Geodesic
Resolutions of toric subvarieties by line bundles and applications
Forum of Mathematics, Pi, Tome 12 (2024)

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Given any toric subvariety Y of a smooth toric variety X of codimension k, we construct a length k resolution of ${\mathcal O}_Y$ by line bundles on X. Furthermore, these line bundles can all be chosen to be direct summands of the pushforward of ${\mathcal O}_X$ under the map of toric Frobenius. The resolutions are built from a stratification of a real torus that was introduced by Bondal and plays a role in homological mirror symmetry.As a corollary, we obtain a virtual analogue of Hilbert’s syzygy theorem for smooth projective toric varieties conjectured by Berkesch, Erman and Smith. Additionally, we prove that the Rouquier dimension of the bounded derived category of coherent sheaves on a toric variety is equal to the dimension of the variety, settling a conjecture of Orlov for these examples. We also prove Bondal’s claim that the pushforward of the structure sheaf under toric Frobenius generates the derived category of a smooth toric variety and formulate a refinement of Uehara’s conjecture that this remains true for arbitrary line bundles. 
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Andrew Hanlon; Jeff Hicks; Oleg Lazarev. Resolutions of toric subvarieties by line bundles and applications. Forum of Mathematics, Pi, Tome 12 (2024). doi: 10.1017/fmp.2024.21

[Abo09] Abouzaid, M., ‘Morse homology, tropical geometry, and homological mirror symmetry for toric varieties’, Selecta Math. 15(2) (2009), 189–270.Google Scholar | DOI

[Ach15] Achinger, P., ‘A characterization of toric varieties in characteristic p’, Int. Math. Res. Not. 2015(16) (2015), 6879–6892.Google Scholar | DOI

[Alt+20] Altmann, K., Buczyński, J., Kastner, L. and Winz, A.-L., ‘Immaculate line bundles on toric varieties’, Pure Appl. Math. 16(4) (2020), 1147–1217.Google Scholar

[BC23] Bai, S. and Côté, L., ‘On the Rouquier dimension of wrapped Fukaya categories and a conjecture of Orlov’, Compos. Math. 159(3) (2023), 437–487.Google Scholar | DOI

[BDM19] Ballard, M. R., Duncan, A., and Mcfaddin, P. K., ‘The toric Frobenius morphism and a conjecture of Orlov’, Eur. J. Math. 5(3) (2019), 640–645.Google Scholar | DOI

[BE24] Brown, M. K. and Erman, D., ‘Tate resolutions on toric varieties’, J. Eur. Math. Soc. (2024), to appear.Google Scholar | DOI

[Bei78] Beilinson, A. A., ‘Coherent sheaves on and problems of linear algebra’, Funct. Anal. Appl. 12(3) (1978), 214–216.Google Scholar | DOI

[BES20] Berkesch, C., Erman, D. and Smith, G. G., ‘Virtual resolutions for a product of projective spaces’, Algebr. Geom. 7(4) (2020), 460–481.Google Scholar

[BF12] Ballard, M. and Favero, D., ‘Hochschild dimensions of tilting objects’, Int. Math. Res. Not. 2012(11) (2012), 2607–2645.Google Scholar

[BFK19] Ballard, M., Favero, D. and Katzarkov, L., ‘Variation of geometric invariant theory quotients and derived categories’, J. Reine Angew. Math. 2019(746) (2019), 235–303.Google Scholar | DOI

[BG03] Bruns, W. and Gubeladze, J., ‘Divisorial linear algebra of normal semigroup rings’, Algebr. Represent. Theory 6 (2003), 139–168.Google Scholar | DOI

[BH09] Borisov, L. and Hua, Z., ‘On the conjecture of King for smooth toric Deligne–Mumford stacks’, Adv. Math. 221(1) (2009), 277–301.Google Scholar | DOI

[Bøg98] Bøgvad, R., ‘Splitting of the direct image of sheaves under the Frobenius’, Proc. Amer. Math. Soc. 126(12) (1998), 3447–3454.Google Scholar | DOI

[Bon06] Bondal, A., ‘Derived categories of toric varieties’, in Oberwolfach Reports: Convex and Algebraic Geometry, ed. by Altmann, K., Batyrev, V. V. and Teissier, B., vol. 3 (EMS Press, 2006), 284–286.Google Scholar

[BPS01] Bayer, D., Popescu, S., and Sturmfels, B., ‘Syzygies of unimodular Lawrence ideals’, J. Reine Angew. Math. 5344 (2001), 169–186.Google Scholar

[Bru05] Bruns, W., ‘Conic divisor classes over a normal monoid algebra’, Commut. Algebra Algebr. Geom. (2005), 63–71.Google Scholar | DOI

[BS22] Brown, M. K. and Sayrafi, M., ‘A short resolution of the diagonal for smooth projective toric varieties of Picard rank 2’, Algebra & Number Theory 18(10) (2024), 1923–1943 Google Scholar | DOI

[BT09] Bernardi, A. and Tirabassi, S., ‘Derived categories of toric Fano 3-folds via the Frobenius morphism’, Matematiche 64(2) (2009), 117–154.Google Scholar

[CK08] Canonaco, A. and Karp, R. L., ‘Derived autoequivalences and a weighted Beilinson resolution’, J. Geom. Phys. 58(6) (2008), 743–760.Google Scholar | DOI

[CMR10] Costa, L. and Miró-Roig, R. M., ‘Frobenius splitting and derived category of toric varieties’, Illinois J. Math. 54(2) (2010), 649–669.Google Scholar | DOI

[CMR12] Costa, L. and Miró-Roig, R. M., ‘Derived category of toric varieties with small Picard number’, Open Math. 10(4) (2012), 1280–1291.Google Scholar

[DLM09] Dey, A., Lason, M. and Michalek, M., ‘Derived Category of toric varieties with Picard number three’, Matematiche 64(2) (2009), 99–116.Google Scholar

[EES15] Eisenbud, D., Erman, D. and Schreyer, F.-O., ‘Tate resolutions for products of projective spaces’, Acta Math. Vietnam. 40 (2015), 5–36.Google Scholar | DOI

[Efi14] Efimov, A. I., ‘Maximal lengths of exceptional collections of line bundles’, J. London Math. Soc. 90(2) (2014), 350–372.Google Scholar | DOI

[Fan+11] Fang, B., Liu, C.-C. M., Treumann, D. and Zaslow, E., ‘A categorification of Morelli’s theorem’, Invent. Math. 186(1) (2011), 79–114.Google Scholar | DOI

[Fan+12] Fang, B., Liu, C.-C. M., Treumann, D. and Zaslow, E., ‘T-duality and homological mirror symmetry for toric varieties’, Adv. Math. 229(3) (2012), 1873–1911.Google Scholar | DOI

[Fan+14] Fang, B., Liu, C.-C. M., Treumann, D. and Zaslow, E., ‘The coherent–constructible correspondence for toric Deligne–Mumford stacks’, Int. Math. Res. Not. 2014(4) (2014), 914–954.Google Scholar | DOI

[FH22] Favero, D. and Huang, J., ‘Homotopy path algebras’, Preprint, 2022, .Google Scholar | arXiv

[FH23] Favero, D. and Huang, J., ‘Rouquier dimension is Krull dimension for normal toric varieties’, Eur. J. Math. 9(4) (2023), 91.Google Scholar | DOI

[FMS19] Faber, E., Muller, G. and Smith, K. E., ‘Non-commutative resolutions of toric varieties’, Adv. Math. 351 (2019), 236–274.Google Scholar | DOI

[For98] Forman, R., ‘Morse theory for cell complexes’, Adv. Math. 134(1) (1998), 90–145.Google Scholar | DOI

[GPS24a] Ganatra, S., Pardon, J. and Shende, V., ‘Microlocal Morse theory of wrapped Fukaya categories’, Ann. Math. 199(3) (2024), 943–1042.Google Scholar | DOI

[GPS24b] Ganatra, S., Pardon, J. and Shende, V., ‘Sectorial descent for wrapped Fukaya categories’, J. Amer. Math. Soc. 37 (2024), 499–635.Google Scholar

[GS15] Geraschenko, A. and Satriano, M., ‘Toric stacks I: The theory of stacky fans’, Trans. Amer. Math. Soc. 367(2) (2015), 1033–1071.Google Scholar | DOI

[HH22] Hanlon, A. and Hicks, J., ‘Aspects of functoriality in homological mirror symmetry for toric varieties’, Adv. Math. 401 (2022), 108317.Google Scholar | DOI

[HHL23] Hanlon, A., Hicks, J. and Lazarev, O., ‘Relating categorical dimensions in topology and symplectic geometry’, Preprint, 2023, .Google Scholar | arXiv

[HK00] Hu, Y. and Keel, S., ‘Mori dream spaces and GIT’, Michigan Math. J. 48(1) (2000), 331–348.Google Scholar | DOI

[HP06] Hille, L. and Perling, M., ‘A counterexample to King’s conjecture’, Compos. Math. 142(6) (2006), 1507–1521.Google Scholar | DOI

[Kaw06] Kawamata, Y., ‘Derived categories of toric varieties’, Michigan Math. J. 54(3) (2006), 517–536.Google Scholar | DOI

[Kaw13] Kawamata, Y., ‘Derived categories of toric varieties II’, Michigan Math. J. 62(2) (2013), 353–363.Google Scholar | DOI

[Kon95] Kontsevich, M., ‘Homological algebra of mirror symmetry’, in Proceedings of the International Congress of Mathematicians (Springer, 1995), 120–139.Google Scholar | DOI

[Kuw20] Kuwagaki, T., ‘The nonequivariant coherent-constructible correspondence for toric stacks’, Duke Math. J. 169(11) (2020), 2125 –2197.Google Scholar | DOI

[LM11] Lasoń, M. and Michałek, M., ‘On the full, strongly exceptional collections on toric varieties with Picard number three’, Collectanea Mathematica 62(3) (2011), 275–296.Google Scholar | DOI

[Orl09] Orlov, D., ‘Remarks on generators and dimensions of triangulated categories’, Mosc. Math. J. 9(1) (2009), 513–519.Google Scholar

[OU13] Ohkawa, R. and Uehara, H., ‘Frobenius morphisms and derived categories on two dimensional toric Deligne–Mumford stacks’, Adv. Math. 244 (2013), 241–267.Google Scholar | DOI

[PN17] Prabhu-Naik, N., ‘Tilting bundles on toric Fano fourfolds’, J. Algebra 471 (2017), 348–398.Google Scholar | DOI

[Rou08] Rouquier, R., ‘Dimensions of triangulated categories’, J. K-Theory 1(2) (2008), 193–256.Google Scholar | DOI

[Skö06] Sköldberg, E., ‘Morse theory from an algebraic viewpoint’, Trans. Amer. Math. Soc. 358(1) (2006), 115–129.Google Scholar | DOI

[Sta22] The Stacks project authors, ‘The Stacks project’, https://stacks.math.columbia.edu. 2022.Google Scholar

[Tho00] Thomsen, J. F., ‘Frobenius direct images of line bundles on toric varieties’, J. Algebra 226(2) (2000), 865–874.Google Scholar | DOI

[Ueh14] Uehara, H., ‘Exceptional collections on toric Fano threefolds and birational geometry’, Internat. J. Math. 25(07) (2014), 1450072.Google Scholar | DOI

[Yan21] Yang, J., ‘Virtual resolutions of monomial ideals on toric varieties’, Proc. Amer. Math. Soc. Ser. B 8(9) (2021), 100–111.Google Scholar | DOI

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