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\newcommand{\Cat}{{\rm Cat}} \newcommand{\A}{\mathcal A} \newcommand{\freestar}{ \framebox[7pt]{$\star$} }

7. Orlov spectrum and Rouquier dimension

Definition: The Orlov spectrum of X is the image of Ob(D^b(X))\to \mathbb{N}\cup \{\infty\} which sends F^{\bullet} to the minimal generating time for D^b(X) using F^{\bullet} to generate.


    1. Add remark

      Problem 7.1.

      What can be said about the Orlov spectrum of a toric variety (or any X)? (e.g. about gaps, sequences, etc.)

      Known:
      • OrSpec(\mathbb{P}^1)=\{1,2\}
      • On a toric surface, there exists full arbitrary large consecutive sequence.
      • More genrally, by replacing D^b(X) with \mathcal{C}, there exists a notion of OrSpec(\mathcal{C}) and OrSpec(Rep (Q)) (Q is a quiver), computed when Q is of A/D types.

        • Rdim of D^b Sing


          Add remark

          Problem 7.2.

          What about Rouquier dimension or orlov spectrum of D^b(Sing(X))=D^b(Coh(X))/D^b(Perf(X))?


            • Add remark

              Problem 7.3.

              Same questions as above for Fuk(X) where X is compact symplectic.

                •     There are two known ways to bound Rouquier dimension of a category
                  1. bound length of a resolution of a diagonal
                  2. the minimal depth of a presentation of \mathcal{C} as hocolim (cats with Rouquier dimension Q). Meaning if F:I^{\leq}\to Cat diagram with Rdim F(i)=0, then Rdim(hocolim F i)\leq depth(I)

                  Add remark

                  Problem 7.4.

                  Can (1) be used to bound (2) or vice-versa?


                    • Add remark

                      Problem 7.5.

                      What can be said about the Rouquier dimension of non-toric varieties and/for their symplectic mirrors?
                      • Toric varieties of dim n \longleftrightarrow(mirror) (T^*T^n,some stop)
                      • What known symplecticly for (T^*M,stops) can be find in (Bai-Cote, Hanlon-Hicks-Lazarev, Favero-Huang)
                      e.g. is there a symplectic (X^{2n},f) such that Rdim \ Fuk(X,f)\lneq n? (for instance by finding (X,f) mirror to a singular Y with Rdim(D^b(Y))>dim(Y))

                          Cite this as: AimPL: Syzygies and mirror symmetry, available at http://aimpl.org/syzygyms.