaces is always finite. We obtain some partial results on this conjecture, using a tower-theoretic interpretation of the Gray index.
We prove that if a holomorphic one-form Ω in a neighborhood of a closed euclidian ball B 2n C n , in the n-dimensional complex affine space, defines a distribution transverse to the boundary sphere S 2n-1 = B 2n , then n is even and Ω admits a sole singularity q B 2n . Moreover, this singularity is simple.
For any closed oriented surface Σ g of genus g>=3, we prove the existence of foliated Σ g -bundles over surfaces such that the signatures of the total spaces are non-zero. We can arrange that the total holonomy of the horizontal foliations preserve a prescribed symplectic form ω on the fiber. We relate the cohomology class represented by the transverse symplectic form to a crossed...
New relations among the genus-zero Gromov–Witten invariants of a complex projective manifold X are exhibited. When the cohomology of X is generated by divisor classes and classes “with vanishing one-point invariants,” the relations determine many-point invariants in terms of one-point invariants.
We develop foundations of a general approach for calculating p-primary v1-periodic homotopy groups of spaces using their p-adic KO-cohomologies and K-cohomologies with particular attention to the case p=2. As a main application, we derive a method for calculating v1-periodic homotopy groups of simply connected compact Lie groups using their complex, real, and quaternionic representation theories....
We make an estimation of the value of the Gromov norm of the Cartesian product of two surfaces. Our method uses a connection between these norms and the minimal size of triangulations of the products of two polygons. This allows us to prove that the Gromov norm of this product is between 32 and 52 when both factors have genus 2. The case of arbitrary genera is easy to deduce from this one.
Fulton and MacPherson (Ann. Math. 139 (1994) 183) found a Sullivan dg-algebra model for the space of n-configurations of a smooth complex projective variety X. Kříž (Ann. Math. 139 (1994) 227) gave a simpler model, En(H), depending only on the cohomology ring, H≔H*X.We construct an even simpler and smaller model, Jn(H). We then define another new dg-algebra, En(H∘), and use Jn(H) to prove that En(H∘)...
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