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We provide a generalization of the Deligne sheaf construction of intersection homology theory and a corresponding generalization of Poincaré duality on pseudomanifolds such that the Goresky–MacPherson, Goresky–Siegel, and Cappell–Shaneson duality theorems all arise as particular cases. Unlike classical intersection homology theory, our duality theorem holds with ground coefficients in an arbitrary PID and with no local cohomology conditions on the underlying space. Self-duality does require local conditions, but our perspective leads to a new class of spaces more general than the Goresky–Siegel IP spaces on which upper-middle perversity intersection homology is self-dual. We also examine torsion-sensitive t-structures and categories of perverse sheaves that contain our torsion-sensitive Deligne sheaves as intermediate extensions.
We study Kähler manifolds-with-boundary, not necessarily compact, with weakly pseudoconvex boundary, each component of which is compact. If such a manifold has boundary components (possibly ), then it has the first Betti number at least , and the Levi form of any boundary component is zero. If has pseudoconvex boundary components and at least one nonparabolic end, then the first Betti number of is at least . In either case, any boundary component has a nonvanishing first Betti number. If has one pseudoconvex boundary component with vanishing first Betti number, then the first Betti number of is also zero. Especially significant are applications to Kähler ALE manifolds and to Kähler 4-manifolds. This significantly extends prior results in this direction (e.g., those of Kohn and Rossi) and uses substantially simpler methods.
We prove that for any infinite-type orientable surface , there exists a collection of essential curves in such that any homeomorphism that preserves the isotopy classes of the elements of is isotopic to the identity. The collection is countable and has an infinite complement in , the curve complex of . As a consequence, we obtain that the natural action of the extended mapping class group of on is faithful.
We show that the Kontsevich space of rational curves of degree at most roughly on a general hypersurface of degree is equidimensional of expected dimension and has two components: one consisting generically of smooth embedded rational curves and the other consisting of multiple covers of a line. This proves more cases of a conjecture of Coskun, Harris, and Starr and shows that the Gromov–Witten invariants in these cases are enumerative.
In this paper, we provide a new Bennequin-type inequality for the Rasmussen–Beliakova–Wehrli invariant, featuring the numerical transverse braid invariants (the -invariants) introduced by the author. From the Bennequin type-inequality and a combinatorial bound on the value of the -invariants we deduce a new computable bound on the Rasmussen invariant.
In the first part of this paper, we consider, in the context of an arbitrary weighted hyperplane arrangement, the map from compactly supported cohomology to the usual cohomology of a local system. We obtain a formula (i.e., an explicit algebraic de Rham representative) for a generalized version of this map.
In the second part, we apply these results to invariant theory: Schechtman and Varchenko connect invariant theoretic objects to the cohomology of local systems on complements of hyperplane arrangements. The first part of this paper is then used, following and completing arguments of Looijenga, to determine the image of invariants in cohomology. In suitable cases (e.g., corresponding to positive integral levels) the space of invariants acquires a mixed Hodge structure over a cyclotomic field. We investigate the Hodge filtration on the space of invariants and characterize the subspace of conformal blocks in Hodge theoretic terms.
A reciprocal linear space is the image of a linear space under coordinatewise inversion. These fundamental varieties describe the analytic centers of hyperplane arrangements and appear as part of the defining equations of the central path of a linear program. Their structure is controlled by an underlying matroid. This provides a large family of hyperbolic varieties, recently introduced by Shamovich and Vinnikov. Here we give a definite determinantal representation to the Chow form of a reciprocal linear space. One consequence is the existence of symmetric rank-one Ulrich sheaves on reciprocal linear spaces. Another is a representation of the entropic discriminant as a sum of squares. For generic linear spaces, the determinantal formulas obtained are closely related to the Laplacian of the complete graph and generalizations to simplicial matroids. This raises interesting questions about the combinatorics of hyperbolic varieties and connections with the positive Grassmannian.
The purpose of this paper is to construct and study equivariant Khovanov homology, a version of Khovanov homology theory for periodic links. Since our construction works regardless of the characteristic of the coefficient ring, it generalizes a previous construction by Chbili. We establish invariance under equivariant isotopies of links and study algebraic properties of integral and rational version of the homology theory. Moreover, we construct a skein spectral sequence converging to equivariant Khovanov homology and use this spectral sequence to compute, as an example, equivariant Khovanov homology of torus links .