On cyclic strong exceptional collections of line bundles on surfaces

Elagin, Alexey; Xu, Junyan; Zhang, Shizhuo

doi:10.1007/s40879-020-00417-w

Cited by 4 publications

(2 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Lemma 2.7 in Ref. [33]). On del Pezzo surfaces the same statement holds, with the exception that there are no −2 curves.…”

Section: Application To Generalised Del Pezzo Surfacesmentioning

confidence: 95%

Cohomology Chambers on Complex Surfaces and Elliptically Fibered Calabi-Yau Three-folds

Brodie,

Constantin

2020

Preprint

View full text Add to dashboard Cite

We determine several classes of smooth complex projective surfaces on which Zariski decomposition can be combined with vanishing theorems to yield cohomology formulae for all line bundles. The obtained formulae express cohomologies in terms of divisor class intersections, and are adapted to the decomposition of the effective cone into Zariski chambers. In particular, we show this occurs on generalised del Pezzo surfaces, toric surfaces, and K3 surfaces. In the second part we use these surface results to derive formulae for all line bundle cohomology on a simple class of elliptically fibered Calabi-Yau three-folds. Computing such quantities is a crucial step in deriving the massless spectrum in string compactifications.

show abstract

“…Lemma 2.7 in Ref. [33]). On del Pezzo surfaces the same statement holds, with the exception that there are no −2 curves.…”

Section: Application To Generalised Del Pezzo Surfacesmentioning

confidence: 95%

Cohomology Chambers on Complex Surfaces and Elliptically Fibered Calabi-Yau Three-folds

Brodie,

Constantin

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…The product of varieties from (iii) [Yan16]. The product of two Fermat elliptic curves with the Fermat surface [BFK14, Theorem 1.6]. Smooth toric Fano varieties of dimension or [BDM19, Propositions 2.15 and 2.16]. Certain blow-ups of the projective space [Pir19, Theorem 4.1]. Some weak del Pezzo surfaces [EXZ21, Corollary 7.5].…”

Section: Introductionmentioning

confidence: 99%

On the Rouquier dimension of wrapped Fukaya categories and a conjecture of Orlov

Bai¹,

Côté²

2023

Compositio Math.

View full text Add to dashboard Cite

We study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds and pairs, and apply this invariant to various problems in algebraic and symplectic geometry. On the algebro-geometric side, we introduce a new method based on symplectic flexibility and mirror symmetry to bound the Rouquier dimension of derived categories of coherent sheaves on certain complex algebraic varieties and stacks. These bounds are sharp in dimension at most $3$ . As an application, we resolve a well-known conjecture of Orlov for new classes of examples (e.g. toric $3$ -folds, certain log Calabi–Yau surfaces). We also discuss applications to non-commutative motives on partially wrapped Fukaya categories. On the symplectic side, we study various quantitative questions including the following. (1) Given a Weinstein manifold, what is the minimal number of intersection points between the skeleton and its image under a generic compactly supported Hamiltonian diffeomorphism? (2) What is the minimal number of critical points of a Lefschetz fibration on a Liouville manifold with Weinstein fibers? We give lower bounds for these quantities which are to the best of the authors’ knowledge the first to go beyond the basic flexible/rigid dichotomy.

show abstract

Mutations of numerically exceptional collections on surfaces

Krah

2024

Math. Z.

View full text Add to dashboard Cite

A conjecture of Bondal–Polishchuk states that, in particular for the bounded derived category of coherent sheaves on a smooth projective variety, the action of the braid group on full exceptional collections is transitive up to shifts. We show that the braid group acts transitively on the set of maximal numerically exceptional collections on rational surfaces up to isometries of the Picard lattice and twists with line bundles. Considering the blow-up of the projective plane in up to 9 points in very general position, these results lift to the derived category. More precisely, we prove that, under these assumptions, a maximal numerically exceptional collection consisting of line bundles is a full exceptional collection and any two of them are related by a sequence of mutations and shifts. The former extends a result of Elagin–Lunts and the latter a result of Kuleshov–Orlov, both concerning del Pezzo surfaces. In contrast, we show in concomitant work (Krah in Invent Math 235(3):1009–1018, 2024) that the blow-up of the projective plane in 10 points in general position admits a non-full exceptional collection of maximal length consisting of line bundles.

show abstract

On cyclic strong exceptional collections of line bundles on surfaces

Cited by 4 publications

References 21 publications

Cohomology Chambers on Complex Surfaces and Elliptically Fibered Calabi-Yau Three-folds

Cohomology Chambers on Complex Surfaces and Elliptically Fibered Calabi-Yau Three-folds

On the Rouquier dimension of wrapped Fukaya categories and a conjecture of Orlov

Mutations of numerically exceptional collections on surfaces

Contact Info

Product

Resources

About