Grassmannian.info

References

Chronological bibliography. Last updated on 28 October 2026.

Topic all topics Exceptional collections HPD and Lefschetz collections k-Fanoness Projective duality Quantum cohomology

1. Beilinson, Alexander. Coherent sheaves on $𝐏^n$ and problems in linear algebra. Funct. Anal. Appl. 12 (1978), 214–216. MR509388.
   Full collections on projective spaces.
2. Kapranov, M. M. On the derived categories of coherent sheaves on some homogeneous spaces. Invent. Math. 92 (1988), 479–508. MR939472.
   Schur-bundle collections on ordinary Grassmannians; spinor collections on quadrics.
3. J. M. Landsberg; L. Manivel. The projective geometry of Freudenthal's magic square. J. Algebra 239 (2001), 477–512. arXiv:math/9908039.
   Cubic and quartic dual hypersurfaces in the magic square.
4. Samokhin, A. V. The derived category of coherent sheaves on $LG_3^{\mathbb C}$. Russian Math. Surveys 56 (2001), 592–594. MR1859740.
   A full collection on the Lagrangian Grassmannian LG(3,6).
5. W. Fulton; C. Woodward. On the quantum product of Schubert classes. J. Algebraic Geom. 13 (2004), 641–661. arXiv:math/0112183.
   The quantum Chevalley formula used for the exact small-quantum certificates (Theorem 10.1).
6. Evgueni Tevelev. Projective Duality and Homogeneous Spaces. Encyclopaedia of Mathematical Sciences 133, Springer (2005). arXiv:math/0112028.
   Classical projective duality, dual defects, and homogeneous examples.
7. Kuznetsov, A. G. Hyperplane sections and derived categories. Izv. Math. 70 (2006), 447–547. MR2238172.
   Linear sections, the spinor tenfold, and the G₂ Grassmannian.
8. Alexander Kuznetsov. Homological projective duality for Grassmannians of lines. Preprint (2006). arXiv:math/0610957.
   Noncommutative Pfaffian HP duals of Gr(2,6) and Gr(2,7).
9. Alexander Kuznetsov. Homological Projective Duality. Publ. Math. IHÉS 105 (2007), 157–220. arXiv:math/0507292.
   The general theory of homological projective duality and linear embeddings.
10. Kuznetsov, Alexander. Exceptional collections for Grassmannians of isotropic lines. Proc. Lond. Math. Soc. (3) 97 (2008), 155–182. MR2434094.
    Lefschetz collections for ordinary and isotropic two-plane Grassmannians.
11. Pierre-Emmanuel Chaput; Laurent Manivel; Nicolas Perrin. Quantum cohomology of minuscule homogeneous spaces III : semi-simplicity and consequences. Canad. J. Math. 62 (2010), 1246–1263. arXiv:0710.1224.
    Semisimplicity at q=1 for minuscule and cominuscule homogeneous spaces.
12. Chaput, P. E.; Perrin, N. On the quantum cohomology of adjoint varieties. Proc. Lond. Math. Soc. (3) 103 (2011), 294–330. arXiv:0904.4824. MR2821244.
    Semisimplicity for adjoint non-coadjoint varieties; table of small quantum cohomology on p. 326.
13. Polishchuk, Alexander; Samokhin, Alexander. Full exceptional collections on the Lagrangian Grassmannians $LG(4,8)$ and $LG(5,10)$. J. Geom. Phys. 61 (2011), 1996–2014. MR2822466.
    Full collections on LG(4,8) and LG(5,10).
14. Anton Fonarev. On minimal Lefschetz decompositions for Grassmannians. Izv. Math. 77 (2013), 1044–1065. arXiv:1108.2292.
    Full rectangular Lefschetz collections in the coprime case, and further constructions.
15. Faenzi, Daniele; Manivel, Laurent. On the derived category of the Cayley plane II. Proc. Amer. Math. Soc. 143 (2015), 1057–1074. MR3293722.
    Fullness of a Lefschetz collection on the Cayley plane.
16. Kuznetsov, Alexander; Polishchuk, Alexander. Exceptional collections on isotropic Grassmannians. J. Eur. Math. Soc. (JEMS) 18 (2016), 507–574. MR3463417.
    Exceptional blocks and collections of maximal length in the classical types.
17. John Alexander Cruz Morales; Alexander Kuznetsov; Anton Mellit; Nicolas Perrin; Maxim Smirnov. On quantum cohomology of Grassmannians of isotropic lines, unfoldings of $A_n$-singularities, and Lefschetz exceptional collections. Ann. Inst. Fourier 69 (2019), 955–991. arXiv:1705.01819.
    Big quantum cohomology, singularities, and Lefschetz collections for IGr(2,2n).
18. Jørgen Vold Rennemo; Ed Segal. Hori-mological projective duality. Duke Math. J. 168 (2019), 2127–2205. arXiv:1609.04045.
    Pfaffian HPD in odd ambient dimension.
19. Lyalya Guseva. On the derived category of $IGr(3,8)$. Sb. Math. 211 (2020), 922–955. arXiv:1810.07777.
    A full Lefschetz collection on IGr(3,8).
20. Alexander Kuznetsov; Maxim Smirnov. On residual categories for Grassmannians. Proc. Lond. Math. Soc. 120 (2020), 617–641. arXiv:1802.08097.
    Residual categories and the relation with quantum cohomology.
21. Pieter Belmans. An update for Grassmannian.info: quantum spectra for horospherical varieties. Research note, 24 February 2021.
    Additional semisimple cases detected by quantum multiplication. Exact certificates for the cases used here are included with the site.
22. Bradley Doyle. Homological Projective Duality for the Plücker embedding of the Grassmannian. Preprint (2021). arXiv:2110.10589.
    A matrix-factorization HPD for Gr(k,N) when gcd(k,N)=1.
23. Alexander Kuznetsov; Alexander Perry. Homological projective duality for quadrics. J. Algebraic Geom. 30 (2021), 457–476. arXiv:1902.09832.
    HPD for smooth quadrics with their spinor Lefschetz structures.
24. Alexander Kuznetsov; Maxim Smirnov. Residual categories for (co)adjoint Grassmannians in classical types. Compos. Math. 157 (2021), 1172–1206. arXiv:2001.04148.
    Full collections and residual categories for OGr(2,2n).
25. Maxim Smirnov. On the derived category of the adjoint Grassmannian of type F. Accepted for Algebraic Geometry and Physics (volume in honour of Yuri I. Manin); preprint revised 2026. arXiv:2107.07814.
    A full rectangular collection on F₄/P₁.
26. Carolina Araujo; Roya Beheshti; Ana-Maria Castravet; Kelly Jabbusch; Svetlana Makarova; Enrica Mazzon; Libby Taylor; Nivedita Viswanathan. Higher Fano Manifolds. Rev. Un. Mat. Argentina 64 (2022), 103–125. arXiv:2110.02339.
    Theorems 1.3 and 1.4 classify 2-Fano and 3-Fano homogeneous spaces of Picard rank one.
27. Anton Fonarev. On the bounded derived category of $𝖨𝖦𝗋(3, 7)$. Transform. Groups 27 (2022), 89–112. arXiv:1804.06946.
    A full minimal Lefschetz collection on the odd symplectic Grassmannian IGr(3,7).
28. Anton Fonarev. Full exceptional collections on Lagrangian Grassmannians. Int. Math. Res. Not. 2022 (2), 1081–1122. arXiv:1911.08968.
    Fullness of the Kuznetsov–Polishchuk collections on all Lagrangian Grassmannians.
29. Gonzales, Richard; Pech, Clélia; Perrin, Nicolas; Samokhin, Alexander. Geometry of horospherical varieties of Picard rank one. Int. Math. Res. Not. 2022 (12), 8916–9012. arXiv:1803.05063.
    Quantum cohomology and exceptional collections on horospherical varieties of Picard rank one.
30. Pieter Belmans; Alexander Kuznetsov; Maxim Smirnov. Derived categories of the Cayley plane and the coadjoint Grassmannian of type F. Transform. Groups 28 (2023), 9–34. arXiv:2005.01989.
    A full Lefschetz collection on F₄/P₄ and the residual categories of F₄/P₄ and the Cayley plane.
31. Warren Cattani. On the derived category of IGr(3, 9). Preprint (2023). arXiv:2305.06867.
    A minimal Lefschetz collection on the odd symplectic Grassmannian IGr(3,9).
32. Hiroshi Iritani; Yuki Koto. Quantum cohomology of projective bundles. Preprint (2023). arXiv:2307.03696.
    A projective-bundle decomposition theorem, with applications to big quantum cohomology.
33. Anton Fonarev. Derived Categories of Grassmannians: a Survey. Russian Math. Surveys 79 (2024), 807–845. arXiv:2407.07455.
    An overview of constructions and explicit Lefschetz shapes through 2024.
34. Pirozhkov, Dmitrii. Towards homological projective duality for $Gr(2, 2n)$. Preprint (2024). arXiv:2402.14754.
    A candidate HPD construction for Gr(2,2n); the equivalence is conjectural.
35. Alexander Samokhin; Wilberd van der Kallen. Highest weight category structures on $\operatorname{rep}(B)$ and full exceptional collections on generalized flag varieties over $\mathbb Z$. Preprint (2024). arXiv:2407.13653.
    Full exceptional collections on all split flag schemes G/P, including every Grassmannian in the table.
36. Vladimiro Benedetti; Daniele Faenzi; Maxim Smirnov. Derived category of the spinor 15-fold. J. Algebra 677 (2025), 460–496. arXiv:2310.01090.
    A full collection on the spinor 15-fold; a proposed analogue for the Freudenthal variety.
37. Sergey Galkin; Naichung Conan Leung; Changzheng Li; Rui Xiong; appendix by Pieter Belmans and Markus Reineke. A-D-E diagrams, Hodge–Tate hyperplane sections and semisimple quantum cohomology. Preprint (2025). arXiv:2509.01101.
    Theorem 1.2 gives an obstruction from the Fano index and Betti numbers. Appendix A is by Belmans–Reineke.
38. Novikov, A. A. Minimal Lefschetz collections on isotropic Grassmannians $IGr(3,2n)$. Russian Math. Surveys 80 (2025), 543–545. MR4961293.
    Full minimal Lefschetz collections on every IGr(3,2n), rectangular for n congruent to 0 or 2 modulo 3.
39. Nicolas Perrin; Maxim Smirnov. On the big quantum cohomology of coadjoint varieties. Selecta Math. 31 (2025), article 30. arXiv:2112.12436.
    Big semisimplicity for coadjoint varieties; small semisimplicity for IGr(3,8) (Remark 1.2).
40. Lyalya Guseva; Alexander Novikov. Full exceptional collections on the symplectic isotropic Grassmannians. Advances in Mathematics (2026), article 111211. arXiv:2505.13005.
    The Kuznetsov–Polishchuk collections on all symplectic isotropic Grassmannians are full and consist of vector bundles.
41. Daniel Holmes; Giosuè Muratore. Effective Gromov–Witten computations for GKM spaces and a semisimple application. Preprint (2026). arXiv:2610.37273.
    Semisimplicity of small quantum cohomology for E₇/P₂, E₇/P₄, E₇/P₅ and E₈/P₆ (Theorem 1.2), completing the exceptional cases.
42. Changzheng Li; Zhaoyang Liu; Tony Yue Yu; Chi Zhang. Convergence and semisimplicity of big quantum cohomology of flag varieties. Preprint (2026). arXiv:2610.28418.
    Convergence near the large-radius limit and generic semisimplicity of big quantum cohomology for every G/P.