Equivalences between Calabi–Yau manifolds and roofs of projective bundles


Marco Rampazzo


birational transformations, derived equivalences, Kanemitsu, Calabi-Yau


It is conjectured that many birational transformations, called K-inequalities, have a categorical counterpart in terms of an embedding of derived categories. In the special case of simple K-equivalence (or more generally K-equivalence), a derived equivalence is expected: we propose a method to prove derived equivalence for a wide class of such cases. This method is related to the construction of roofs of projective bundles introduced by Kanemitsu. Such roofs can be related to candidate pairs of derived equivalent, L-equivalent and non isomorphic Calabi–Yau varieties, we prove such claims in some examples of this construction.

In the same framework, we show that a similar approach applies to prove derived equivalence of pairs of Calabi–Yau fibrations, we provide some working examples and we relate them to gauged linear sigma model phase transitions.

Author Biography

Marco Rampazzo

University of Stavanger
Faculty of Science and Technology, Department of Mathematics and Physics


[Arr96] Arrondo. Subvarieties of Grassmannians. 1996. http://www.mat.ucm.es/arrondo/trento.pdf

[ADM19] Nicholas Addington, Will Donovan, and Ciaran Meachan. Mukai flops and P-twists. Journal für die Reine und Angewandte Mathematik. 748, 227-240 (2019).


[ADS15] Nicolas Addington, Will Donovan, Ed Segal. The Pfaffian-Grassmannian equivalence revisited. Alg. Geom. 2(3):332-364, 2015. e-Print:ArXiv:1401.3661.


[Bal06] Vikraman Balaji. Lectures on principal bundles. 3rd cycle. Guanajuato (Mexique), 2006, pp.17. cel-00392133

[Bei78] Alexander Beilinson. Coherent sheaves on Pn and problems in linear algebra (Russian) Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 68-69.


[Böh06] Christian Böhning. Derived categories of coherent sheaves on rational homogeneous manifolds. Doc. Math. 11 (2006), 261-331 (electronic). MR 2262935 (2008f:14032

[Bon89] Alexiei Bondal. Representations of associative algebras and coherent sheaves. Izv. Akad. Nauk SSSR, Ser. Mat., 53 (1989), 25-44 translation in Math. USSR-Izv., 34 (1990), 23-42.


[Bo94] Francis Borceux. Handbook of Categorical Algebra, Vol. 1: Basic Category Theory. Cambridge University Press (1994)


[Bor91] Armand Borel. Linear Algebraic Groups. Springer- Verlag New York. 1991.


[Bri02] Tom Bridgeland. Flops and derived cate- gories. Invent. math. 147, 613-632 (2002).


[BC08] Lev Borisov, Andrei Căldăraru. The Pfaffian- Grassmannian derived equivalence J. Algebraic Geom. 18 (2009), 201-222


[BCP20] Lev A. Borisov, Andrei Caldararu, Alexander Perry. Intersections of two Grassmannians in P9. Journal für die reine und angewandte Mathematik, Volume 2020, Issue 760, Pages 133-162, eISSN 1435-5345, ISSN 0075- 4102, e- Print:arXiv:1707.00534.


[BFM19] Marcello Bernardara, Enrico Fatighenti, Laurent Manivel. Nested varieties of K3 type. e-Print: arXiv:1912.03144

[BFMT17] Vladimiro Bemnedetti, Sara A. Filippini, Laurent Manivel, Fabio Tanturri. Orbital degeneraci loci and applications. Annali della Scuola normale superiore di Pisa, Classe di scienze. DOI: 10.2422/2036- 2145.201804 016.

[BK] Alexey I. Bondal, Mikhail M. Kapranov. Representable functors, Serre functors, and mutations. Izv. Akad. Nauk SSSR Ser. Mat., 1989, Volume 53, Issue 6, Pages 1183-1205 (Mi izv1153)

[BM02] Tom Bridgeland, Antony Maciocia. Fourier-Mukai transforms for K3 and elliptic fibrations. Journal: J. Algebraic Geom. 11 (2002), 629-657


[BO01] Alexey I. Bondal, Dmitri Orlov. Reconstruction of a Va- riety from the Derived Category and Groups of Autoequivalences. Compositio Mathematica 125, 327-344 (2001).


[BO02] Alexei Bondal and Dmitri Orlov. Derived categories of coherent sheaves. In International Congress of Mathematicians, page 47, 2002.

[BO95] Alexei Bondal, Dmitri Orlov. Semiorthogonal de- composition for algebraic varieties. e-Print:arXiv:alg- geom/9506012v1. 1995.

[BV03] Alexey I. Bondal, Michel van den Bergh. (2003). Gen- erators and representability of functors in commutative and noncommutative geometry. Moscow Mathematical Journal, 3(1), 1-36.


[Cal05] Andrei Caldararu. Derived categories of sheaves: a skim- ming. e-Print:arXiv:https://arxiv.org/abs/math/0501094

[Cor97] John F. Cornwell. Group Theory in Physics: An Introduction. Academic Press, 1997.

[CDGP91] Philip Candelas, Xenia de la Ossa, Paul S. Green, Linda Parkes. A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory. Nuclear PhysicsB359 (1991) 21-74 North-Holland.


[CFFGKS08] Ionut Ciocan-Fontanine, David Favero, Jérémy Guéré, Bumsig Kim, Mark Shoemaker. Fundamental Factorization of a GLSM, Part I: Construction. e- Print:ArXiv:1802.05247. (2008).

[CGP15] Brian Conrad, Ofer Gabber, Gopal Prasad. (2015). Pseudo-reductive Groups (2nd ed., New Mathematical Monographs). Cambridge: Cambridge University Press.


[CLNS18] Antoine Chambert-Loir, Johannes Nicaise, Julien Sebag. Motivic integration. Progress in Mathematics, Volume 325.


[Don77] Ron Y. Donagi. On the geometry of Grass-mannians. Duke Math. J. 44 (1977), no. 4, 795-837. doi:10.1215/S0012-7094-77-04436-2. https://projecteuclid.org/euclid.dmj/1077312543


[DK20] Olivier Debarre, Alexander Kuznetsov. Gushel-Mukai varieties: intermediate Jacobians. e-Print:arXiv:2002.05620 (2020)


[DR19] Ivan Dimitrov, Mike Roth. Intersection multiplicity one for classical groups. Transformation Groups (2019).


[Efi18] Alexander I. Efimov. Some remarks on L-equivalence of algebraic varieties. Selecta Math. (N.S.), 24:4 (2018), 3753-3762.


[Fon19] Anton Fonarev. Full exceptional collections on La- grangian Grassmannians. e-Print:arXiv:1911.08968

[FI73] Robert M. Fossum, Birger Iversen. On Picard groups of algebraic fibre spaces. Journal of Pure and Applied Algebra, Volume 3, Issue 3, September 1973, Pages 269-280


[Ful98] William Fulton. Intersection theory. (1998). Springer- Verlag, Ergebnisse der Mathematik und ihrer Grenzge- biete, 3. Folge, Volume 2

[GM03] Sergei I. Gelfand, Yuri I. Manin. methods of ho- mological algebra. Springer Monographs in Math- ematics, Ed. 2, Springer-Verlag, Berlin, 2003.


[GS14] Sergei Galkin, Evgeny Shinder. The Fano variety of lines and rationality problem for a cubic hypersurface. e-Print:arXiv:1405.5154

[GS19] Daniel R. Grayson, Michael E. Stillman, Macaulay2- a software system for research in alge- braic geometry (version 1.12), available at: http://www.math.uiuc.edu/Macaulay2/, 2019.

[Hal04] Mihai Halic. Quotients of affine spaces for actions of reductive groups. e-Print:arXiv:math/0412278

[Hel62] Sigurdur Helgason. Differential Geometry and Symmet- ric Spaces. 500 pp. Academic Press, New York, 1962. Russian translation, MIR, Moscow, 1964. Published by Amer. Math. Soc. in 2000.

[Huy06] Daniel Huybrechts. Fourier-Mukai Transforms in Al- gebraic Geometry. Oxford Mathematical Monographs, ISSN 0964-9174. Clarendon Press, 2006.


[Huy16] Daniel Huybrechts. Lectures on K3 Surfaces. Cambridge Studies in Advanced Mathematics (2016). Cambridge: Cambridge University Press.


[HL18] Brendan Hassett, Kuan-Wen Lai. Cremona transfor- mations and derived equivalences of K3 surfaces. Compositio Math. 154 (2018) 1508-1533. e-Print: arXiv:1612.07751, 2018.


[MS] Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandharipande, Richard Thomas, Cumrun Vafa, Ravi Vakil, Eric Zaslow. Mirror symmetry. Clay Mathematics Monographs, V. 1. ISBN-10: 0-8218-2955-6

[Har77] Robin Hartshorne. Algebraic Geometry. Graduate Texts in Mathematics, Volume 52, ISSN 0072-5285. Springer Science Business Media, 1977.


[HT07] Kentaro Hori, David Tong. Aspects of Non-Abelian Gauge Dynamics in Two-Dimensional N=(2,2) Theories. JHEP 0705 (2007) 079.


[IMOU16] Atsushi Ito, Makoto Miura, Shinnosuke Okawa, Kazushi Ueda. Calabi-Yau complete intersections in G2-Grassmannians. e-Print: arXiv:1606.04076

[IMOU19] Atsushi Ito, Makoto Miura, Shinnosuke Okawa, Kazushi Ueda. The class of the affine line is a zero divisor in the Grothendieck ring: via G2-Grassmannians. J. Algebraic Geom. 28 (2019), 245-250 https://www.ams.org/journals/jag/2019- 28-02/S1056-3911-2018-00731-3/


[Kan18] Akihiro Kanemitsu. Mukai pairs and simple K - equivalence. e-Print: arXiv:1812.05392, 2018

[Kan19] Akihiro Kanemitsu. Extremal Rays and Nefness of Tan- gent Bundles. Michigan Math. J. Volume 68, Issue 2 (2019), 301-322.


[Kap88] Mikhail Kapranov. On the derived categories of coherent sheaves on some homogeneous spaces. Invent. Math. 92 (1988), no. 3, 479-508.


[Kaw02] Yujiro Kawamata. D-equivalence and K-equivalence. Journal of Differential Geometry, 61(1):147-171, 2002.


[Kaw08] Yujiro Kawamata. Derived categories and birational geometry. Proceedings of Symposia in Pure Mathematics 2009 Volume 80.2


[King94] Alastair King. Moduli of representations of finite dimen- sional algebras. Quarterly J. Math. Oxford 45 (1994) 515-530.


[Klei74] Steven Kleiman. The transversality of a general translate. Compositio Mathematica, tome 28, no 3 (1974), p. 287-297

[Kol96] János Kollár. Rational Curves on Algebraic Varieties. 1996, Springer-Verlag Berlin Heidelberg

[Kuz06b] Alexander Kuznetsov. Homological projective duality for Grassmannians of lines. e-Print: arXiv:math/0610957

[Kuz06] Alexander Kuznetsov. Hyperplane sections and derived categories. Izv. Ross. Akad. Nauk Ser. Mat. 70 (2006), no. 3, 23-128 (Russian)


English translation in Izv. Math. 70 (2006), no. 3, 447-547 MR-2238172


[Kuz07] Alexander Kuznetsov. Homological projective duality. Publ.math.IHES 105, 157-220 (2007).


[Kuz08] Alexander Kuznetsov. Exceptional collections for Grass- mannians of isotropic lines. Proc. London Math. Soc. (3) 97 (2008) 155-182 Ce2008 London Mathematical Society


[Kuz10] Alexander Kuznetsov. Derived categories of cubic fourfolds. "Cohomological and Geometric Approaches to Rationality Problems. New Perspectives" Series: Progress in Mathematics, Vol. 282, Bogomolov, Fedor; Tschinkel, Yuri (Eds.) 2010, X, 314 p. 47 illus., Hard-cover ISBN: 978-0-8176-4933-3 A Birkhauser book

[Kuz18] Alexander Kuznetsov. Derived equivalence of Ito-Miura-Okawa-Ueda Calabi-Yau 3-folds. J. Math. Soc. Japan, Volume 70, Number 3 (2018), 1007-1013.


[KP19] Alexander Kuznetsov, Alexander Perry. Categorical joins. e-Print:arXiv:1804.00144

[Kuz11] Alexander Kuznetsov. Base change for semiorthogonal decompositions. Compositio Math. 147 (2011), 852-876.


[KKM20] Grzegorz Kapustka, Michał Kapustka, Riccardo Moschetti. Equivalence of K3 surfaces from Verra threefolds. Kyoto J. Math., advance publication, 24 September 2020. https://projecteuclid.org/euclid.kjm/1600934411


[KR17] Michał Kapustka, Marco Rampazzo. Torelli problem for Calabi-Yau threefolds with GLSM description. Commu- nications in Number Theory and Physics 13(4) 2019.


[KR20] Michał Kapustka, Marco Rampazzo. Mukai duality via roofs of projective bundles. e-Print:arXiv:2001.06385

[KS18] Alexander Kuznetsov, Evgeny Shinder. Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics. Selecta Math, V. 24 (2018), No. 4, pp. 3475-3500. e-Print: arXiv:1612.07193, 2016.


[Laz04a] Robert Lazarsfeld. Positivity in Algebraic Geometry I. 2004. A Series of Modern Surveys in Mathematics, vol 48. Springer, Berlin, Heidelberg. http://doi-org- 443.webvpn.fjmu.edu.cn/10.1007/978-3-642-18808-4 7

[Laz04b] Robert Lazarsfeld. Positivity in Algebraic Geometry II. 2004. A Series of Modern Surveys in Mathematics, vol 49. Springer, Berlin, Heidelberg.

[Li19] Duo Li. On certain K-equivalent birational maps. Math. Z. 291, 959-969 (2019).


[LL03] Michael Larsen, Valery A. Lunts,Motivic measures and stable birational geometry. Mosc. Math. J. 3 (2003), no. 1, 85-95, 259.


[LMS75] Alexandru T. Lascu, David Mumford, David Bernard Scott. The self-intersection formula and the 'formule- clef'. Math. Proc. Camb. Phil. Soc. (1975), 78, 117 MPCPS 78-10


[LX19] Naichung Conan Leung, Ying Xie. DK Conjecture for Derived Categories of Grassmannian Flips. 2019. e- Print:arXiv:1911.07715

[Man17] Laurent Manivel. Double spinor Calabi-Yau varieties. Épijournal de Géométrie Algébrique, Volume 3 (April 1, 2019) epiga:5333.


[Mil17] James Milne. Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field. Cambridge Studies in Advanced Mathematics. Cambridge University Press. 2017


[Mor19] Hayato Morimura. Derived equivalences for the flops of type C2 and AG via mutation of semiorthogonal decom- position. e-Print:arXiv:1812.06413

[Mit01] Stephen A. Mitchell. Notes on principal bun- dles and classifying spaces. Available at http://math.mit.edu/ mbehrens/18.906spring10/prin.pdf

[Muk87] Shigeru Mukai. Curves, K3 surfaces and Fano 3-folds of genus 10. Algebraic Geometry and Commutative Al- gebra in Honor of Masayoshi NAGATA, Kinokuniya, Tokyo, 1987, pp. 357-377.


[Muk88] Shigeru Mukai. Problems on characterization of the com- plex projective space. Birational Geometry of Algebraic Varieties, Open Problems, Katata the 23rd Int'l Symp., Taniguchi Foundation, 1988, pp. 57-60.

[Muk98] Shigeru Mukai. Duality of polarized K3 surfaces. Pro- ceedings of Euroconference on Algebraic Geometry, (K. Hulek and M. Reid ed.), Cambridge University Press, 1998, 107122.


[Mum12] David Mumford. Abelian varieties. Tata Institute of Fun- damental Research Publications Volume: 13; 2012

[MFK94] David Mumford, John Fogarty, Frances Kirwan. Geometric invariant theory. Springer-Verlag Berlin Heidelberg. 1994.


[MM86] T. Matsusaka, David Mumford. Two fundamental theorems on deformations of polarized varieties. Amer. J. Math., 86, pp. 668-684.


[Nam03] Yoshinori Namikawa. Mukai flops and derived categories. J. reine angew. Math. 560 (2003), 65-76


[Nor82] Madhav V. Nori. The fundamental group-scheme. Proc. Indian Acad. Sci. (Math. Sci.) 91, 73-122 (1982).


[Occ99] Gianluca Occhetta. Extremal rays of smooth projective varieties. PhD Thesis, 1999. Available at https://core.ac.uk/download/pdf/150080249.pdf

[Orl92] Dmitri Orlov. Projective bundles, monoidal transforma- tions, and derived categories of coherent sheaves. Izv. Akad. Nauk SSSR Ser. Mat., 56 (1992): 852-862

English transl., Russian Acad. Sci. Izv. Math., 41 (1993): 133-141.


[Orl03] Dmitri Orlov. Triangulated categories of singularities and equivalences between Landau-Ginzburg models. Sbornik: Mathematics 197.12 (2006): 1827.00-224.


[Ogu01] Keiji Oguiso. K3 surfaces via almost-primes. Math. Res. Lett., 9(1):47-63, 2002.


[Orl97] Dmitri O. Orlov. Equivalences of derived categories and K3 surfaces. J. Math. Sci. 84 (1997), 1361- 1381.


[Orl03] Dmitri O. Orlov. Derived categories of coherent sheaves and equivalences between them. Russian Math. Surveys 58:3 511-591.


[Ott88] Giorgio Ottaviani. Spinor bundles on quadrics. Transactions of the American mathematical society, volume 307, number 1, May 1988.


[Ott95] Giorgio Ottaviani. Rational Homogeneous Varieties. Course notes in algebraic geometry for SMI, Cortona, Italy. http://web.math.unifi.it/users/ottavian/rathomo/rathomo.pdf, 1995.

[OR17] John C. Ottem, Jørgen V. Rennemo. A counterexample to the birational Torelli problem for Calabi-Yau threefolds. J. London Math. Society 2017 DOI:10.1112/jlms.12111. e-Print: arXiv:1706.09952, 2017.


[ORS20] Gianluca Occhetta, Luis E. Solá Conde, Eleonora A. Romano. Manifolds with two projective bundle structures. 2020. e-Print:arXiv:2001.06215

[Rød98] Einar A. Rødland. The Pfaffian Calabi-Yau, its Mirror, and their Link to the Grassmannian G(2,7). Compositio Mathematica (2000) 122: 135. 2000.


[RS17] Jørgen Vold Rennemo Ed Segal. Hori-mological projective duality. Duke Math. J. 168 (2019), no. 11, 2127-2205. https://projecteuclid.org/euclid.dmj/1563328950


[RS00] Kristian Ranestad, Frank-Olaf Schreyer. Varieties of sums of powers. J. Reine Angew. Math. 525 (2000) 147-181


[Sam06] Alexander Samokhin. Some remarks on the derived cate- gories of coherent sheaves on homogeneous spaces. Jour- nal of the London Mathematical Society 76(1) 2007.


[Seg15] Ed Segal. A new 5-fold flop and derived equivalence. Bulletin of the London Mathematical Society 48(3). 2015.


[Shi12] Ian Shipman. (2012). A geometric approach to Orlov's theorem. Compositio Mathematica, 148(5), 1365-1389.


[Shi] Evgeny Shinder. Derived Categories: Lecture 2. https://e-shinder.staff.shef.ac.uk/Mainz-Lecture-2.pdf

[Toen11] Bertrand Toën. Lectures on DG-categories. Topics in Al- gebraic and Topological K-Theory (2011) 243-301


[Ued19] Kazushi Ueda. G2−Grassmannians and derived equivalences. Manuscripta Mathematica. 159(3−4), 549−559 (2019).


[Voi10] Claire Voisin. Hodge theory and complex algebraic geometry. Cambridge University Press. 2002. ISBN:9780511615344.


[Wey03] Jerzy Weyman. Cohomology of vector bundles and syzy- gies. Cambridge University Press. 2003.


[Wit93] Edward Witten. Phases of N=2 Theories In Two Dimensions. Nucl.Phys.B403:159-222,1993


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