30,453 research outputs found
Augmentations are Sheaves
We show that the set of augmentations of the Chekanov-Eliashberg algebra of a
Legendrian link underlies the structure of a unital A-infinity category. This
differs from the non-unital category constructed in [BC], but is related to it
in the same way that cohomology is related to compactly supported cohomology.
The existence of such a category was predicted by [STZ], who moreover
conjectured its equivalence to a category of sheaves on the front plane with
singular support meeting infinity in the knot. After showing that the
augmentation category forms a sheaf over the x-line, we are able to prove this
conjecture by calculating both categories on thin slices of the front plane. In
particular, we conclude that every augmentation comes from geometry.Comment: 109 pages; v2: added Legendrian mirror example in section 4.4.4,
corrected typos and other minor changes; v3: accepted versio
Refined curve counting on complex surfaces
We define refined invariants which "count" nodal curves in sufficiently ample
linear systems on surfaces, conjecture that their generating function is
multiplicative, and conjecture explicit formulas in the case of K3 and abelian
surfaces. We also give a refinement of the Caporaso-Harris recursion, and
conjecture that it produces the same invariants in the sufficiently ample
setting. The refined recursion specializes at y = -1 to the
Itenberg-Kharlamov-Shustin recursion for Welschinger invariants. We find
similar interactions between refined invariants of individual curves and real
invariants of their versal families.Comment: 53 pages, 1 figure. (v2 updated to match published version.
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