Volume 2013, Issue 681

Abstract. We formulate the Asymptotic Expansion Conjecture for the Witten–Reshetikhin–Turaev quantum invariants of closed oriented 3-manifolds. For of finite order mapping tori, we study these quantum invariants via the geometric gauge theory approach to the corresponding quantum representations and prove, using a version of the Lefschetz–Riemann–Roch theorem due to Baum, Fulton, MacPherson & Quart, that the quantum invariants can be expressed as a sum over the components of the moduli space of flat connections on the mapping torus. Moreover, we show that the term corresponding to a component is a polynomial in the level  k , weighted by a complex phase, which is k times the Chern–Simons invariant corresponding to the component. We express the coefficients of these polynomials in terms of cohomological pairings on the fixed point set of the moduli space of flat connections on the surface. We explicitly describe the fixed point set in terms of moduli spaces of the quotient orbifold Riemann surface and for the smooth components we express the aforementioned coefficients in terms of the known generators of the cohomology ring. We provide an explicit formula in terms of the Seifert invariants of the mapping torus for the contributions from each of the smooth components. We further establish that the Asymptotic Expansion Conjecture for manifolds and we verify the Growth Rate Conjecture for these finite order mapping tori.

Abstract. We study some aspects of the relationship between 𝔸 1 -homotopy theory and birational geometry. We study the so-called 𝔸 1 -singular chain complex and zeroth 𝔸 1 -homology sheaf of smooth algebraic varieties over a field k . We exhibit some ways in which these objects are similar to their counterparts in classical topology and similar to their motivic counterparts (the (Voevodsky) motive and zeroth Suslin homology sheaf). We show that if k is infinite, the zeroth 𝔸 1 -homology sheaf is a birational invariant of smooth proper varieties, and we explain how these sheaves control various cohomological invariants, e.g., unramified étale cohomology. In particular, we deduce a number of vanishing results for cohomology of 𝔸 1 -connected varieties. Finally, we give a partial converse to these vanishing statements by giving a characterization of 𝔸 1 -connectedness by means of vanishing of unramified invariants.

Abstract. Let  be the (open) set of  divergence free vector fields generating global smooth solutions to the three-dimensional incompressible Navier–Stokes equations. We prove that any element of  can be perturbed by an arbitrarily large, smooth divergence free vector field which varies slowly in one direction, and the resulting vector field (which remains arbitrarily large) is an element of  if the variation is slow enough. This result implies that through any point in  passes an uncountable number of arbitrarily long segments included in .

Abstract. C. Deninger and A. Werner (2005, 2007) developed a partial p -adic analogue of the classical Narasimhan-Seshadri correspondence between vector bundles and representations of the fundamental group. We use this theory to prove a connectedness result for the algebraic monodromy groups of vector bundles on a smooth projective curve. We apply this result to the restriction of certain stable vector bundles on the projective space and compute the Tannaka dual groups in this case.

Abstract. For a proper smooth variety of even dimension over a field of characteristic different from 2 or ℓ, the second Stiefel–Whitney class of the ℓ-adic cohomology and the second Hasse–Witt class of the de Rham cohomology are both defined in the second Galois cohomology. We state a conjecture on their relation and give several evidences.

Abstract. We treat a relative version of the main theorem in “A transcendental approach to Kollár's injectivity theorem” (Osaka J. Math., to appear). More explicitly, we give a curvature condition that implies Kollár type cohomology injectivity theorems in the relative setting. To carry out this generalization, we use the Ohsawa–Takegoshi twisted version of Nakano's identity.

Abstract. We use the Asymptotic Large Sieve and Levinson's method to obtain lower bounds for the proportion of simple zeros on the critical line of the twists by primitive Dirichlet characters of a fixed L -function of degree 1, 2, or 3.

Abstract. We study the asymptotic growth of the p -primary component of the Šafarevič–Tate group in the cyclotomic direction at any odd prime of good supersingular reduction, generalizing work of Kobayashi. As an application, we explain formulas obtained by Kurihara, Perrin-Riou, and Nasybullin in terms of Iwasawa invariants of modified Selmer groups.

Abstract. We give a definition of the action of a tensor triangulated category on a triangulated category . In the case that is rigidly-compactly generated and is compactly generated we show this gives rise to a notion of supports which categorifies work of Benson, Iyengar, and Krause and extends work of Balmer and Favi. We prove that a suitable version of the local-to-global principle holds very generally. A relative version of the telescope conjecture is formulated and we give a sufficient condition for it to hold.