2 research outputs found
Algebraic Geometry Over Hyperrings
We develop basic notions and methods of algebraic geometry over the algebraic
objects called hyperrings. Roughly speaking, hyperrings generalize rings in
such a way that an addition is `multi-valued'. This paper largely consisits of
two parts; algebraic aspects and geometric aspects of hyperrings. We first
investigate several technical algebraic properties of a hyperring. In the
second part, we begin by giving another interpretation of a tropical variety as
an algebraic set over the hyperfield which canonically arises from a totally
ordered semifield. Then we define a notion of an integral hyperring scheme
and prove that for any
integral affine hyperring scheme .Comment: 37 page
The quantum Johnson homomorphism and symplectomorphism of 3-folds
We consider the action of symplectic monodromy on chain-level enhancements of
quantum cohomology. First, we construct a family version of
-structure on quantum cohomology (this should morally correspond to
Hochschild cohomology of a "family of Fukaya categories over the circle").
Following Kaledin, we look at the obstruction class of this structure, and
argue that it can be related to a quantum version of Massey products on the one
hand (which, in nice cases, can be related to actual counts of rational curves)
and to the classical Andreadakis-Johnson theory of the Torelli group on the
other hand.
In the second part of the paper, we go hunting for exotic symplectomorphism:
these are elements of infinite order in the kernel of the
forgetful map from the symplectic mapping class group to the ordinary MCG. We
demonstrate how we can apply the theory above to prove the existence of such
elements for certain a Fano 3-fold obtained by blowing-up
at a genus 4 curve. Unlike the four-dimensional case, no power
of a Dehn twist around Lagrangian 3-spheres can be exotic (because they have
infinite order in smooth MCG).
In the final part of the paper, the classical connection between our Fano
varieties and cubic 3-folds allows us to prove the existence of a new
phenomena: "exotic relations" in the subgroup generated by all Dehn twists.
Namely, it turns out we can factor some power of in into 3-dimensional Dehn twists. So the isotopy class
of the product in the ordinary MCG is torsion, but of infinite order in the
symplectic MCG.Comment: v1: 197 pages. Feedback and comments are welcome! This paper will
(eventually) be split into two papers. arXiv admin note: text overlap with
arXiv:1205.0713, arXiv:0904.1474, arXiv:math/0610004, arXiv:1405.0744,
arXiv:1007.0265, arXiv:math/0702887, arXiv:1109.5669 by other author