Duality and kernels in microlocal geometry

Abstract

We study the dualizability of sheaves on manifolds with isotropic singular supports ${\mathrm{Sh}}_{\Lambda }(M)$ and microsheaves with isotropic supports ${\mathrm{\mu sh}}_{\Lambda }(\Lambda )$ and obtain a classification result of colimit-preserving functors by convolutions of sheaf kernels. Moreover, for sheaves with isotropic singular supports and compact supports ${\mathrm{Sh}}_{\Lambda }^b(M{)}_0$, the standard categorical duality and Verdier duality are related by the wrap-once functor, which is the inverse Serre functor in proper objects, and we thus show that the Verdier duality extends naturally to all compact objects ${\mathrm{Sh}}_{\Lambda }^c(M{)}_0$ when the wrap-once functor is an equivalence, for instance, when $\Lambda$ is a full Legendrian stop or a swappable Legendrian stop.

33 pages, 3 figures. This article extends certain results from arXiv:2210.06643, which no longer includes them since v4. v3: Sec 1 the discussion of miraculous duality is modified, Thm 1.8 is removed and Cor 1.8 on the relation to toric mirror symmetry is added; Sec 2.1 & 2.3 the part regarding microsheaves is modified; Sec 4.3 the relation between duality and wrapping is added. Published in IMRN

References in corpus (13)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• A Duality Exact Sequence for Legendrian Contact Homology
• Sheaf quantization of Hamiltonian isotopies and applications to non displaceability problems
• The nonequivariant coherent-constructible correspondence for toric stacks
• A categorification of Morelli's theorem
• Duality for Legendrian contact homology
• Quantization of conic Lagrangian submanifolds of cotangent bundles
• Functors of wrapped Fukaya categories from Lagrangian correspondences
• Definable triangulations with regularity conditions
• Wrapped sheaves
• The six operations in topology
• Duality for automorphic sheaves with nilpotent singular support
• On the Hochschild cohomology of Tamarkin categories