| Summary: | <p>This thesis studies derived equivalences between total spaces of vector bundles and dg-quivers.</p> <p>A dg-quiver is a graded quiver whose path algebra is a dg-algebra. A quiver with superpotential is a dg-quiver whose differential is determined by a "function" Φ. It is known that the bounded derived category of representations of quivers with superpotential with finite dimensional cohomology is a Calabi- Yau triangulated category. Hence quivers with superpotential can be viewed as noncommutative Calabi- Yau manifolds.</p> <p>One might then ask if there are derived equivalences between Calabi-Yau manifolds and quivers with superpotential. In this thesis, we answer this question and, generalizing Bridgeland [15], give a recipe on how to construct such derived equivalences.</p> <p>Let π : <em>V</em> &rightarrow; <em>X</em> be an anti-semiample vector bundle over a smooth projective variety <em>X</em>, i.e., <em>S<sup>k</sup>V</em><sup>V</sup> is globally generated for <em>k</em> » 0. Given a full exceptional sequence <em>E</em> on D<sup>b</sup>(Coh (<em>X</em>)), under some cohomological vanishing conditions, we construct a dg-quiver <em>Q<sub>E</sub></em> in terms of the dual exceptional sequence of <em>E</em> such that D<sup><em>b</em></sup>(Coh (<em>V</em>)) ≅ D<sup><em>b</em></sup><sub style="position: relative; left: -.5em;"><em>fg</em></sub>(Rep(<em>Q<sub>E</sub></em> )). Moreover, this equivalence restricts to an equivalence between D<sup><em>b</em></sup><sub style="position: relative; left: -.5em;"><em>cs</em></sub>(Coh (<em>V</em>)), the full subcategory containing complexes of compact support, and D<sup><em>b</em></sup><sub style="position: relative; left: -.5em;"><em>fd</em></sub>(Rep(<em>Q<sub>E</sub></em> )), the full subcategory containing complexes with finite dimensional cohomology. If <em>V</em> is non-compact Calabi- Yau, we show that <em>Q<sub>E</sub></em> is equipped with a superpotential Φ, i.e., the differential on <em>Q<sub>E</sub></em> is determined by the "function" Φ. In this case, the triangulated categories D<sup><em>b</em></sup><sub style="postion: relative; left: -.5em;"><em>cs</em></sub>(Coh (<em>V</em>)) and D<sup><em>b</em></sup><sub style="position: relative; left: -.5em;"><em>fd</em></sub>(Rep(<em>Q<sub>E</sub></em> )) are both Calabi-Yau.</p> <p>We can also construct derived equivalences equivariantly. Suppose a finite group <em>G</em> acts on <em>X</em> and this action lifts to <em>V</em> , endowing π : <em>V</em> &rightarrow; <em>X</em> the structure of an equivariant vector bundle. Suppose further that each object in the exceptional sequence <em>E</em> is equipped with a <em>G</em>-linearization. Then we can construct a quotient dg-quiver <em>Q<sub>E</sub></em>/<em>G</em> from <em>Q<sub>E</sub></em> , generalizing the construction of the McKay quiver, such that D<sup><em>b</em></sup>(Coh <sup><em>G</em></sup>(<em>V</em> )) ≅ D<sup><em>b</em></sup>(Rep<sub>fg</sub>(<em>Q<sub>E</sub></em>/<em>G</em>)). If <em>V</em> is non-compact Calabi-Yau equivariantly, then <em>Q<sub>E</sub></em>/<em>G</em> is also equipped with a superpotential.</p> <p>We also give a product construction for derived equivalences. Suppose we have vector bundles π<sub><em>V</em></sub> : <em>V</em> &rightarrow; <em>X</em> and π<sub><em>W</em></sub> : <em>W</em> &rightarrow; <em>Y</em> , with full exceptional sequences <em>E</em> on D<sup><em>b</em></sup>(Coh (<em>V</em> )) (resp. <em>F</em> on D<sup><em>b</em></sup>(Coh (<em>W</em>))), then we can construct a product dg-quiver <em>Q<sub>E</sub></em> x <em>Q<sub>F</sub></em> such that D<sup><em>b</em></sup>(Coh (<em>V</em> x <em>W</em>)) ≅ D<sup><em>b</em></sup>(Rep<sub>fg</sub>(<em>Q<sub>E</sub></em> x <em>Q<sub>F</sub></em>)). If both <em>V</em> and <em>W</em> are Calabi-Yau, then <em>Q<sub>E</sub></em> x <em>Q<sub>F</sub></em> is also equipped with a superpotential. Using these constructions, we can produce a lot of beautiful pictures of quivers with superpotential derived equivalent to the total spaces of vector bundles which are Calabi-Yau. Examples include <em>T</em><sup>V</sup><sub style="position: relative; left: -.5em;"><sub>ℙ</sub>2</sub> , <em>K</em><sub><sub>ℙ</sub><em>n</em></sub>, and <em>O</em><sub><sub>ℙ</sub>2</sub> (-1) ⊕ <em>O</em><sub><sub>ℙ</sub>2</sub> (-2) etc.</p> <p>Finally, we try to connect quivers with superpotential to the recent work by Pantev, Toën, Vaquiée and Vezzosi [58] and Ben-Bassat, Brav, Bussi and Joyce [4] on shifted symplectic structures. We outline a strategy of proof for the existence of shifted symplectic structures in a standard 'Darboux form' on the derived moduli stack of representations of quivers with superpotential.</p>
|
|---|