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Abstract: Abstract We show that any toric Kähler cone with smooth compact cross-section admits a family of Calabi–Yau cone metrics with conical singularities along its toric divisors. The family is parametrized by the Reeb cone and the angles are given explicitly in terms of the Reeb vector field. The result is optimal, in the sense that any toric Calabi–Yau cone metric with conical singularities along the toric divisor (and smooth elsewhere) belongs to this family. We also provide examples and interpret our results in terms of Sasaki–Einstein metrics. PubDate: 2022-05-04

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Abstract: Abstract We formulate the geometric P=W conjecture for singular character varieties. We establish it for compact Riemann surfaces of genus one, and obtain partial results in arbitrary genus. To this end, we employ non-Archimedean, birational and degeneration techniques to study the topology of the dual boundary complex of character varieties. We also clarify the relation between the geometric and the cohomological P=W conjectures. PubDate: 2022-05-04

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Abstract: Abstract In this paper we will present a homological model for Coloured Jones Polynomials. For each colour \(N \in {\mathbb {N}}\) , we will describe the invariant \(J_N(L,q)\) as a graded intersection pairing of certain homology classes in a covering of a configuration space on the punctured disc. This construction is based on the Lawrence representation and a result due to Kohno that relates quantum representations and homological representations of the braid groups. PubDate: 2022-04-30

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Abstract: Abstract For a Koszul Artin-Schelter regular algebra (also called twisted Calabi-Yau algebra), we show that it has a “twisted" bi-symplectic structure, which may be viewed as a noncommutative and twisted analog of the shifted symplectic structure introduced by Pantev, Toën, Vaquié and Vezzosi. This structure gives a quasi-isomorphism between the tangent complex and the twisted cotangent complex of the algebra, and may be viewed as a DG enhancement of Van den Bergh’s noncommutative Poincaré duality; it also induces a twisted symplectic structure on its derived representation schemes. PubDate: 2022-04-28

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Abstract: Abstract For every one-relator monoid \(M = \langle A \mid u=v \rangle \) with \(u, v \in A^*\) we construct a contractible M-CW complex and use it to build a projective resolution of the trivial module which is finitely generated in all dimensions. This proves that all one-relator monoids are of type \(\mathrm{FP}_{\infty }\) , answering positively a problem posed by Kobayashi in 2000. We also apply our results to classify the one-relator monoids of cohomological dimension at most 2, and to describe the relation module, in the sense of Ivanov, of a torsion-free one-relator monoid presentation as an explicitly given principal left ideal of the monoid ring. In addition, we prove the topological analogues of these results by showing that all one-relator monoids satisfy the topological finiteness property \(\mathrm{F}_\infty \) , and classifying the one-relator monoids with geometric dimension at most 2. These results give a natural monoid analogue of Lyndon’s Identity Theorem for one-relator groups. PubDate: 2022-04-27

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Abstract: Abstract This article introduces a new method to construct volume-filling symplectic embeddings of 4-dimensional ellipsoids by employing polytope mutations in toric and almost toric varieties. The construction uniformly recovers the full sequences for the Fibonacci Staircase of McDuff–Schlenk, the Pell Staircase of Frenkel–Müller and the Cristofaro-Gardiner–Kleinman Staircase, and adds new infinite sequences of ellipsoid embeddings. In addition, we initiate the study of symplectic-tropical curves for almost toric fibrations and emphasize the connection to quiver combinatorics. PubDate: 2022-04-27

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Abstract: Abstract Let (X, o) be a complex normal surface singularity with rational homology sphere link and let \(\widetilde{X}\) be one of its good resolutions. Consider an effective cycle Z supported on the exceptional curve and the isomorphism classes \(\mathrm{Pic}(Z)\) of line bundles on Z. The set of possible values \(h^1(Z,\mathcal {L})\) for \(\mathcal {L}\in \mathrm{Pic}(Z)\) can be understood in terms of the dimensions of the images of the Abel maps, as subspaces of \(\mathrm{Pic}(Z)\) . In this note we present two algorithms, which provide these dimensions. Usually, the dimension of \(\mathrm{Pic}(Z)\) and of the dimension of the image of the Abel maps are not topological. However, we provide combinatorial formulae for them in terms of the resolution graph whenever the analytic structure on \(\widetilde{X}\) is generic. PubDate: 2022-04-27

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Abstract: Abstract Given a connected reductive algebraic group G and a Borel subgroup \(B \subseteq G\) , we study B-normalized one-parameter additive group actions on affine spherical G-varieties. We establish basic properties of such actions and their weights and discuss many examples exhibiting various features. We propose a construction of such actions that generalizes the well-known construction of normalized one-parameter additive group actions on affine toric varieties. Using this construction, for every affine horospherical G-variety X we obtain a complete description of all G-normalized one-parameter additive group actions on X and show that the open G-orbit in X can be connected with every G-stable prime divisor via a suitable choice of a B-normalized one-parameter additive group action. Finally, when G is of semisimple rank 1, we obtain a complete description of all B-normalized one-parameter additive group actions on affine spherical G-varieties having an open orbit of a maximal torus \(T \subseteq B\) . PubDate: 2022-04-27

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Abstract: Abstract We consider the Hodge filtration on the sheaf of meromorphic functions along free divisors for which the logarithmic comparison theorem holds. We describe the Hodge filtration steps as submodules of the order filtration on a cyclic presentation in terms of a special factor of the Bernstein–Sato polynomial of the divisor and we conjecture a bound for the generating level of the Hodge filtration. Finally, we develop an algorithm to compute Hodge ideals of such divisors and we apply it to some examples. PubDate: 2022-04-18

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Abstract: Abstract We prove the height two case of a conjecture of Hovey and Strickland that provides a K(n)-local analogue of the Hopkins–Smith thick subcategory theorem. Our approach first reduces the general conjecture to a problem in arithmetic geometry posed by Chai. We then use the Gross–Hopkins period map to verify Chai’s Hope at height two and all primes. Along the way, we show that the graded commutative ring of completed cooperations for Morava E-theory is coherent, and that every finitely generated Morava module can be realized by a K(n)-local spectrum as long as \(2p-2>n^2+n\) . Finally, we deduce consequences of our results for descent of Balmer spectra. PubDate: 2022-04-10

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Abstract: Abstract Motivated by the intersection theory of moduli spaces of curves, we introduce psi classes in matroid Chow rings and prove a number of properties that naturally generalize properties of psi classes in Chow rings of Losev-Manin spaces. We use these properties of matroid psi classes to give new proofs of (1) a Chow-theoretic interpretation for the coefficients of the reduced characteristic polynomials of matroids, (2) explicit formulas for the volume polynomials of matroids, and (3) Poincaré duality for matroid Chow rings. PubDate: 2022-04-01

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Abstract: Abstract We study categorical primitive forms for Calabi–Yau \(A_\infty \) categories with semi-simple Hochschild cohomology. We classify these primitive forms in terms of certain grading operators on the Hochschild homology. We use this result to prove that, if the Fukaya category \({{\textsf {Fuk}}}(M)\) of a symplectic manifold M has semi-simple Hochschild cohomology, then its genus zero Gromov–Witten invariants may be recovered from the \(A_\infty \) -category \({{\textsf {Fuk}}}(M)\) together with the closed-open map. An immediate corollary of this is that in the semi-simple case, homological mirror symmetry implies enumerative mirror symmetry. PubDate: 2022-03-25

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Abstract: Abstract We study the birational self-maps of the projective plane over finite fields that induce permutations on the set of rational points. As a main result, we prove that no odd permutation arises over a non-prime finite field of characteristic two, which completes the investigation initiated by Cantat about which permutations can be realized this way. Main ingredients in our proof include the invariance of parity under groupoid conjugations by birational maps, and a list of generators for the group of such maps. PubDate: 2022-03-23

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Abstract: Abstract We prove that p-determinants of a certain class of differential operators can be lifted to power series over \(\mathbb {Q}\) . We compute these power series in terms of monodromy of the corresponding differential operators. PubDate: 2022-03-23

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Abstract: Abstract The generalized Miller–Morita–Mumford classes of a manifold bundle with fiber M depend only on the underlying \(\tau _M\) -fibration, meaning the family of vector bundles formed by the tangent bundles of the fibers. This motivates a closer study of the classifying space for \(\tau _M\) -fibrations, \(Baut(\tau _M)\) , and its cohomology ring, i.e., the ring of characteristic classes of \(\tau _M\) -fibrations. For a bundle \(\xi \) over a simply connected Poincaré duality space, we construct a relative Sullivan model for the universal \(\xi \) -fibration with holonomy in a given connected monoid, together with explicit cocycle representatives for the characteristic classes of the canonical bundle over its total space. This yields tools for computing the rational cohomology ring of \(Baut(\xi )\) as well as the subring generated by the generalized Miller–Morita–Mumford classes. To illustrate, we carry out sample computations for spheres and complex projective spaces. We discuss applications to tautological rings of simply connected manifolds and to the problem of deciding whether a given \(\tau _M\) -fibration comes from a manifold bundle. PubDate: 2022-03-22

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Abstract: Abstract Muttalib–Borodin determinants are generalizations of Hankel determinants and depend on a parameter \(\theta >0\) . In this paper, we obtain large n asymptotics for \(n \times n\) Muttalib–Borodin determinants whose weight possesses an arbitrary number of Fisher–Hartwig singularities. As a corollary, we obtain asymptotics for the expectation and variance of the real and imaginary parts of the logarithm of the underlying characteristic polynomial, several central limit theorems, and some global bulk rigidity upper bounds. Our results are valid for all \(\theta > 0\) . PubDate: 2022-03-08

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Abstract: Abstract We prove a result characterizing conditions for the existence and uniqueness of solutions of a certain Diophantine equation, then using techniques from equivariant bifurcation theory, we apply the result to prove symmetric Hopf bifurcation type theorems for both dissipative and non-dissipative autonomous wave equations, for a large set of spatial dimensions. For the latter only the classical implicit function theorem is used. The set of admissible spatial dimensions is the union of the perfect squares together with finitely many non-perfect squares. PubDate: 2022-03-01 DOI: 10.1007/s00029-022-00761-7

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Abstract: Abstract In this paper we construct the Poincare line bundle for the stack of Higgs bundles on smooth projective curves and show that it induces a fully-faithful Fourier-Mukai transform on the category of quasi-coherent sheaves. PubDate: 2022-03-01 DOI: 10.1007/s00029-022-00763-5

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Abstract: Abstract We introduce an operad \({{\,\mathrm{Patch}\,}}\) which acts on the Gerstenhaber–Schack complex of a prestack as defined by Dinh Van and Lowen, and which in particular allows us to endow this complex with an underlying \(L_{\infty }\) -structure. We make use of the operad \({{\,\mathrm{Quilt}\,}}\) which was used by Hawkins in order to solve the presheaf case. Due to the additional difficulty posed by the presence of twists, we have to use \({{\,\mathrm{Quilt}\,}}\) in a fundamentally different way (even for presheaves) in order to allow for an extension to prestacks. PubDate: 2022-02-28 DOI: 10.1007/s00029-022-00759-1

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Abstract: Abstract For a moduli space \({\mathsf M}\) of stable sheaves over a K3 surface X, we propose a series of conjectural identities in the Chow rings \(CH_\star ({\mathsf M}\times X^\ell ),\, \ell \ge 1,\) generalizing the classic Beauville–Voisin identity for a K3 surface. We emphasize consequences of the conjecture for the structure of the tautological subring \(R_\star ({\mathsf M}) \subset CH_\star ({\mathsf M}).\) The conjecture places all tautological classes in the lowest piece of a natural filtration emerging on \(CH_\star ({\mathsf M})\) , which we also discuss. We prove the proposed identities when \({\mathsf M}\) is the Hilbert scheme of points on a K3 surface. PubDate: 2022-02-16 DOI: 10.1007/s00029-021-00729-z