Recent constructions have shown that interesting behaviours can be observed in the finiteness properties of Kähler groups and their subgroups. In this work, we push this further and exhibit, for each integer , new hyperbolic groups admitting surjective homomorphisms to and to , whose kernel is of type but not of type . By a fibre product construction, we also find examples of non-normal subgroups of Kähler groups with exotic finiteness properties.
We describe a new algebro-geometric perspective on the study of the Milnor fibration and, as a first step toward putting it into practice, prove powerful criteria for a deformation of a holomorphic function germ to admit a stratification on its domain partially satisfying the Thom condition and, more generally, to respect the Milnor fibration of the original germ in an appropriate sense. As corollaries, we obtain a method of partitioning the space of homogeneous polynomials of a fixed degree into finitely many locally closed subsets such that the fiber diffeomorphism type of the Milnor fibration is constant along each subset and a criterion under which deformations of a function with critical locus a complete intersection will be well-behaved.
We prove that homological mirror symmetry for very affine hypersurfaces respects certain natural symplectic operations (as functors between partially wrapped Fukaya categories), verifying conjectures of Auroux. These conjectures concern compatibility between mirror symmetry for a very affine hypersurface and its complement, itself also a very affine hypersurface. We find that the complement of a very affine hypersurface has, in fact, two natural mirrors, one of which is a derived scheme. These two mirrors are related via a nongeometric equivalence mediated by Knörrer periodicity; Auroux's conjectures require some modification to take this into account. Our proof also introduces new techniques for presenting Liouville manifolds as gluings of Liouville sectors.
We characterize a certain neck-pinching degeneration of (marked) -structures on a closed oriented surface of genus at least two. In a more general setting, we take a path of -structures on that leaves every compact subset in its deformation space, such that the holonomy of converges in the -character variety as . Then, it is well known that the complex structure
This paper has two main goals. First, we prove nonabelian refinements of basechange theorems in étale cohomology (i.e., prove analogues of the classical statements for sheaves of spaces). Second, we apply these theorems to prove a number of results about the étale homotopy type. Specifically, we prove nonabelian refinements of the smooth basechange theorem, Huber–Gabber affine analogue of the proper basechange theorem, and Fujiwara–Gabber rigidity theorem. Our methods also recover Chough's nonabelian refinement of the proper basechange theorem. Transporting an argument of Bhatt–Mathew to the nonabelian setting, we apply nonabelian proper basechange to show that the profinite étale homotopy type satisfies arc-descent. Using nonabelian smooth and proper basechange and descent, we give rather soft proofs of a number of Künneth formulas for the étale homotopy type.
We show that a large class of satellite operators are rank-expanding; that is, they map some rank-one subgroup of the concordance group onto an infinite linearly independent set. Our work constitutes the first systematic study of this property in the literature and partially affirms a conjecture of the second author and Pinzón-Caicedo. More generally, we establish a Floer-theoretic condition for a family of companion knots to have infinite-rank image under satellites from this class. The methods we use are amenable to patterns that act trivially in topological concordance and are capable of handling a surprisingly wide variety of companions. For instance, we give an infinite linearly independent family of Whitehead doubles whose companion knots all have negative -invariant. Our results also recover and extend several theorems in this area established using instanton Floer homology.
The paper computes the Witt-sheaf cohomology rings of partial flag varieties in type A in terms of the Pontryagin classes of the subquotient bundles. The proof is based on a Leray–Hirsch-type theorem for Witt-sheaf cohomology for the maximal rank cases, and a detailed study of cohomology ring presentations and annihilators of characteristic classes for the general case. The computations have consequences for the topology of real flag manifolds: we show that all torsion in the integral cohomology is 2-torsion, which was not known in full generality previously. This allows for example to compute the Poincaré polynomials of complete flag varieties for cohomology with twisted integer coefficients. The computations also allow to describe the Chow–Witt rings of flag varieties, and we sketch an enumerative application to counting flags satisfying multiple incidence conditions to given hypersurfaces.
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