Timeline for Why believe Kontsevich cosheaf conjecture?
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8 events
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Dec 30, 2020 at 19:17 | answer | added | Jonny Evans | timeline score: 12 | |
Dec 29, 2020 at 10:26 | comment | added | DamienC | @Jonny: your comments are great, and would deserve to be turned into an answer! | |
Feb 5, 2019 at 21:47 | comment | added | Jonny Evans | In fact, Kontsevich explains the motivation for the conjecture in his own words in his paper "Symplectic geometry of homological algebra". | |
Feb 5, 2019 at 20:04 | comment | added | Jonny Evans | ...So what is mirror to the Lagrangian torus fibration of the cotangent bundle by cotangent fibres? The zero section itself. Quite how you figure out that the derived category should be replaced by the category of microlocal sheaves, I don't know. | |
Feb 5, 2019 at 20:04 | comment | added | Jonny Evans | Maybe more speculative: 3. In mirror symmetry, you could think of R^n as a torus with very large radius (like how R is a circle with very large radius). When taking the dual Lagrangian torus fibration, a non-compact R^n fibre should therefore be dual to a point. You see this in recent work by Lekili and Polishchuk, where they find mirrors to punctured surfaces which are nodal curves: the structure sheaf at the node is mirror to a wrapped Lagrangian brane which is is a non-compact R going off to the puncture... | |
Feb 5, 2019 at 20:03 | comment | added | Jonny Evans | Some naive reasons why it's reasonable to expect local information on the skeleton to know everything 1. It's true for cotangent bundles and represents a nice categorical enhancement of many historical results about cotangent bundles, starting with Viterbo/Abbondandolo-Schwarz results that symplectic homology of a cotangent bundle is the homology of the loopspace of the zero section. 2. Liouville flow retracts a Weinstein manifold onto its skeleton, so everything about symplectomorphism topology of the completion should be determined by an infinitesimal neighborhood of the skeleton. | |
Feb 5, 2019 at 18:20 | review | First posts | |||
Feb 5, 2019 at 18:41 | |||||
Feb 5, 2019 at 18:18 | history | asked | rori | CC BY-SA 4.0 |