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.
We construct a compact simply connected manifold with holonomy G2 that is nonformal. This manifold is the resolution of a flat G2 orbifold with Z2 isotropy, which can be resolved using both the generalized Kummer construction by D. D. Joyce and the method developed by Joyce and S. Karigiannis. A non-vanishing triple Massey product is obtained by arranging the singular locus in a particular configuration.
We investigate the Gopakumar-Vafa (GV) theory of local curves, namely, the total spaces of rank two vector bundles with canonical determinant on smooth projective curves. Under a certain genericity condition on the rank two bundles, we propose a general mechanism to compute the degree two GV invariants of local curves. In particular, we determine all the degree two GV invariants when the base curve has genus two. Combined with previous work by Bryan and Pandharipande, we obtain the GV/GW correspondence in this case. When the base curve has genus greater than two, we calculate GV invariants for some extremal genera, providing evidence for the GV/GW conjecture for curves of higher genus.
We exhibit moduli spaces of slope stable vector bundles on general polarized HK varieties (X,h) of type K3^[2] which have an irreducible component of dimension 2a^2+2, with a an arbitrary integer greater than 1. This is done by studying the case X=S^[2] where S is an elliptic K3 surface. We show that in this case there is an irreducible component of the moduli space of stable vector bundles on S^[2] which is birational to a moduli space of sheaves on S. We expect that if the moduli space of sheaves on S is a smooth HK variety (necessarily of type K3^[a^2+1]) then the following more precise version holds: the closure of the moduli space of slope stable vector bundles on (X,h) in the moduli space of Gieseker-Maruyama semistable sheaves with its GIT polarization is a general polarized HK variety of type K3^[a^2+1].
Grothendieck's formal functions theorem states that the coherent cohomology of a Noetherian scheme can be recovered from that of a blowup, and the infinitesimal thickenings of the center and of the exceptional divisor of the blowup. We prove an analogous descent result, called "pro-cdh descent", for certain cohomological invariants of arbitrary quasicompact, quasiseparated derived schemes. Our results in particular apply to algebraic K-theory, topological Hochschild and cyclic homology, and the cotangent complex. As an application, we deduce that K-n(X) = 0 when n <-d for quasicompact, quasiseparated derived schemes X of valuative dimension d. This generalizes Weibel's conjecture, which was originally stated for Noetherian (nonderived) X of Krull dimension d, and proved in this form in 2018 by Kerz, Strunk, and the third author.
Let X be a Fano manifold of dimension at least 2 and D be a smooth divisor in a multiple of the anticanonical class, 1/alpha(-K-X) with alpha>1. It is well known that Kahler-Einstein metrics on X with conic singularities along D may exist only if the angle 2 pi beta is bigger than some positive limit value 2 pi beta(*). Under the hypothesis that the automorphisms of D are induced by the automorphisms of the pair (X, D), we prove that for beta > beta(*) close enough to beta(*), such Kahler-Einstein metrics do exist. We identify the limits at various scales when beta -> beta(*) and, in particular, we exhibit the appearance of the Tian-Yau metric of X\D.
We develop a new boundary condition for the weak inverse mean curvature flow, which gives canonical and non-trivial solutions in bounded domains. Roughly speaking, the boundary of the domain serves as an outer obstacle, and the evolving hypersurfaces are assumed to stick tangentially to the boundary upon contact. In smooth bounded domains, we prove an existence and uniqueness theorem for weak solutions, and establish C^1,α regularity of the level sets up to the obstacle. The proof combines various techniques, including elliptic regularization, blow-up analysis, and certain parabolic estimates. As an analytic application, we address the well-posedness problem for the usual weak inverse mean curvature flow, showing that the initial value problem always admits a unique maximal (or innermost) weak solution.
We establish a pseudoisotopy result for embedding spaces in the line of that of Weiss and Williams for diffeomorphism groups. In other words, for $P\subset M$ a codimension at least three embedding, we describe the difference in a range of homotopical degrees between the spaces of block and ordinary embeddings of $P$ into $M$ as a certain infinite loop space involving the relative algebraic $K$-theory of the pair $(M,M-P)$. This range of degrees is the so-called concordance embedding stable range, which, by recent developments of Goodwillie--Krannich--Kupers, is far beyond that of the aforementioned theorem of Weiss--Williams. As an application, we give a full description of the homotopy type (away from 2 and up to the concordance embedding stable range) of the space of long knots of codimension at least 3.
We prove that the (2n, 1)-cable of the figure-eight knot is not smoothly slice when n is odd, by using the real Seiberg-Witten Fr & oslash;yshov invariant of Konno, Miyazawa and Taniguchi. For the computation, we develop an O (2)-equivariant version of the lattice homotopy type, originally introduced by Dai, Sasahira and Stoffregen. This enables us to compute the real Seiberg-Witten Floer homotopy type for a certain class of knots. Additionally, we present some computations of Miyazawa's real framed Seiberg-Witten invariant for 2-knots.
We prove 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 new 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. More generally, we show that if G is a RFRS group of cohomological dimension n and of type FP_n-1, 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.
We prove that the examples by Smith and McMullen-Taubes provide infinitely many counterexamples to one direction of Donaldson's 4-6 question and the closely related Stabilising Conjecture. These are the first known counterexamples. In the other direction, we show that the Gromov-Witten invariants of two simply-connected closed symplectic $4$-manifolds, whose products with $(S^2,\omega_{\text{std}})$ are deformation equivalent, agree. In particular, when $b_2^+ \geq 2$, these $4$-manifolds have the same Seiberg-Witten invariants. Furthermore, one can replace $(S^2,\omega_{\text{std}})$ by $(S^2,\omega_{\text{std}})^k$ for any $k \geq 1$ in both results.
Using Ravenel's Thom spectrum X(n), we introduce the concept of chromatic defect, which measures how far a spectrum is from being complex-orientable. We compute the chromatic defect of various examples of interest, such as finite spectra, the Real Johnson-Wilson spectra ER(n), fixed points of Morava E-theories (with respect to finite subgroups of the Morava stabilizer group), and the connective image of J spectrum. Moreover, an obstruction theory is developed for determining chromatic defect. Having finite chromatic defect is closely related to the existence of analogues of the classical Wood equivalence. We show that such equivalences exist in a wide generality and use them to construct Z-indexed Adams-Novikov towers.
A classical and beautiful story in geometric representation theory is the construction by Springer of an action of the Weyl group on the cohomology of the fibres of the Springer resolution of the nilpotent cone. We establish a natural extension of Springer's theory to arbitrary symplectic resolutions of conical symplectic singularities. We analyse features of the action in the case of affine quiver varieties, constructing Weyl group actions on the cohomology of ADE quiver varieties, and also consider "symplectically dual" examples arising from slices in the affine Grassmannian. Along the way, we document some basic features of the symplectic geometry of quiver varieties.
Motivated by G2-manifolds with coassociative fibrations in the adiabatic limit, Donaldson and Scaduto conjectured the existence of associative submanifolds homeomorphic to a three-holed 3-sphere with three asymptotically cylindrical ends in the G2-manifold X x R-3, or equivalently similar special Lagrangians in the Calabi-Yau 3-fold X x C, where X is an A2-type ALE hyperkahler 4-manifold. We prove this conjecture by solving a real Monge-Ampere equation with a singular right-hand side, which produces a potentially singular special Lagrangian. Then, we prove the smoothness and asymptotic properties for the special Lagrangian using inputs from geometric measure theory. The method produces many other asymptotically cylindrical U(1)-invariant special Lagrangians in X x C, where X arises from the Gibbons-Hawking construction.
We compute the twisted cohomology of the mapping class group with level structures, with coefficients in the r-tensor powers of the Prym representations for any positive integer r. When r >= 2, we show that the cohomology exhibits instability for large genus, whereas it remains stable for r = 0 or r = 1. As a corollary, we prove that the symplectic Prym representation associated with any finite abelian regular cover of a nonclosed finite-type surface is infinitesimally rigid.
We prove the 3-fold DT/PT correspondence for K-theoretic vertices via wall-crossing techniques. We provide two different setups, following Mochizuki and following Joyce; both reduce the problem to q-combinatorial identities on word rearrangements. An important technical step is the construction of symmetric almost-perfect obstruction theories (APOTs) on auxiliary moduli stacks, e.g. master spaces, from the symmetric DT or PT obstruction theory. For this, we introduce symmetrized pullbacks of symmetric obstruction theories along smooth morphisms of Artin stacks.
We prove an upper bound on the filling radius of complete, spin manifold with uniformly positive scalar curvature using the quantitative operator $K$-theory and index theory.
