Homological projective duality

Alexander Kuznetsov

Publications Mathématiques de l'IHÉS (2007)

  • Volume: 105, page 157-220
  • ISSN: 0073-8301

Abstract

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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.

How to cite

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Kuznetsov, Alexander. "Homological projective duality." Publications Mathématiques de l'IHÉS 105 (2007): 157-220. <http://eudml.org/doc/104222>.

@article{Kuznetsov2007,
abstract = {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.},
author = {Kuznetsov, Alexander},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {projective duality; hyperplane sections; derived categories of singularities},
language = {eng},
pages = {157-220},
publisher = {Springer},
title = {Homological projective duality},
url = {http://eudml.org/doc/104222},
volume = {105},
year = {2007},
}

TY - JOUR
AU - Kuznetsov, Alexander
TI - Homological projective duality
JO - Publications Mathématiques de l'IHÉS
PY - 2007
PB - Springer
VL - 105
SP - 157
EP - 220
AB - 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.
LA - eng
KW - projective duality; hyperplane sections; derived categories of singularities
UR - http://eudml.org/doc/104222
ER -

References

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  3. 3. A. Bondal and D. Orlov, Semiorthogonal decomposition for algebraic varieties, preprint math.AG/9506012. 
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  9. 9. K. Hori and C. Vafa, Mirror Symmetry, arXiv:hep-th/0404196. Zbl1044.14018
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  11. 11. A. Kuznetsov, Hyperplane sections and derived categories (Russian), Izv. Ross. Akad. Nauk, Ser. Mat., 70 (2006), 23-128 Zbl1133.14016MR2238172
  12. 12. A. Kuznetsov, Derived categories of quadric fibrations and intersections of quadrics, preprint math.AG/0510670. 
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  14. 14. A. Kuznetsov, Homological projective duality for Grassmannians of lines, preprint math.AG/0610957. 
  15. 15. D. Orlov, Projective bundles, monoidal transformations, and derived categories of coherent sheaves (Russian), Izv. Ross. Akad. Nauk, Ser. Mat., 56 (1992), 852-862 Zbl0798.14007MR1208153
  16. 16. D. Orlov, Equivalences of derived categories and K3 surfaces, algebraic geometry, 7, J. Math. Sci., New York, 84 (1997), 1361-1381 Zbl0938.14019MR1465519
  17. 17. D. Orlov, Triangulated categories of singularities and D-branes in Landau–Ginzburg models (Russian), Tr. Mat. Inst. Steklova, 246 (2004), 240-262 Zbl1101.81093MR2101296
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