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},
pages = {157--220},
publisher = {Springer},
volume = {105},
year = {2007},
doi = {10.1007/s10240-007-0006-8},
mrnumber = {2354207},
zbl = {1131.14017},
language = {en},
url = {http://archive.numdam.org/articles/10.1007/s10240-007-0006-8/}
}
TY - JOUR AU - Kuznetsov, Alexander TI - Homological projective duality JO - Publications Mathématiques de l'IHÉS PY - 2007 SP - 157 EP - 220 VL - 105 PB - Springer UR - http://archive.numdam.org/articles/10.1007/s10240-007-0006-8/ DO - 10.1007/s10240-007-0006-8 LA - en ID - PMIHES_2007__105__157_0 ER -
Kuznetsov, Alexander. Homological projective duality. Publications Mathématiques de l'IHÉS, Tome 105 (2007), pp. 157-220. doi : 10.1007/s10240-007-0006-8. http://archive.numdam.org/articles/10.1007/s10240-007-0006-8/
1. , Representations of associative algebras and coherent sheaves (Russian), Izv. Akad. Nauk SSSR, Ser. Mat., 53 (1989), 25-44 | MR | Zbl
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 | Zbl
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 | Zbl
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 | Zbl
7. and , Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J., 3 (2003), 1-36, 258. | MR
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
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 | Zbl
11. , Hyperplane sections and derived categories (Russian), Izv. Ross. Akad. Nauk, Ser. Mat., 70 (2006), 23-128 | MR | Zbl
12. A. Kuznetsov, Derived categories of quadric fibrations and intersections of quadrics, preprint math.AG/0510670. | Zbl
13. A. Kuznetsov, Exceptional collections for Grassmannians of isotropic lines, preprint math.AG/0512013. | Zbl
14. , Homological projective duality for Grassmannians of lines, preprint math.AG/0610957.
15. , Projective bundles, monoidal transformations, and derived categories of coherent sheaves (Russian), Izv. Ross. Akad. Nauk, Ser. Mat., 56 (1992), 852-862 | MR | Zbl
16. , Equivalences of derived categories and K3 surfaces, algebraic geometry, 7, J. Math. Sci., New York, 84 (1997), 1361-1381 | MR | Zbl
17. , Triangulated categories of singularities and D-branes in Landau-Ginzburg models (Russian), Tr. Mat. Inst. Steklova, 246 (2004), 240-262 | MR | Zbl
18. D. Orlov, Triangulated categories of singularities and equivalences between Landau-Ginzburg models, preprint math.AG/0503630. | Zbl
Cité par Sources :
