Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-22T01:57:03.510Z Has data issue: false hasContentIssue false
  • English
  • Français

Homogeneity of cohomology classes associated with Koszul matrix factorizations

Part of: Quantum field theory; related classical field theories Homological methods Local theory

Published online by Cambridge University Press:  20 July 2016

Alexander Polishchuk
Alexander Polishchuk*
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR 97405, USA email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this work we prove the so-called dimension property for the cohomological field theory associated with a homogeneous polynomial $W$ with an isolated singularity, in the algebraic framework of [A. Polishchuk and A. Vaintrob, Matrix factorizations and cohomological field theories, J. Reine Angew. Math. 714 (2016), 1–122]. This amounts to showing that some cohomology classes on the Deligne–Mumford moduli spaces of stable curves, constructed using Fourier–Mukai-type functors associated with matrix factorizations, live in prescribed dimension. The proof is based on a homogeneity result established in [A. Polishchuk and A. Vaintrob, Algebraic construction of Witten’s top Chern class, in Advances in algebraic geometry motivated by physics (Lowell, MA, 2000) (American Mathematical Society, Providence, RI, 2001), 229–249] for certain characteristic classes of Koszul matrix factorizations of $0$ . To reduce to this result, we use the theory of Fourier–Mukai-type functors involving matrix factorizations and the natural rational lattices in the relevant Hochschild homology spaces, as well as a version of Hodge–Riemann bilinear relations for Hochschild homology of matrix factorizations. Our approach also gives a proof of the dimension property for the cohomological field theories associated with some quasihomogeneous polynomials with an isolated singularity.

Keywords

matrix factorizationFan–Jarvis–Ruan–Witten theoryHochschild homology

MSC classification

Primary: 81T70: Quantization in field theory; cohomological methods
Secondary: 14B05: Singularities 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 10 , October 2016 , pp. 2071 - 2112
Copyright
© The Author 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Atiyah, M., Complex analytic connections in fibre bundles , Trans. Amer. Math. Soc. 85 (1957), 181–207.Google Scholar
Ballard, M., Deliu, D., Favero, D., Isik, M. U. and Katzarkov, L., Resolutions in factorization categories , Adv. Math. 295 (2016), 195–249.CrossRefGoogle Scholar
Bergh, D., Lunts, V. A. and Schnürer, O. M., Geometricity for derived categories of algebraic stacks, Preprint (2016), arXiv:1601.04465.Google Scholar
Blanc, A., Topological K-theory of complex noncommutative spaces , Compos. Math. 152 (2016), 489–555.Google Scholar
Bondal, A. and Kapranov, M., Representable functors, Serre functors, and reconstructions , Math. USSR-Izv. 35 (1990), 519–541.CrossRefGoogle Scholar
Caldararu, A., The Mukai pairing. II. The Hochschild–Kostant–Rosenberg isomorphism , Adv. Math. 194 (2005), 34–66.Google Scholar
Caldararu, A. and Tu, J., Curved A ∞ -algebras and Landau–Ginzburg models , New York J. Math. 19 (2013), 305–342.Google Scholar
Caldararu, A. and Willerton, S., The Mukai pairing I: a categorical approach , New York J. Math. 16 (2010), 61–98.Google Scholar
Chang, H.-L., Li, J. and Li, W.-P., Witten’s top Chern class via cosection localization , Invent. Math. 200 (2015), 1015–1063.CrossRefGoogle Scholar
Chiodo, A., Witten’s top Chern class via K-theory , J. Algebraic Geom. 15 (2006), 681–707.CrossRefGoogle Scholar
Chiodo, A., Iritani, H. and Ruan, Y., Landau–Ginzburg/Calabi–Yau correspondence, global mirror symmetry and Orlov equivalence , Publ. Math. Inst. Hautes Études Sci. 119 (2014), 127–216.Google Scholar
Chiodo, A. and Ruan, Y., LG/CY correspondence: the state space isomorphism , Adv. Math. 227 (2011), 2157–2188.CrossRefGoogle Scholar
Deligne, P., Hodge cycles on abelian varieties , in Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900 (Springer, Berlin–New York, 1982).Google Scholar
Dyckerhoff, T., Compact generators in categories of matrix factorizations , Duke Math. J. 159 (2011), 223–274.Google Scholar
Efimov, A. and Positselski, L., Coherent analogues of matrix factorizations and relative singularity categories , Algebra Number Theory 9 (2015), 1159–1292.Google Scholar
Fan, H., Jarvis, T. and Ruan, Y., The Witten equation, mirror symmetry and quantum singularity theory , Ann. of Math. (2) 178 (2013), 1–106.Google Scholar
Guéré, J., A Landau–Ginzburg mirror theorem without concavity, Preprint (2013), arXiv:1307.5070.Google Scholar
Hartshorne, R., Residues and duality, Lecture Notes in Mathematics, vol. 20 (Springer, Berlin, 1966).CrossRefGoogle Scholar
Kontsevich, M., XI Solomon Lefschetz Memorial Lecture series: Hodge structures in non-commutative geometry (notes by Ernesto Lupercio) , in Non-commutative geometry in mathematics and physics (American Mathematical Society, Providence, RI, 2008), 1–21.Google Scholar
Kontsevich, M. and Manin, Yu. I., Gromov–Witten classes, quantum cohomology, and enumerative geometry , Comm. Math. Phys. 164 (1994), 525–562.CrossRefGoogle Scholar
Kresch, A., On the geometry of Deligne–Mumford stacks , in Algebraic geometry (Seattle 2005), part 1 (American Mathematical Society, Providence, RI, 2009), 259–271.Google Scholar
Kresch, A. and Vistoli, A., On coverings of Deligne–Mumford stacks and surjectivity of the Brauer map , Bull. Lond. Math. Soc. 36 (2004), 188–192.Google Scholar
Kuznetsov, A., Hochschild homology and semiorthogonal decompositions, Preprint (2009), arXiv:0904.4330.Google Scholar
Kuznetsov, A., Base change for semiorthogonal decompositions , Compos. Math. 147 (2011), 852–876.Google Scholar
Lin, K. and Pomerleano, D., Global matrix factorizations , Math. Res. Lett. 20 (2013), 91–106.Google Scholar
Macrí, E. and Stellari, P., Infinitesimal derived Torelli theorem for K3 surfaces , Int. Math. Res. Not. IMRN 17 (2009), 3190–3220.Google Scholar
Moore, W. F., Piepmeyer, G., Spiroff, S. and Walker, M. E., Hochster’s theta invariant and the Hodge–Riemann bilinear relations , Adv. Math. 226 (2011), 1692–1714.Google Scholar
Orlov, D., Triangulated categories of singularities and D-branes in Landau–Ginzburg models , Proc. Steklov Inst. Math. 246 (2004), 227–248.Google Scholar
Orlov, D., Derived categories of coherent sheaves and triangulated categories of singularities , in Algebra, arithmetic, and geometry: in honor of Yu. I. Manin, Vol. II (Birkhauser, Boston, MA, 2009), 503–531.Google Scholar
Orlov, D., Matrix factorizations for nonaffine LG-models , Math. Ann. 353 (2012), 95–108.CrossRefGoogle Scholar
Orlov, D., Smooth and proper noncommutative schemes and gluing of DG categories, Preprint (2014), arXiv:1402.7364.Google Scholar
Polishchuk, A., Lefschetz type formulas for dg-categories , Selecta Math. 20 (2014), 885–928.Google Scholar
Polishchuk, A. and Vaintrob, A., Algebraic construction of Witten’s top Chern class , in Advances in algebraic geometry motivated by physics (Lowell, MA, 2000) (American Mathematical Society, Providence, RI, 2001), 229–249.CrossRefGoogle Scholar
Polishchuk, A. and Vaintrob, A., Matrix factorizations and singularity categories for stacks , Ann. Inst. Fourier 61 (2011), 2609–2642.CrossRefGoogle Scholar
Polishchuk, A. and Vaintrob, A., Chern characters and Hirzebruch–Riemann–Roch formula for matrix factorizations , Duke Math. J. 161 (2012), 1863–1926.CrossRefGoogle Scholar
Polishchuk, A. and Vaintrob, A., Matrix factorizations and cohomological field theories , J. Reine Angew. Math. 714 (2016), 1–122.CrossRefGoogle Scholar
Ramadoss, A. C., The relative Riemann–Roch theorem from Hochschild homology , New York J. Math. 14 (2008), 643–717.Google Scholar
Shipman, I., A geometric approach to Orlov’s theorem , Compos. Math. 148 (2012), 1365–1389.CrossRefGoogle Scholar
Shklyarov, D., Hirzebruch–Riemann–Roch-type theorem for DG algebras , Proc. Lond. Math. Soc. (3) 106 (2013), 1–32.Google Scholar
Thomason, R. W., Algebraic K-theory of group scheme actions , in Algebraic topology and algebraic K-theory (Princeton, NJ, 1983) (Princeton University Press, Princeton, NJ, 1987), 539–563.Google Scholar
Toën, B., The homotopy theory of dg-categories and derived Morita theory , Invent. Math. 167 (2007), 615–667.Google Scholar
Toën, B. and Vaquié, M., Moduli of objects in dg-categories , Ann. Sci. Éc. Norm. Supér. (4) 40 (2007), 387–444.CrossRefGoogle Scholar
Wells, R. O., Differential analysis on complex manifolds (Springer, New York, 1980).Google Scholar