Given an extension of locally compact groups, with abelian, and a compatible essentially bijective -cocycle , we define a dual unitary -cocycle on and show that the associated deformation of is a cocycle bicrossed product defined by a matched pair of subgroups of . We also discuss an interpretation of our construction from the point of view of Kac cohomology for matched pairs. Our setup generalizes that of Etingof and Gelaki for finite groups and its extension due to Ben David and Ginosar, as well as our earlier work on locally compact groups satisfying the dual orbit condition. In particular, we get a locally compact quantum group from every involutive nondegenerate set-theoretical solution of the Yang–Baxter equation, or more generally, from every brace structure. On the technical side, the key new points are constructions of an irreducible projective representation of on and a unitary quantization map of Kohn–Nirenberg type.
The number of singularities, counted with multiplicity, of a generic codimension one holomorphic distribution on a compact toric orbifold is determined. As a consequence, we provide a classification for regular distributions on rational normal scrolls and weighted projective spaces. Additionally, under specific conditions, we prove that the singular set of a codimension one holomorphic foliation on a compact toric orbifold admits at least one irreducible component of codimension two, and we also present a Darboux–Jouanolou type integrability theorem for codimension one holomorphic foliations. Our results are exemplified through various illustrative examples.
This paper obtains weak-type estimates, limiting weak-type behaviors for variation operators associated with functions. Besides, we give a new characterization of Hardy space via the boundedness of variation operators associated with functions.
Let be a smooth quasi-projective curve. We previously constructed a Deligne–Hitchin moduli space with Hecke gauge groupoid for connections of rank . We extend this construction to the case of any rank , although still keeping a genericity hypothesis. The formal neighborhood of a preferred section corresponding to a tame harmonic bundle is governed by a mixed twistor structure.
We give a survey of the implosion construction, extending some of its aspects relating to hypertoric geometry from type to a general reductive group, and interpret it in the context of the Moore–Tachikawa category. We use these ideas to discuss how the contraction construction in symplectic geometry can be generalized to the hyperkähler or complex symplectic situation.
Let be a function in the complex Sobolev space , where is an open subset in . We show that the complement of the set of Lebesgue points of is pluripolar. The key ingredient in our approach is to show that for is locally bounded from above by a plurisubharmonic function.
We prove a simple necessary and sufficient condition for a two-bridge knot to be quasipositive, based on the continued fraction expansion of . As an application, coupled with some classification results in contact and symplectic topology, we give a new proof of the fact that smoothly slice two-bridge knots are non-quasipositive. Another proof of this fact using methods within the scope of knot theory is presented in Appendix A, by Stepan Orevkov.
We demonstrate that various metrizability problems for Finsler sprays can be reformulated in terms of the geodesic invariance of two tensors, namely the metric and angular tensors. We show that a spray is the geodesic spray of some Finsler metric if and only if its metric tensor is geodesically invariant. Moreover, we establish that gyroscopic sprays constitute the largest class of sprays characterized by a geodesic-invariant angular metric. Scalar functions associated with these geodesically invariant tensors will also be invariant, thereby providing first integrals for the given spray.
In this paper, we study several closely related invariants associated to Dirac operators on odd-dimensional manifolds with boundary with an action of the compact group of isometries. In particular, the equality between equivariant winding numbers, equivariant spectral flow and equivariant Maslov indices is established. We also study equivariant -invariants which play a fundamental role in the equivariant analog of Getzler’s spectral flow formula. As a consequence, we establish a relation between equivariant -invariants and equivariant Maslov triple indices in the splitting of manifolds.
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