We establish a connection between the theory of Ulrich sheaves and A1-homotopy theory. For instance, we prove that the A1-degree of a morphism between projective varieties, that is relatively oriented by an Ulrich sheaf, is constant on the target even when it is not A1-chain connected or A1-connected. Further if an embedded projective variety is the support of a symmetric Ulrich sheaf of rank one, the A1-degree of all its linear projections can be read off in an explicit way from the free resolution of the Ulrich sheaf. Finally, we construct an Ulrich sheaf on the secant variety of a curve and use this to define an arithmetic version of Viro's encomplexed writhe for curves in P3. This can be considered to be an arithmetic analogue of a knot invariant. Namely, we define a notion of algebraic isotopy under which the arithmetic writhe is invariant. For rational curves of degree at most four in P3 we obtain a complete classification up to algebraic isotopies.
For a group G of type FP(R) for a ring R and a homomorphism chi: G -> Z, we show that cd(R)(ker chi) = cd(R)(G)-1 if the top-dimensional cohomology of G with coefficients in the Novikov rings (R[G]) over cap (+/-chi) vanishes. This criterion is applied to show that if G is a finitely generated RFRS group of cohomological dimension 2, then G is virtually free-by-cyclic if and only if b(2)((2)) (G) = 0. This answers a question of Wise, and generalises and gives a significantly shorter proof of a recent theorem of Kielak and Linton, where the same result is obtained under the additional hypotheses that G is virtually compact special and hyperbolic. A consequence of the result is that all virtually RFRS groups of rational cohomological dimension 2 with vanishing second l(2)-Betti number are coherent. More generally, we show that if G is a RFRS group of cohomological dimension n and of type FPn-1(Q), then G admits a virtual map to Z with kernel of rational cohomological dimension n-1 if and only if b(n)((2)) (G) = 0.
Let X be a compact Gromov-Hausdorff limit space of a collapsing sequence of compact n-manifolds, M_i, of Ricci curvature Ric_M_i≥ -(n-1) and all points in M_i are (δ,ρ)-local rewinding Reifenberg points, or sectional curvature sec_M_i≥ -1, respectively. We conjecture that if M_i is an aspherical manifold of fundamental group satisfying a certain condition (e.g., a nilpotent group), then X is a differentiable, or topological aspherical manifold, respectively. A main result in this paper asserts that if M_i a diffeomorphic or homeomorphic to a nilmanifold, then X is diffeomorphic or homeomorphic to a nilmanifold, respectively.
Any hyper-K & auml;hler variety K of generalized Kummer type is associated via Hodge theory with a K3 surface SK. We show how they are related geometrically through a moduli space of sheaves on SK. As a consequence, building fundamentally on the works of O'Grady, Markman, Voisin and Varesco, we establish the Hodge conjecture for all powers of any of these K3 surfaces as well as for all abelian fourfolds of Weil type with discriminant 1 and their powers, strengthening a result of Markman.
