zbMATH — the first resource for mathematics

Noncommutative deformations and flops. (English) Zbl 1346.14031
The study of birational geometry of algebraic varieties via the minimal model program depends on the geometry of certain modifications of codimension 2 known as flips and flops. The geometry of these is very intricate, and is a contemporary object of study where a central problem is to classify flips and flops such that appropriate invariants can be constructed.
This article gives a new invariant to every flipping or flopping curve in a 3-fold, using noncommutative deformation theory. In fact, the idea is to replace all the invariants by a new noncommutative geometric object represented by the noncommutative deformation algebra \(A_{\mathrm{con}}\) associated to the curve. This algebra is finite dimensional, and can be associated to any contractible rational curve in any 3-fold, singular or not. A very nice property of this algebra is that it recovers the classical invariants in a natural way. The main difference from a separate set of invariants is that the fact that it is an algebra, makes it possible to give an intrinsic description of a derived autoequivalence associated to a general flopping curve.
The authors give the background on 3-folds, the simplest example being the Atiyah flop. Then the normal bundle is \(\mathcal O(-1)\oplus\mathcal O(-1)\) and the curve is rigid (i.e. the orbit is finite under the action of the groups of projective transformations), and the flop can be factored as a blowup of the curve followed by a blowdown. For a general flopping irreducible rational curve \(C\) in a smooth 3-fold \(X\), the authors recall the classical invariants: The normal bundle \((a,b):=\mathcal O(a)\oplus\mathcal O(b)\) which must be \((-1,-1),\;(-2,0)\), or \((-3,1)\), the width, the Dynkin type, the length, and the Normal bundle sequence: The flop \(f:X\dashrightarrow X^\prime\) factors into a sequence of blowups in centers \(C_1,\dots,C_n\), followed by blowdowns, and the normal bundles of these curves form the \(\mathcal N\)-sequence.
The authors give a table of relations between these invariants which proves that none of these classify all analytic equivalence types of flopping curves.
The new invariant, i.e. the deformation algebra, of flopping and flipping rational curves \(C\) in a 3-fold \(X\) is constructed by noncommutatively deforming the associated sheaf \(E:=\mathcal O_C(-1)\). Infinitesimal deformations are controlled by \(\mathrm {Ext}^1_X(E,E)\), and its dimension is determined by the normal bundle \(\mathcal N_{C|X}.\) If \(\dim_{\mathbb C}\mathrm {Ext}^1_X(E,E)\leq 1\), then the curve \(C\) deforms over an Artinian base \(\mathbb C[x]/x^n\). When \(\mathrm {Ext}^1_X(E,E)\geq 2\) the commutative deformations of \(C\) does not induce enough invariants.
The problem is solved by applying noncommutative deformation theory. The commutative deformation functor \(c\mathcal Def_E:\mathsf{CArt}\rightarrow\mathsf{Sets}\) is given by \(R\mapsto\{\mathrm{flat }R\)-families of coherent sheaves deforming \(E\)

14D15 Formal methods and deformations in algebraic geometry
14E30 Minimal model program (Mori theory, extremal rays)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
16S38 Rings arising from noncommutative algebraic geometry
18E30 Derived categories, triangulated categories (MSC2010)
Full Text: DOI Euclid arXiv
[1] R. Anno and T. Logvinenko, Orthogonally spherical objects and spherical fibrations , Adv. Math. 286 (2016), 338-386. · Zbl 1435.14019
[2] R. Anno and T. Logvinenko, Spherical DG-functors , preprint, [math.AG]. arXiv:1309.5035v2 · Zbl 1435.14019
[3] P. S. Aspinwall and D. R. Morrison, Quivers from matrix factorizations , Comm. Math Phys. 313 (2012), 607-633. · Zbl 1250.81076
[4] C. Bartocci, U. Bruzzo, and D. Hernández Ruipérez, Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics , Progr. Math. 276 , Birkhäuser, Boston, 2009.
[5] T. Bridgeland, Flops and derived categories , Invent. Math. 147 (2002), 613-632. · Zbl 1085.14017
[6] T. Bridgeland, A. King, and M. Reid, The McKay correspondence as an equivalence of derived categories , J. Amer. Math. Soc. 14 (2001), 535-554. · Zbl 0966.14028
[7] R.-O. Buchweitz and L. Hille, Hochschild (co-)homology of schemes with tilting object , Trans. Amer. Math. Soc. 365 , no. 6 (2013), 2823-2844. · Zbl 1274.14018
[8] J.-C. Chen, Flops and equivalences of derived categories for threefolds with only terminal Gorenstein singularities , J. Differential Geom. 61 (2002), 227-261. · Zbl 1090.14003
[9] H. Clemens, J. Kollár, and S. Mori, Higher-Dimensional Complex Geometry , Astérisque 166 , Soc. Math. France, Paris, 1988.
[10] W. Donovan and M. Wemyss, Contractions and deformations , preprint, [math.AG]. arXiv:1511.0040
[11] A. I. Efimov, V. A. Lunts, and D. O. Orlov, Deformation theory of objects in homotopy and derived categories, I: General theory , Adv. Math. 222 (2009), 359-401. · Zbl 1180.18006
[12] A. I. Efimov, V. A. Lunts, and D. O. Orlov, Deformation theory of objects in homotopy and derived categories, II: Pro-representability of the deformation functor , Adv. Math. 224 (2010), 45-102. · Zbl 1197.18003
[13] A. I. Efimov, V. A. Lunts, and D. O. Orlov, Deformation theory of objects in homotopy and derived categories, III: Abelian categories , Adv. Math. 226 (2011), 3857-3911. · Zbl 1225.18009
[14] D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations , Trans. Amer. Math. Soc. 260 , no. 1 (1980), 35-64. · Zbl 0444.13006
[15] D. Eisenbud, Commutative Algebra: With a View toward Algebraic Geometry , Grad. Texts in Math. 150 , Springer, New York, 1995. · Zbl 0819.13001
[16] E. Eriksen, “An introduction to noncommutative deformations of modules” in Noncommutative Algebra and Geometry , Lect. Notes Pure Appl. Math. 243 , Chapman and Hall/CRC, Boca Raton, Fla., 2006, 90-125. · Zbl 1104.16025
[17] E. Eriksen, Computing noncommutative deformations of presheaves and sheaves of modules , Canad. J. Math. 62 (2010), 520-542. · Zbl 1196.14016
[18] V. Ginzburg, Calabi-Yau algebras , preprint, [math.AG]. · Zbl 1138.01321
[19] D. Hernández Ruipérez, A. C. López Martín, and F. Sancho de Salas, Fourier-Mukai transforms for Gorenstein schemes , Adv. Math. 211 (2007), 594-620. · Zbl 1118.14022
[20] D. Hernández Ruipérez, A. C. López Martín, and F. Sancho de Salas, Relative integral functors for singular fibrations and singular partners , J. Eur. Math. Soc. (JEMS) 11 (2009), 597-625. · Zbl 1221.18010
[21] D. Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry , Oxford Math. Monogr., Oxford Univ. Press, Oxford, 2006. · Zbl 1095.14002
[22] O. Iyama and I. Reiten, Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras , Amer. J. Math. 130 (2008), 1087-1149. · Zbl 1162.16007
[23] O. Iyama and M. Wemyss, The classification of special Cohen-Macaulay modules , Math. Z. 265 (2010), 41-83. · Zbl 1192.13012
[24] O. Iyama and M. Wemyss, Maximal modifications and Auslander-Reiten duality for non-isolated singularities , Invent. Math. 197 (2014), 521-586. · Zbl 1308.14007
[25] O. Iyama and M. Wemyss, Singular derived categories of \(\mathbb{Q}\)-factorial terminalizations and maximal modification algebras , Adv. Math. 261 (2014), 85-121. · Zbl 1326.14033
[26] S. Katz and D. R. Morrison, Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups , J. Algebraic Geom. 1 (1992), 449-530. · Zbl 0788.14036
[27] Y. Kawamata, General hyperplane sections of nonsingular flops in dimension \(3\) , Math. Res. Lett. 1 (1994), 49-52. · Zbl 0834.32007
[28] J. Kollár and S. Mori, Classification of three-dimensional flips , J. Amer. Math. Soc. 5 (1992), 533-703. · Zbl 0773.14004
[29] O. A. Laudal, Noncommutative deformations of modules , Homology Homotopy Appl. 4 (2002), 357-396. · Zbl 1013.16018
[30] W. Lowen and M. Van den Bergh, Deformation theory of abelian categories , Trans. Amer. Math. Soc. 358 , no. 12 (2006), 5441-5483. · Zbl 1113.13009
[31] H. C. Pinkham, “Factorization of birational maps in dimension \(3\)” in Singularities, Part 2 (Arcata, Calif., 1981) , Proc. Sympos. Pure Math. 40 , Amer. Math. Soc., Providence, 1983, 343-371.
[32] M. Reid, “Minimal models of canonical \(3\)-folds” in Algebraic Varieties and Analytic Varieties (Tokyo, 1981) , Adv. Stud. Pure Math. 1 , North-Holland, Amsterdam, 1983, 131-180.
[33] E. Segal, The \(A_{\infty}\) deformation theory of a point and the derived categories of local Calabi-Yaus , J. Algebra 320 (2008), 3232-3268. · Zbl 1168.18005
[34] P. Seidel and R. Thomas, Braid group actions on derived categories of coherent sheaves , Duke Math. J. 108 (2001), 37-108. · Zbl 1092.14025
[35] A. Smoktunowicz, Growth, entropy and commutativity of algebras satisfying prescribed relations , Selecta Math. (N.S.) 20 (2014), 1197-1212. · Zbl 1320.16009
[36] Y. Toda, On a certain generalization of spherical twists , Bull. Soc. Math. France 135 (2007), 119-134. · Zbl 1155.18010
[37] Y. Toda, Non-commutative width and Gopakumar-Vafa invariants , Manuscripta Math. 148 (2015), 521-533. · Zbl 1348.14040
[38] Y. Toda, Stability conditions and crepant small resolutions , preprint, [math.AG]. · Zbl 1225.14030
[39] M. Van den Bergh, Three-dimensional flops and noncommutative rings , Duke Math. J. 122 (2004), 423-455. · Zbl 1074.14013
[40] M. Wemyss, Aspects of the homological minimal model program , preprint, [math.AG]. arXiv:1411.7189v1
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.