×

Tilting bundles on toric Fano fourfolds. (English) Zbl 1386.14015

Consider a smooth variety \(X\) over \(\mathbb C\). A tilting object is an object \(\mathcal T\in\mathcal D^b(X)\) inducing an equivalence of bounded derived categories \(\text{RHom}_X(\mathcal T,-):\mathcal D^b(X)\simeq\mathcal D^b(A):(-){\mathop\otimes\limits^L}_A T,\) where \(A=\text{End}(\mathcal T)\). If a tilting object \(\mathcal T\) is actually a sheaf on \(X\), it is called a tilting sheaf, (or a tilting bundle if it is locally free).
The main study of this article is the search for varieties where tilting objects exists, and then the possible constructions of tilting objects.
For a projective curve one can construct tilting sheaves by full strong exceptional collections of sheaves \(\{E_i\}_{i\in I}\). Given such a collection, \(\mathcal T=\bigoplus_{i\in I}E_i\) is a tilting sheaf. Conversely, if \(\mathcal T\) is a tilting sheaf, it comes from a full strong exceptional collection. The exposition is modelled on Beilinson’s classical result that \(\mathcal O\oplus\mathcal O(1)\oplus\cdots\oplus\mathcal O(n)\) is a tilting bundle for \(\mathbb P^n\).
Next to projective varieties are the toric varieties. On these varieties it is possible to control the collections of sheaves and to control when they are strong exceptional. Also, if they are, the endomorphism ring can be explicitly computed.
This article concentrates on the particular case of toric Fano varieties. It is a finite number of such varieties in each dimension, and they are classified up to dimension \(5\). There exists a general classification algorithm. Some results have already appeared in this direction. A. King [“Tilting bundles on some rational surfaces”, http://www.maths.bath.ac.uk/~masadk/papers/tilt.pdf] has given full strong exceptional collections of line bundles for the 5 smooth toric Fano surfaces, and H. Uehara [Int. J. Math. 25, No. 7, Article ID 1450072, 32 p. (2014; Zbl 1310.14025)] has given full strong exceptional collections of line bundles for the 18 smooth toric Fano threefolds.
We state the two main result of the article verbatim:
{Theorem 7.4.} Let \(X\) be one of the 124 smooth toric Fano fourfolds. Then one can construct explicitly a full strong exceptional collection of line bundles on \(X\), a database of which is contained in the computer package QuiversToricVarieties for Macaulay2.
{Theorem 7.8} Let \(X\) be an \(n\)-dimensional smooth toric Fano variety for \(n\leq 4\), \(\mathcal L=\{ L_0,\dots,L_r\}\) be the full strong exceptional collection on \(X\) from the database and \(\pi:Y:=\text{tot}(\omega_X)\rightarrow X\) be the bundle map. Then \(Y\) has a tilting bundle that decomposes as a sum of the line bundles, given by \(\bigoplus_{i=1}^r\pi^\ast(L_i).\)
To prove that a given collection on a toric Fano variety \(X\) is strong exceptional the author uses the construction of the not necessarily non-vanishing cohomology cones (nnnvc-cones) in the Picard lattice for \(X\) given by Eisenbud, Mustata, and Stillman [D. Eisenbud et al., J. Symb. Comput. 29, No. 4–5, 583–600 (2000; Zbl 1044.14028)]. Then the strong exceptional condition can be verified by an algorithm, implemented in QuiversToricVarieties.
It is a more involved theory to prove that a given strong exceptional collection \(\mathcal L\) on \(X\) generates \(\mathcal D^b(X).\) The author uses two methods for proving that \(\mathcal L\) is full. The first is similar to Uehara’s method on the toric Fano threefolds. This uses Frobenious pushforward and obtains a set of line bundles that generates \(\mathcal D^b(X)\), and using exacts sequences with these bundles and \(\mathcal L \), \(\mathcal L\) is proved to generate \(\mathcal D^b(X)\). The second method uses the same set of line bundles to obtain a resolution of the structure sheaf of the diagonal \(\mathcal O_\Delta\). This last method includes the toric cell complex introduced by A. Craw and A. Quintero Vélez [Adv. Math. 229, No. 3, 1516–1554 (2012; Zbl 1280.16023)]. This is needed to obtain a minimal projective \(A-A\) bimodule resolution of the endomorphism algebra \(A=\text{End}(\bigoplus_i L_i^{-1})\). The pullback of \(\mathcal L\) to \(Y=\text{tot}(\omega_X)\) is a CY5 algebra for which the 0th, 1st, and 2nd terms of its minimal bimodule resolution is known. The symmetric structure of the CY5 algebra works to guess the 3rd, 4rt and 5th terms, the result is sheafified and restricted to \(X\) and the resulting sequence of sheaves \(S^\bullet\) is proved to be a resoulution of \(\mathcal O_\Delta\) by quiver moduli.
Let \(X\) be a smooth toric Fano threefold or one of the 88 smooth toric Fano fourfoulds such that the given full strong exceptional collection \(\mathcal L\) in the database has a corresponding exact sequence of sheaves \(S^\bullet\in\mathcal D^b(X\times X)\). Let \(B\) denote the rolled up helix algebra of \(A=\text{End}(\bigoplus_{L\in\mathcal L}L^{-1})\). The calculations above leads the author to conjecture: The toric cell complex of \(B\) exists and is supported on a real four or five-dimensional torus respectively. Moreover: The cellular resolution exists in the sense of Craw-Quintero Vélez [Zbl 1280.16023], thereby producing the minimal projective bimodule resolution of \(B\), and when \(\mathcal T=\bigoplus_{L\in\mathcal L}L^{-1}\) the object \(S^\bullet\) is quasi-isomorphic to the exterior product \(T^{\vee}{\mathop\boxtimes\limits^{\mathbf{L}}}_A\mathcal T\in\mathcal D^b(X\times X).\)
The birational geometry of the toric Fano fourfolds together with collections \(\mathcal L\) from a special set of line bundles (as Uehara did for threefolds) gives that the pushforward of \(\mathcal L\) onto a torus-invariant divisorial contraction is automatically full. This is a very useful technical result stated as proposition 7.2 in the article. Using this, full exceptional collections on many of the toric Fano fourfolds are obtained from the pushforward of collections on the birationally maximal examples. A database of the full strong exceptional collections on \(n\)-dimensional smooth toric Fano varieties for \(1\leq n\leq 4,\) and many of the algorithms used is contained in the QuiversToricVarieties package in Macaulay2.
The article is a nice application of the theory of quivers and tilting bundles applied to toric varieties. The general theory described in various sources for projective varieties is pushed to fit in the toric setting, and is computed in the toric Fano examples. The article illustrates techniques that can be used in other (similar) settings, and the algorithms implemented in Macaulay2 can be used. Also, the article contains several important computational results on full strong exceptional collections of sheaves.

MSC:

14A22 Noncommutative algebraic geometry
13F60 Cluster algebras
22F30 Homogeneous spaces
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Baer, Dagmar, Tilting sheaves in representation theory of algebras, Manuscripta Math., 60, 3, 323-347 (1988) · Zbl 0639.16018
[2] Batyrev, V. V., Boundedness of the degree of multidimensional toric Fano varieties, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1, 22-27 (1982), 76-77 · Zbl 0505.14033
[3] Batyrev, V. V., Toroidal Fano 3-folds, Math. USSR, Izv., 19, 13-25 (1982) · Zbl 0495.14027
[4] Batyrev, V. V., On the classification of toric Fano 4-folds, Algebraic Geometry, vol. 9. Algebraic Geometry, vol. 9, J. Math. Sci. (N. Y.), 94, 1, 1021-1050 (1999) · Zbl 0929.14024
[5] Bocklandt, Raf; Craw, Alastair; Quintero Vélez, Alexander, Geometric Reid’s recipe for dimer models, Math. Ann., 1-35 (2014) · Zbl 1331.14022
[6] Beĭlinson, A. A., Coherent sheaves on \(P^n\) and problems in linear algebra, Funktsional. Anal. i Prilozhen., 12, 3, 68-69 (1978) · Zbl 0402.14006
[7] Bondal, A. I., Helices, representations of quivers and Koszul algebras, (Helices and Vector Bundles. Helices and Vector Bundles, London Math. Soc. Lecture Note Ser., vol. 148 (1990), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 75-95 · Zbl 0742.14010
[8] Bondal, Alexey, Derived categories of toric varieties, (Convex and Algebraic Geometry. Convex and Algebraic Geometry, Oberwolfach Conf. Rep., vol. 3 (2006), EMS Publishing House), 284-286
[9] Bridgeland, Tom; Stern, David, Helices on del Pezzo surfaces and tilting Calabi-Yau algebras, Adv. Math., 224, 4, 1672-1716 (2010) · Zbl 1193.14022
[10] Bocklandt, Raf; Schedler, Travis; Wemyss, Michael, Superpotentials and higher order derivations, J. Pure Appl. Algebra, 214, 9, 1501-1522 (2010) · Zbl 1219.16016
[11] Bernardi, Alessandro; Tirabassi, Sofia, Derived categories of toric Fano 3-folds via the Frobenius morphism, Matematiche (Catania), 64, 2, 117-154 (2009) · Zbl 1195.14021
[12] Cox, David A.; Little, John B.; Schenck, Henry K., Toric Varieties, Grad. Stud. Math., vol. 124 (2011), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1223.14001
[13] Costa, L.; Miró-Roig, R. M., Tilting sheaves on toric varieties, Math. Z., 248, 4, 849-865 (2004) · Zbl 1071.14022
[14] Craw, Alastair; Maclagan, Diane; Thomas, Rekha R., Moduli of McKay quiver representations. I. The coherent component, Proc. Lond. Math. Soc. (3), 95, 1, 179-198 (2007) · Zbl 1140.14046
[15] Cox, David A., The homogeneous coordinate ring of a toric variety, J. Algebraic Geom., 4, 1, 17-50 (1995) · Zbl 0846.14032
[16] Craw, Alastair; Quintero Vélez, Alexander, Cellular resolutions of noncommutative toric algebras from superpotentials, Adv. Math., 229, 3, 1516-1554 (2012) · Zbl 1280.16023
[17] Craw, Alastair; Smith, Gregory G., Projective toric varieties as fine moduli spaces of quiver representations, Amer. J. Math., 130, 6, 1509-1534 (2008) · Zbl 1183.14066
[18] Dey, Arijit; Lasoń, Michał; Michałek, Mateusz, Derived category of toric varieties with Picard number three, Matematiche (Catania), 64, 2, 99-116 (2009) · Zbl 1207.14025
[19] Efimov, Alexander I., Maximal lengths of exceptional collections of line bundles, J. Lond. Math. Soc., 90, 2, 350-372 (2014) · Zbl 1318.14047
[20] Eisenbud, David; Mustaţǎ, Mircea; Stillman, Mike, Cohomology on toric varieties and local cohomology with monomial supports, Symbolic Computation in Algebra, Analysis, and Geometry. Symbolic Computation in Algebra, Analysis, and Geometry, Berkeley, CA, 1998. Symbolic Computation in Algebra, Analysis, and Geometry. Symbolic Computation in Algebra, Analysis, and Geometry, Berkeley, CA, 1998, J. Symbolic Comput., 29, 4-5, 583-600 (2000) · Zbl 1044.14028
[21] Fulton, William, Introduction to Toric Varieties: The William H. Roever Lectures in Geometry, Ann. of Math. Stud., vol. 131 (1993), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0813.14039
[22] Grayson, Daniel R.; Stillman, Michael E., Macaulay2, a software system for research in algebraic geometry, available at
[23] Hille, Lutz, Toric quiver varieties, (Algebras and Modules, II. Algebras and Modules, II, Geiranger, 1996. Algebras and Modules, II. Algebras and Modules, II, Geiranger, 1996, CMS Conf. Proc., vol. 24 (1998), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 311-325 · Zbl 0937.14039
[24] Hille, Lutz; Perling, Markus, A counterexample to King’s conjecture, Compos. Math., 142, 6, 1507-1521 (2006) · Zbl 1108.14040
[25] Huybrechts, D., Fourier-Mukai Transforms in Algebraic Geometry, Oxford Math. Monogr. (2006), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press Oxford · Zbl 1095.14002
[26] Kawamata, Yujiro, Derived categories of toric varieties, Michigan Math. J., 54, 3, 517-535 (2006) · Zbl 1159.14026
[27] King, A. D., Moduli of representations of finite-dimensional algebras, Q. J. Math. Oxford Ser. (2), 45, 180, 515-530 (1994) · Zbl 0837.16005
[28] King, Alastair, Tilting bundles on some rational surfaces (1997), available as a preprint at
[29] Kreuzer, Maximilian; Nill, Benjamin, Classification of toric Fano 5-folds, Adv. Geom., 9, 1, 85-97 (2009) · Zbl 1193.14067
[30] Lasoń, Michał; Michałek, Mateusz, On the full, strongly exceptional collections on toric varieties with Picard number three, Collect. Math., 62, 3, 275-296 (2011) · Zbl 1238.14039
[31] Øbro, M., An algorithm for the classification of smooth Fano polytopes (2007)
[32] Oda, Tadao, Convex bodies and algebraic geometry, (An Introduction to the Theory of Toric Varieties. An Introduction to the Theory of Toric Varieties, Ergeb. Math. Grenzgeb. (3), vol. 15 (1988), Springer-Verlag: Springer-Verlag Berlin), translated from the Japanese · Zbl 0628.52002
[33] Prabhu-Naik, Nathan, Generationtoricfanofourfolds - a file for Macaulay2, available at
[34] Prabhu-Naik, Nathan, Strongexccollsmaxfourfolds - a file for Macaulay2, available at
[35] Prabhu-Naik, Nathan, Quiverstoricvarieties: a package to construct quivers of sections on complete toric varieties (2015)
[36] Stein, W. A., Sage Mathematics Software (Version 6.1.1) (2015), The Sage Development Team
[37] Sato, Hiroshi, Toward the classification of higher-dimensional toric Fano varieties, Tohoku Math. J. (2), 52, 3, 383-413 (2000) · Zbl 1028.14015
[38] Thomsen, Jesper Funch, Frobenius direct images of line bundles on toric varieties, J. Algebra, 226, 2, 865-874 (2000) · Zbl 0957.14036
[39] Uehara, Hokuto, Exceptional collections on toric Fano threefolds and birational geometry, Internat. J. Math., 25, 7, Article 1450072 pp. (2014) · Zbl 1310.14025
[40] Van den Bergh, Michel, Three-dimensional flops and noncommutative rings, Duke Math. J., 122, 3, 423-455 (2004) · Zbl 1074.14013
[41] Watanabe, Keiichi; Watanabe, Masayuki, The classification of Fano 3-folds with torus embeddings, Tokyo J. Math., 5, 1, 37-48 (1982) · Zbl 0581.14028
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.