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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.