We introduce a notion of homological projective duality for smooth algebraic varieties in dual projective spaces, a homological extension of the classical projective duality. If algebraic varieties X and Y in dual projective spaces are homologically projectively dual, then we prove that the orthogonal linear sections of X and Y admit semiorthogonal decompositions with an equivalent nontrivial component. In particular, it follows that triangulated categories of singularities of these sections are equivalent. We also investigate homological projective duality for projectivizations of vector bundles.
@article{PMIHES_2007__105__157_0,
author = {Kuznetsov, Alexander},
title = {Homological projective duality},
journal = {Publications Math\'ematiques de l'IH\'ES},
volume = {106},
year = {2007},
pages = {157-220},
doi = {10.1007/s10240-007-0006-8},
mrnumber = {2354207},
zbl = {1131.14017},
language = {en},
url = {http://dml.mathdoc.fr/item/PMIHES_2007__105__157_0}
}
Kuznetsov, Alexander. Homological projective duality. Publications Mathématiques de l'IHÉS, Tome 106 (2007) pp. 157-220. doi : 10.1007/s10240-007-0006-8. http://gdmltest.u-ga.fr/item/PMIHES_2007__105__157_0/
1. , Representations of associative algebras and coherent sheaves (Russian), Izv. Akad. Nauk SSSR, Ser. Mat., 53 (1989), 25-44 | MR 992977 | Zbl 0692.18002
2. and , Representable functors, Serre functors, and reconstructions (Russian), Izv. Akad. Nauk SSSR, Ser. Mat., 53 (1989), 1183-1205, 1337; translation in Math. USSR-Izv., 35 (1990), 519-541. | MR 1039961 | Zbl 0703.14011
3. A. Bondal and D. Orlov, Semiorthogonal decomposition for algebraic varieties, preprint math.AG/9506012.
4. and , Derived categories of coherent sheaves, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pp. 47-56, Higher Ed. Press, Beijing, 2002. | MR 1957019 | Zbl 0996.18007
5. A. Bondal and D. Orlov, private communication.
6. , , Reconstruction of a variety from the derived category and groups of autoequivalences, Compos. Math., 125 (2001), 327-344 | MR 1818984 | Zbl 0994.18007
7. and , Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J., 3 (2003), 1-36, 258. | MR 1996800 | Zbl pre02069670
8. , Residues and Duality. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne., Springer, Berlin, New York (1966) | MR 222093
9. K. Hori and C. Vafa, Mirror Symmetry, arXiv:hep-th/0404196.
10. , Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pp. 120-139, Birkhäuser, Basel, 1995. | MR 1403918 | Zbl 0846.53021
11. , Hyperplane sections and derived categories (Russian), Izv. Ross. Akad. Nauk, Ser. Mat., 70 (2006), 23-128 | MR 2238172 | Zbl 1133.14016 | Zbl pre05229544
12. A. Kuznetsov, Derived categories of quadric fibrations and intersections of quadrics, preprint math.AG/0510670. | Zbl 1168.14012
13. A. Kuznetsov, Exceptional collections for Grassmannians of isotropic lines, preprint math.AG/0512013. | Zbl 1168.14032
14. , Homological projective duality for Grassmannians of lines, preprint math.AG/0610957. | Zbl 1131.14017
15. , Projective bundles, monoidal transformations, and derived categories of coherent sheaves (Russian), Izv. Ross. Akad. Nauk, Ser. Mat., 56 (1992), 852-862 | MR 1208153 | Zbl 0798.14007
16. , Equivalences of derived categories and K3 surfaces, algebraic geometry, 7, J. Math. Sci., New York, 84 (1997), 1361-1381 | MR 1465519 | Zbl 0938.14019
17. , Triangulated categories of singularities and D-branes in Landau-Ginzburg models (Russian), Tr. Mat. Inst. Steklova, 246 (2004), 240-262 | MR 2101296 | Zbl 1101.81093
18. D. Orlov, Triangulated categories of singularities and equivalences between Landau-Ginzburg models, preprint math.AG/0503630. | Zbl 1161.14301