## Duke Mathematical Journal

### Chern characters and Hirzebruch–Riemann–Roch formula for matrix factorizations

#### Abstract

We study the category of matrix factorizations for an isolated hypersurface singularity. We compute the canonical bilinear form on the Hochschild homology of this category. We find explicit expressions for the Chern character and the boundary-bulk maps and derive an analogue of the Hirzebruch–Riemann–Roch formula for the Euler characteristic of the $\operatorname{Hom}$-space between a pair of matrix factorizations. We also establish $G$-equivariant versions of these results.

#### Article information

Source
Duke Math. J., Volume 161, Number 10 (2012), 1863-1926.

Dates
First available in Project Euclid: 27 June 2012

https://projecteuclid.org/euclid.dmj/1340801626

Digital Object Identifier
doi:10.1215/00127094-1645540

Mathematical Reviews number (MathSciNet)
MR2954619

Zentralblatt MATH identifier
1249.14001

#### Citation

Polishchuk, Alexander; Vaintrob, Arkady. Chern characters and Hirzebruch–Riemann–Roch formula for matrix factorizations. Duke Math. J. 161 (2012), no. 10, 1863--1926. doi:10.1215/00127094-1645540. https://projecteuclid.org/euclid.dmj/1340801626

#### References

• [1] M. Auslander, “Functors and morphisms determined by objects” in Representation Theory of Algebras (Proc. Conf., Temple Univ., Philadelphia, Pa., 1976), Lect. Notes Pure Appl. Math. 37, Dekker, New York, 1978, 1–244.
• [2] P. Bressler, R. Nest, and B. Tsygan, Riemann-Roch theorems via deformation quantization, I, Adv. Math. 167 (2002), 1–25.
• [3] R. Buchweitz, Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings, preprint, 1986.
• [4] R.-O. Buchweitz, G.-M. Greuel, and F.-O. Schreyer, Cohen-Macaulay modules on hypersurface singularities, II, Invent. Math. 88 (1987), 165–182.
• [5] D. Burghelea, The cyclic homology of the group rings, Comment. Math. Helv. 60 (1985), 354–365.
• [6] A. Căldăraru and S. Willerton, The Mukai paring, I: A categorical approach, New York J. Math. 16 (2010), 61–98.
• [7] K. Costello, Topological conformal field theories and Calabi-Yau categories, Adv. Math. 210 (2007), 165–214.
• [8] S. D. Cutkosky and H. Srinivasan, Equivalence and finite determinancy of mappings, J. Algebra 188 (1997), 16–57.
• [9] H. Dao, Decent intersection and Tor-rigidity for modules over local hypersurfaces, to appear in Trans. Amer. Math. Soc., preprint, arXiv:math/0611568 [math.AC]
• [10] T. Dyckerhoff, Compact generators in categories of matrix factorizations, Duke Math. J. 159 (2011), 223–274.
• [11] A. I. Efimov, Homological mirror symmetry for curves of higher genus, Adv. Math. 230 (2012), 493–530.
• [12] D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260, no. 1 (1980), 35–64.
• [13] H. Fan, T. J. Jarvis, and Y. Ruan, The Witten equation, mirror symmetry and quantum singularity theory, preprint, arXiv:0712.4021 [math.AG]
• [14] B. L. Feĭgin and B. L. Tsygan, “Cyclic homology of algebras with quadratic relations, universal enveloping algebras and group algebras” in K-Theory, Arithmetic and Geometry (Moscow, 1984–1986), Lecture Notes in Math. 1289, Springer, Berlin, 1987, 210–239.
• [15] N. Ganter and M. Kapranov, Representation and character theory in 2-categories, Adv. Math. 217 (2008), 2268–2300.
• [16] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Pure Appl. Math., Wiley-Interscience, New York, 1978.
• [17] R. Hartshorne, Residues and Duality, with an appendix by P. Deligne, Lecture Notes in Math. 20, Springer, Berlin, 1966.
• [18] H. Hironaka, “On the equivalence of singularities, I” in Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper & Row, New York, 1965, 153–200.
• [19] A. Kapustin and Y. Li, D-branes in Landau-Ginzburg models and algebraic geometry, J. High Energy Phys. 2003, no. 12, art. ID 005.
• [20] A. Kapustin and Y. Li, Topological correlators in Landau-Ginzburg models with boundaries, Adv. Theor. Math. Phys. 7 (2003), 727–749.
• [21] A. Kapustin and L. Rozansky, On the relation between open and closed topological strings, Comm. Math. Phys. 252 (2004), 393–414.
• [22] L. Katzarkov, M. Kontsevich, and T. Pantev, “Hodge theoretic aspects of mirror symmetry” in From Hodge Theory to Integrability and TQFT tt*-geometry, Proc. Sympos. Pure Math. 78, Amer. Math. Soc., Providence, 2008, 87–174.
• [23] B. Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), 63–102.
• [24] B. Keller, On the cyclic homology of ringed spaces and schemes, Doc. Math. 3 (1998), 231–259.
• [25] B. Keller, On the cyclic homology of exact categories, J. Pure Appl. Algebra 136 (1999), 1–56.
• [26] B. Keller, “On differential graded categories” in International Congress of Mathematicians, Vol. II, Eur. Math. Soc., Zürich, 2006, 151–190.
• [27] B. Keller, Derived invariance of higher structure on the Hochschild complex, preprint, available at http://www.math.jussieu.fr/~keller/publ/dih.pdf (accessed 24 April 2012).
• [28] M. Khovanov and L. Rozansky, Matrix factorizations and link homology, Fund. Math. 199 (2008), 1–91.
• [29] M. Kontsevich and Y. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), 525–562.
• [30] M. Kontsevich and Y. Soibelman, “Notes on $A_{\infty}$-algebras, $A_{\infty}$-categories and non-commutative geometry” in Homological Mirror Symmetry, Lecture Notes in Phys. 757, Springer, Berlin, 2009, 153–219.
• [31] E. J. N. Looijenga, Isolated Singular Points on Complete Intersections, London Math. Soc. Lecture Notes Ser. 77, Cambridge Univ. Press, Cambridge, 1984.
• [32] M. Lorenz, On the homology of graded algebras, Comm. Algebra 20 (1992), 489–507.
• [33] J. N. Mather, Stability of $C^{\infty}$ mappings, III: Finitely determined mapgerms, Inst. Hautes Études Sci. Publ. Math. 35 (1968), 279–308.
• [34] W. F. Moore, G. Piepmeyer, S. Spiroff, and M. E. Walker, Hochster’s theta invariant and the Hodge-Riemann bilinear relations, Adv. Math. 226 (2011), 1692–1714.
• [35] D. Murfet, Residues and duality for singularity categories of isolated Gorenstein singularities, preprint, arXiv:0912.1629 [math.AC]
• [36] D. O. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models (in Russian), Tr. Mat. Inst. Steklova 246, no. 3 (2004), 240–262; English translation in Proc. Steklov Inst. Math. 246 (2004), 227–248.
• [37] D. O. Orlov, “Derived categories of coherent sheaves and triangulated categories of singularities” in Algebra, Arithmetic, and Geometry: In Honor of Yu. I. Manin, II, Birkhäuser, Boston, 2009, 503–531.
• [38] D. O. Orlov, Formal completions and idempotent completions of triangulated categories of singularities, Adv. Math. 226 (2011), 206–217.
• [39] A. Polishchuk and A. Vaintrob, Matrix factorizations and cohomological field theories, preprint, arXiv:1105.2903v1 [math.AG]
• [40] A. Quintero Vélez, McKay correspondence for Landau-Ginzburg models, Commun. Number Theory Phys. 3 (2009), 173–208.
• [41] A. C. Ramadoss, A generalized Hirzebruch Riemann-Roch theorem, C. R. Math. Acad. Sci. Paris 347 (2009), 289–292.
• [42] M. Roberts, Characterisations of finitely determined equivariant map germs, Math. Ann. 275 (1986), 583–597.
• [43] E. Segal, The closed state space of affine Landau-Ginzburg B-models, to appear in J. Noncommut. Geom., preprint, arXiv:0904.1339v2 [math.AG]
• [44] D. Shklyarov, Hirzebruch-Riemann-Roch theorem for DG algebras, preprint, arXiv:0710.1937v3 [math.KT]
• [45] B. Toën, The homotopy theory of $dg$-categories and derived Morita theory, Invent. Math. 167 (2007), 615–667.
• [46] B. Toën, “Lectures on dg-categories” in Topics in Algebraic and Topological K-Theory, Lecture Notes in Math. 2008, Springer, Berlin, 2011, 243–302.
• [47] B. Toën and M. Vaquié, Moduli of objects in dg-categories, Ann. Sci. École Norm. Sup. (4) 40 (2007), 387–444.
• [48] J. Walcher, Stability of Landau-Ginzburg branes, J. Math. Phys. 46 (2005), no. 082305.
• [49] C. T. C. Wall, A second note on symmetry of singularities, Bull. London Math. Soc. 12 (1980), 347–354.
• [50] C. A. Weibel, An Introduction to Homological Algebra, Cambridge Stud. Adv. Math. 38, Cambridge Univ. Press, Cambridge, 1994.