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We give a short proof of Kaledin’s theorem on the degeneration of the noncommutative Hodge-to-de Rham spectral sequence. Our approach is based on topological Hochschild homology and the theory of cyclotomic spectra. As a consequence, we also obtain relative versions of the degeneration theorem, both in characteristic zero and for regular bases in characteristic .
We prove that the –equivariant mod Eilenberg–Mac Lane spectrum arises as an equivariant Thom spectrum for any finite, –power cyclic group , generalizing a result of Behrens and the second author in the case of the group . We also establish a construction of , and prove intermediate results that may be of independent interest. Highlights include constraints on the Hurewicz images of equivariant spectra that admit norms, and an analysis of the extent to which the nonequivariant arises as the Thom spectrum of a more than double loop map.
Given a smooth projective variety and a smooth divisor , we study relative Gromov–Witten invariants of and the corresponding orbifold Gromov–Witten invariants of the root stack . For sufficiently large , we prove that orbifold Gromov–Witten invariants of are polynomials in . Moreover, higher-genus relative Gromov–Witten invariants of are exactly the constant terms of the corresponding higher-genus orbifold Gromov–Witten invariants of . We also provide a new proof for the equality between genus-zero relative and orbifold Gromov–Witten invariants, originally proved by Abramovich, Cadman and Wise (2017). When is sufficiently large and is a curve, we prove that stationary relative invariants of are equal to the stationary orbifold invariants in all genera.
We study the monopole contribution to the refined Vafa–Witten invariant recently defined by Maulik and Thomas (work in progress). We apply the results of Gholampour and Thomas (to appear in Compos. Math.) to prove a universality result for the generating series of contributions of Higgs pairs with –dimensional weight spaces. For prime rank, these account for the entire monopole contribution by a theorem of Thomas. We use toric computations to determine part of the generating series and find agreement with the conjectures of Göttsche and Kool (Pure Appl. Math. Q. 14 (2018) 467–513) for ranks and .
Let be a closed, connected and oriented –manifold. This article is the first of a five-part series that constructs an isomorphism between the Heegaard Floer homology groups of and the corresponding Seiberg–Witten Floer homology groups of .
This is the second of five papers that construct an isomorphism between the Seiberg–Witten Floer homology and the Heegaard Floer homology of a given compact, oriented –manifold. The isomorphism is given as a composition of three isomorphisms; the first of these relates a version of embedded contact homology on an auxiliary manifold to the Heegaard Floer homology on the original. This paper describes this auxiliary manifold, its geometry and the relationship between the generators of the embedded contact homology chain complex and those of the Heegaard Floer chain complex. The pseudoholomorphic curves that define the differential on the embedded contact homology chain complex are also described here as a first step to relate the differential on the latter complex with that on the Heegaard Floer complex.
This is the third of five papers that construct an isomorphism between the Seiberg–Witten Floer homology and the Heegaard Floer homology of a given compact, oriented –manifold. The isomorphism is given as a composition of three isomorphisms; the first of these relates a version of embedded contact homology on an auxiliary manifold to the Heegaard Floer homology on the original. This paper describes the relationship between the differential on the embedded contact homology chain complex and the differential on the Heegaard Floer chain complex. The paper also describes the relationship between the various canonical endomorphisms that act on the homology groups of these two complexes.
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