Helices on del Pezzo surfaces and tilting Calabi-Yau algebras. (English) Zbl 1193.14022
The authors study tilting for a class of Calabi–Yau algebras associated to helices on Fano varieties.
A helix of sheaves of period \(n\) on a smooth projective variety \(Z\) is an infinite collection of coherent sheaves \({\mathbb{H}} = \{ E_i \}_{i \in {\mathbb{Z}}}\) such that for all \(i \in \mathbb{Z}\), the collection \((E_{i+1}, \ldots, E_{i+n})\) is fully exceptional and \(E_{i-n} = E_i \otimes \omega_Z\). Such helices exist on many Fano varieties, for examples on projective spaces and on Fano surfaces. The main tool to describe helices is the theory of exceptional collections and mutation functors developed in [A. N. Rudakov, (ed.), Helices and vector bundles: Seminaire Rudakov. London Mathematical Society Lecture Note Series, 148. Cambridge etc.: Cambridge University Press. (1990; Zbl 0727.00022)]. To a helix \(\mathbb{H}\) one can associate the graded algebra \(A({\mathbb{H}})\) given by the product of the morphism spaces \(\operatorname{Hom}(E_i,E_j)\) running over all \(j \geq i\). Twisting by \(\omega_Z\) gives a \(\mathbb{Z}\)-action on \(A({\mathbb{H}})\), and \(B({\mathbb{H}})\) is the quotient.
The aim of the authors is studying tilting of Calabi–Yau algebras associated to a special kind of such helices on smooth projective Fano varieties. It is known from T. Bridgeland [\(t\)-structures on some local Calabi–Yau varieties, J. Algebra 289, No. 2, 453-483 (2005; Zbl 1069.14044)] that if \(\mathbb{H}\) is a geometric helix on a smooth projective Fano variety \(Z\) of dimension \(d-1\), the algebra \(B({\mathbb{H}})\) is a graded Calabi–Yau \(d\)-dimensional quiver algebra which is Noetherian and finite over its centre. The bounded derived category \(D(B)\) is equivalent to \(D(Y)\), where \(Y\) is the total space of the canonical bundle \(\omega_Z\).
Let \(Z\) be a del Pezzo surface. The main result of the paper describes completely the tilting process of the algebras \(B({\mathbb{H}})\), for \(\mathbb{H}\) a geometric helix on \(Z\), in term of quiver mutations. Those algebras are three dimensional and their underlying quiver has no loops or oriented two-cycles. Moreover, the tilting of \(B({\mathbb{H}})\) at any vertex is of the form \(B({\mathbb{H}}')\) for some geometric helix \({\mathbb{H}}'\).
A helix of sheaves of period \(n\) on a smooth projective variety \(Z\) is an infinite collection of coherent sheaves \({\mathbb{H}} = \{ E_i \}_{i \in {\mathbb{Z}}}\) such that for all \(i \in \mathbb{Z}\), the collection \((E_{i+1}, \ldots, E_{i+n})\) is fully exceptional and \(E_{i-n} = E_i \otimes \omega_Z\). Such helices exist on many Fano varieties, for examples on projective spaces and on Fano surfaces. The main tool to describe helices is the theory of exceptional collections and mutation functors developed in [A. N. Rudakov, (ed.), Helices and vector bundles: Seminaire Rudakov. London Mathematical Society Lecture Note Series, 148. Cambridge etc.: Cambridge University Press. (1990; Zbl 0727.00022)]. To a helix \(\mathbb{H}\) one can associate the graded algebra \(A({\mathbb{H}})\) given by the product of the morphism spaces \(\operatorname{Hom}(E_i,E_j)\) running over all \(j \geq i\). Twisting by \(\omega_Z\) gives a \(\mathbb{Z}\)-action on \(A({\mathbb{H}})\), and \(B({\mathbb{H}})\) is the quotient.
The aim of the authors is studying tilting of Calabi–Yau algebras associated to a special kind of such helices on smooth projective Fano varieties. It is known from T. Bridgeland [\(t\)-structures on some local Calabi–Yau varieties, J. Algebra 289, No. 2, 453-483 (2005; Zbl 1069.14044)] that if \(\mathbb{H}\) is a geometric helix on a smooth projective Fano variety \(Z\) of dimension \(d-1\), the algebra \(B({\mathbb{H}})\) is a graded Calabi–Yau \(d\)-dimensional quiver algebra which is Noetherian and finite over its centre. The bounded derived category \(D(B)\) is equivalent to \(D(Y)\), where \(Y\) is the total space of the canonical bundle \(\omega_Z\).
Let \(Z\) be a del Pezzo surface. The main result of the paper describes completely the tilting process of the algebras \(B({\mathbb{H}})\), for \(\mathbb{H}\) a geometric helix on \(Z\), in term of quiver mutations. Those algebras are three dimensional and their underlying quiver has no loops or oriented two-cycles. Moreover, the tilting of \(B({\mathbb{H}})\) at any vertex is of the form \(B({\mathbb{H}}')\) for some geometric helix \({\mathbb{H}}'\).
Reviewer: Marcello Bernardara (Essen)
MSC:
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
| 14J25 | Special surfaces |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
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