Duke Mathematical Journal

Matrix factorization of Morse–Bott functions

Constantin Teleman

Abstract

For a function $W\in \mathbb{C}[X]$ on a smooth algebraic variety $X$ with Morse–Bott critical locus $Y\subset X$, Kapustin, Rozansky, and Saulina suggest that the associated matrix factorization category $\operatorname{MF}(X;W)$ should be equivalent to the differential graded category of $2$-periodic coherent complexes on $Y$ (with a topological twist from the normal bundle). We confirm their conjecture in the special case when the first neighborhood of $Y$ in $X$ is split and establish the corrected general statement. The answer involves the full Gerstenhaber structure on Hochschild cochains. This note was inspired by the failure of the conjecture, observed by Pomerleano and Preygel, when $X$ is a general $1$-parameter deformation of a $K3$ surface $Y$.

Article information

Source
Duke Math. J., Volume 169, Number 3 (2020), 533-549.

Dates
Revised: 4 July 2019
First available in Project Euclid: 28 January 2020

https://projecteuclid.org/euclid.dmj/1580202168

Digital Object Identifier
doi:10.1215/00127094-2019-0048

Mathematical Reviews number (MathSciNet)
MR4065148

Zentralblatt MATH identifier
07198460

Citation

Teleman, Constantin. Matrix factorization of Morse–Bott functions. Duke Math. J. 169 (2020), no. 3, 533--549. doi:10.1215/00127094-2019-0048. https://projecteuclid.org/euclid.dmj/1580202168

References

• [1] M. F. Atiyah, R.Bott, and A. Shapiro, Clifford modules, Topology 3 (1964), suppl. 1, 3–38.
• [2] A. Blanc, L. Katzarkov, and P. Pandit, Generators in formal deformations of categories, Compos. Math. 154 (2018), no. 10, 2055–2089.
• [3] R.-O. Buchweitz, D. Eisenbud, and J. Herzog, “Cohen–Macaulay modules on quadrics” in Singularities, Representation of Algebras, and Vector Bundles (Lambrecht, 1985), Lecture Notes in Math. 1273, Springer, Berlin, 1987, 58–116.
• [4] V. Dolgushev, D. Tamarkin, and B. Tsygan, Formality theorems for Hochschild complexes and their applications, Lett. Math. Phys. 90 (2009), nos. 1–3, 103–136.
• [5] P. Donovan and M. Karoubi, Graded Brauer groups and $K$-theory with local coefficients, Inst. Hautes Études Sci. Publ. Math. 38 (1970), 5–25.
• [6] J. Francis, The tangent complex and Hochschild cohomology of $\mathscr{E}_{n}$-rings, Compos. Math. 149 (2013), no. 3, 430–480.
• [7] T. Holm, Hochschild cohomology rings of algebras $k[X]/(f)$, Beiträge Algebra Geom. 41 (2000), no. 1, 291–301.
• [8] A. Kapustin, L. Rozansky, and N. Saulina, Three-dimensional topological field theory and symplectic algebraic geometry, I, Nuclear Phys. B 816 (2009), no. 3, 295–355.
• [9] B. Keller and W. Lowen, On Hochschild cohomology and Morita deformations, Int. Math. Res. Not. IMRN 2009, no. 17, 3221–3235.
• [10] K. H. Lin and D. Pomerleano, Global matrix factorizations, Math. Res. Lett. 20 (2013), no. 1, 91–106.
• [11] D. Orlov, Triangulated categories of singularities and D-branes in Landau–Ginzburg models (in Russian), Tr. Mat. Inst. Steklova 246 (2004), Algebr. Geom. Metody, Svyazi i Prilozh., 240–262; English translation in Proc. Steklov Inst. Math. 2004, no. 3(246), 227–248.
• [12] D. Orlov, Matrix factorizations for nonaffine LG-models, Math. Ann. 353 (2012), no. 1, 95–108.
• [13] D. Pomerleano and A. Preygel, personal communication, 2014.
• [14] A. Preygel, Thom-Sebastiani and duality for matrix factorizations, and results on the higher structures of the Hochschild invariants, PhD dissertation, Massachusetts Institute of Technology, Cambridge, MA, 2012.