Abstract
Toric degeneration in algebraic geometry is a process of degenerating a given projective variety into a toric one. Then one can obtain information about the original variety via analyzing the toric one, which is a much easier object to study. Harada and Kaveh described how one incorporates a symplectic structure into this process, providing a very useful tool for solving certain problems in symplectic geometry. Below we present two applications of this method: questions about the Gromov width, and cohomological rigidity problems.
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Notes
- 1.
A moment map is a T-invariant map \(\mu :M \rightarrow \text {Lie}(T)^*\cong \mathbb {R}^n\) such that for every \(X \in \text {Lie}(T)\) it holds that \(\iota _{X^\sharp } \omega =d\mu ^X\) where \(X^\sharp \) denotes the vector field on M induced by X and \(\mu ^X :M \rightarrow \mathbb {R}\) is defined by \(\mu ^X(p)=\langle \mu (p), X\rangle .\).
- 2.
Recall that a polytope in \(\mathbb {R}^n\) is called rational if the directions of its edges are in \(\mathbb {Z}^n\). It is called smooth if for every vertex the primitive vectors in the directions of edges meeting at that vertex form a \(\mathbb {Z}\)-basis for \(\mathbb {Z}^n\).
- 3.
Recall that the Plücker embedding sends a Grassmannian spanned by vectors \(v,w \in \mathbb {C}^4\) to a point \([x_{12}:\ldots :x_{34}]\in \mathbb {C}\mathbb {P}^5\) with \(x_{ij}\) equal to the determinant of the \(2\times 2\) minor of \([v^T,w^T]\) spanned by rows i and j.
- 4.
Recall that for a graded algebra \(A=\oplus _{j=0}^{\infty }A_j\) the set \({{\,\mathrm{Proj}\,}}A\) is the set of homogeneous prime ideals in A that do not contain all of \(A_+:=\oplus _{j=1}^{\infty }A_j\). The topology on \({{\,\mathrm{Proj}\,}}A\) is defined by setting the closed sets to be \(V(I):=\{J;\ J \subset I\text { is a homogeneous prime ideal of }A \text { not containing all of} A_+\}\), for some homogeneous ideal I of A. For more details see, for example [17, II.2], [9, III.2], and [5, Chap. 7].
- 5.
During the work on the project [18], about complex Grassmannians, Karshon and Tolman looked at various examples of other coadjoint orbits and got the impression that the above value might be the Gromov width of all coadjoint orbits. They never formulated this expectation formally as their conjecture, but they shared this idea with other mathematicians in private communications. This is how this value became to be known as the expected Gromov width for coadjoint orbits.
- 6.
A coadjoint orbit through a point \(\lambda \) in the interior of a chosen positive Weyl chamber is called indecomposable in [29] if there exists a simple positive root \(\alpha \) such that for any positive root \(\alpha '\) there exists a positive integer k such that \(\langle \lambda , \alpha ' \rangle =k \langle \lambda , \alpha \rangle \).
- 7.
The result about \({{\,\mathrm{SO}\,}}(2n+1,\mathbb {C})\) holds only for orbits satisfying a mild technical condition: the point \(\lambda \) of intersection of the orbit and a chosen positive Weyl chamber should not belong to a certain subset of one wall of the chamber; see [26] for more details. In particular, all generic orbits satisfy this condition.
- 8.
In the standard action of \((S^1)^n\) on \((\mathbb {C}\mathbb {P}^1)^n\) each \(S^1\) in \((S^1)^n\) acts on the respective copy of \(\mathbb {C}\mathbb {P}^1\) by \(e^{it}\cdot [(z_0,z_1)]=[(z_0,e^{it}z_1)]\).
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Acknowledgements
First of all, the author would like to thank her collaborators: Xin Fang, Iva Halacheva, Peter Littelmann and Sue Tolman. Results contained in this manuscript were obtained in collaboration with the above mathematicians [11, 14, 27], and therefore all of them could also be considered as the authors of this paper. The author also thanks the organisers of the workshops “Interactions with Lattice Polytopes” for giving her the opportunity to participate and present her results at these workshops. The author is supported by the DFG (Die Deutsche Forschungsgemeinschaft) grant CRC/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics”.
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Pabiniak, M. (2022). Toric Degenerations in Symplectic Geometry. In: Kasprzyk, A.M., Nill, B. (eds) Interactions with Lattice Polytopes. ILP 2017. Springer Proceedings in Mathematics & Statistics, vol 386. Springer, Cham. https://doi.org/10.1007/978-3-030-98327-7_13
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