arXiv subjects
Publications and source records attributed to David Favero.
We introduce a variant of the Cox ring using $\mathbb{Q}$-Cartier divisors and use this to remedy various deficiencies of nonsimplicial toric varieties. Our main applications are: Cox's ideal-variety correspondence, an explicit classification of subschemes and sheaves in terms of multigraded modules, an associated toric Deligne-Mumford stack, and a stacky extension of the GKZ/Mori theory of toric varieties.
We define Kuznetsov and anti-Kuznetsov categories for gauged linear sigma models. We show that for complete intersections of ample divisors in smooth projective toric varieties, the Kuznetsov category is left orthogonal to an exceptional collection. We prove that any complete intersection of $r \ge 2$ ample divisors in a Fano GIT quotient is a Fano visitor and the derived category of its Fano host is equivalent to an anti-Kuznetsov category of a gauged linear sigma model.
We state and prove a realization of King's Conjecture for a category glued from the derived categories of all of the toric varieties arising from a given Cox ring. Our perspective extends ideas of Beilinson and Bondal to all semiprojective toric varieties.
We give a topological description of Ext groups between simple representations of categories via a nerve type construction. We use it to show that the Koszulity of indiscretely based category algebras is equivalent to the locally bouquet property of this nerve. We also provide a class of functors which preserve the Koszulity of category algebras called almost discrete fibrations. Specializing from categories to posets, we show that the equivalence relations of V. Reiner and D. Stamate in arXiv:0904.1683 [math.AC] are exactly almost discrete fibrations and recover their results. As an application, we classify when a shifted dual collection to a full strong exceptional collection of line bundles on a toric variety is strong.
We consider a finite collection of line bundles $\Phi$ introduced by Bondal on a smooth, projective toric variety $X$. For any coherent sheaf $F$ on $X$, we construct minimal resolutions of $F$ by line bundles in $\Phi$, up to twist, with length bounded by the dimension of $X$ and provide explicit formulae for their Betti numbers. For a toric subvariety $Y \subset X$ of codimension $k$, we give a construction of the minimal resolution of $f_{*}\mathcal{O}_{Y}$ of length $k$ by line bundles in $\Phi$ and relate their Betti numbers to the topology of a stratified real torus. Additionally, we recover a (generally non-minimal) cellular resolution of $f_{*}\mathcal{O}_{Y}$ constructed in Hanlon-Hicks-Lazarev. Aspects of our proof run through the Coherent Constructible Correspondence, a form of homological mirror symmetry for toric varieties.
We prove that for any normal toric variety, the Rouquier dimension of its bounded derived category of coherent sheaves is equal to its Krull dimension. Our proof uses the coherent-constructible correspondence to translate the problem into the study of Rouquier dimension for certain categories of constructible sheaves.
We define a basic class of algebras which we call homotopy path algebras. We find that such algebras always admit a cellular resolution and detail the intimate relationship between these algebras, stratifications of topological spaces, and entrance/exit paths. As examples, we prove versions of homological mirror symmetry due to Bondal-Ruan for toric varieties and due to Berglund-H\"ubsch-Krawitz for hypersurfaces with maximal symmetry. We also demonstrate that a form of shellability implies Koszulity and the existence of a minimal cellular resolution. In particular, when the algebra determined by the image of the toric Frobenius morphism is directable, then it is Koszul and admits a minimal cellular resolution.
We construct GLSM invariants for a general choice of stability in both the narrow and broad sector cases and prove they form a Cohomological Field Theory. This is obtained by forming the analogue of a virtual fundamental class which lives in the local cohomology of the twisted Hodge complex. This general construction comes from the use of two new ingredients. First, the use of the Thom-Sullivan and Godement resolutions applied to matrix factorizations are introduced to handle poorly behaved (non-separated) moduli spaces. Second, a localized Chern character map built from the Atiyah class of a matrix factorization is utilized to forgo the use of Hochschild homology.
We show that there exists a cubic threefold defined by an invertible polynomial that, when quotiented by the maximal diagonal symmetry group, has a derived category which does not have a full exceptional collection consisting of line bundles. This provides a counterexample to a conjecture of Lekili and Ueda.
We prove the existence of a full exceptional collection for the derived category of equivariant matrix factorizations of an invertible polynomial with its maximal symmetry group. This proves a conjecture of Hirano--Ouchi. In the Gorenstein case, we also prove a stronger version of this conjecture due to Takahashi. Namely, that the full exceptional collection is strong.
We study a Fourier-Mukai kernel associated to a GIT wall-crossing for arbitrarily singular (not necessarily reduced or irreducible) affine varieties over any field. This kernel is closely related to a derived fiber product diagram for the wall-crossing and simple to understand from the viewpoint of commutative differential graded algebras. However, from the perspective of algebraic varieties, the kernel can be quite complicated, corresponding to a complex with multiple homology sheaves. Under mild assumptions in the Calabi-Yau case, we prove that this kernel provides an equivalence between the category of perfect complexes on the two GIT quotients. More generally, we obtain semi-orthogonal decompositions which show that these categories differ by a certain number of copies of the derived category of the derived fixed locus. The derived equivalence for the Mukai flop is recovered as a very special case.
We develop a generalization of the $Q$-construction of the first author, Diemer, and the third author for Grassmann flips. This generalization provides a canonical idempotent kernel on the derived category of the associated global quotient stack. The idempotent kernel, after restriction, induces a semi-orthogonal decomposition which compares the flipped varieties. Furthermore its image, after restriction to the geometric invariant theory semistable locus, "opens" a canonical "window" in the derived category of the quotient stack. We check this window coincides with the set of representations used by Kapranov to form a full exceptional collection on Grassmannians.
We define enumerative invariants associated to a hybrid Gauged Linear Sigma Model. We prove that in the relevant special cases, these invariants recover both the Gromov-Witten type invariants defined by Chang-Li and Fan-Jarvis-Ruan using cosection localization as well as the FJRW type invariants constructed by Polishchuk-Vaintrob. The invariants are defined by constructing a "fundamental factorization" supported on the moduli space of Landau-Ginzburg maps to a convex hybrid model. This gives the kernel of a Fourier-Mukai transform; the associated map on Hochschild homology defines our theory.
To any affine scheme with a $\mathbb{G}_m$-action, we provide a Bousfield colocalization on the equivariant derived category of modules by constructing, via homotopical methods, an idempotent integral kernel. This endows the equivariant derived category with a canonical semi-orthogonal decomposition. As a special case, we demonstrate that grade-restriction windows appear as a consequence of this construction, giving a new proof of wall-crossing equivalences which works over an arbitrary base. The construction globalizes to yield interesting integral transforms associated to $D$-flips.
We give criteria for the existence of a Serre functor on the derived category of a gauged Landau-Ginzburg model. This is used to provide a general theorem on the existence of an admissible (fractional) Calabi-Yau subcategory of a gauged Landau-Ginzburg model and a geometric context for crepant categorical resolutions. We explicitly describe our framework in the toric setting. As a consequence, we generalize several theorems and examples of Orlov and Kuznetsov, ending with new examples of semi-orthogonal decompositions containing (fractional) Calabi-Yau categories.
We prove a derived analogue to the results of Borisov, Clarke, Kelly, and Shoemaker on the birationality of Berglund-Hubsch-Krawitz mirrors. Heavily bootstrapping off work of Seidel and Sheridan, we obtain Homological Mirror Symmetry for Berglund-Hubsch-Krawitz mirror pencils to hypersurfaces in projective space.
Given the same anti-canonical linear system on two distinct toric varieties, we provide a derived equivalence between partial crepant resolutions of the corresponding stacky hypersurfaces. The applications include: a derived unification of toric mirror constructions, calculations of Picard lattices for linear systems of quartic surfaces, and a birational reduction of Reid's list to 81 families.
We prove equivalences of derived categories for the various mirrors in the Batyrev-Borisov construction. In particular, we obtain a positive answer to a conjecture of Batyrev and Nill. The proof involves passing to an associated category of singularities and toric variation of geometric invariant theory quotients.