Given a quasi-projective algebraic variety, X, with the action of a linear algebraic group, G, there are various (birational) incarnations of the quotient X/G coming from a choice of a G-equivariant ample line bundle. As we vary this choice, there is a semi-orthogonal relationship between the derived categories of the resulting quotients, A and B. Furthermore, if (X,w) is a Landau-Ginzburg model, and w is a G-invariant section of a line bundle on X, then the same holds for "coherent sheaves on" (A,w) and (B,w) (categories of matrix factorizations/categories of singularities/stable derived categories). As a special case, one can reproduce a theorem of Orlov relating categories of coherent sheaves for complete intersections in projective space to the graded category of singularities of the cone, a theorem of Herbst and Walcher demonstrating an equivalence of derived categories between "neighboring" Calabi-Yau complete intersections in toric varieties, and two theorems of Kawamata; one concerning behavior of derived categories of algebraic varieties under simple toroidal flips, the other stating that the derived category of coherent sheaves on any smooth toric variety has a full exceptional collection (in the projective case). If time permits, I will also discuss the relationship with Kuznetsov's homological projective duality.
This talk describes various measures of "badness" of singularities of commutative rings of prime characteristic. The first part of the talk describes the development of tight-closure as a tool to measure singularities. The second part of the talk deals with certain numerical invariants of rings of prime characteristic, namely $F$-jumping coefficients, and their properties.
The BGP reflection is a combinatorial operation on quivers defined at vertices which are sinks or sources and preserves the number of arrows. From a representation-theoretic perspective, it induces derived equivalence between the path algebras. The Fomin-Zelevinsky mutation extends this operation to arbitrary vertices on a combinatorial level, whereas the mutation of quivers with potentials (QP) introduced by Derksen, Weyman and Zelevinsky does this algebraically. However, in general the number of arrows is no longer preserved and the corresponding Jacobian algebras are not necessarily derived equivalent. In this talk we will characterize all the quivers with the property that performing arbitrary sequences of mutations does not change their number of arrows. It turns out that these quivers arise from ideal triangulations of certain marked bordered surfaces in the sense of Fomin, Shapiro and Thurston. This combinatorial property has also a representation-theoretic counterpart: to each such quiver there is a naturally associated potential such that performing arbitrary sequences of QP mutations does not change the derived equivalence class of the corresponding Jacobian algebra. Most of these algebras are finite-dimensional and gentle, but some of them are infinite-dimensional and locally gentle. The latter resemble the 3-Calabi-Yau algebras despite not being so, and their quivers admit a covering by an infinite quiver arising from triangulations of an infinity-gon.
A dimer model on a surface with punctures is an embedded quiver such that its vertices correspond to the punctures and the arrows circle round the faces they cut out. To any dimer model Q we can associate 2 categories: A wrapped Fukaya category F(Q) and a category of matrix factorizations M(Q). In both categories the objects are arrows, which are interpreted as Lagrangian subvarieties in F(Q) and will give us certain matrix factorizations of a potential on the Jacobi algebra of the dimer in M(Q). We show that there is a duality on the set of all dimers such that for consistent dimers the category of matrix factorizations M(Q) is isomorphic to the Fukaya category of its dual, F(D(Q)). We also discuss the connection with classical mirror symmetry.
We present a strategy for proving that full exceptional collections of vector bundles on projective n-space can be constructed by mutation from a standard collection of line bundles, reducing the question of constructibility to the problem of freeness of (derived) monodromy groups of associated families of Calabi-Yau varieties. We use the ping-pong lemma of Fricke-Klein to solve this problem in low dimensions, thus providing a new and more informative proof of constructibility of exceptional collections in some cases. We expect a similar ping-pong argument to give constructibility on projective n-space and on some other Fano varieties of Picard rank one. This is joint work in progress with Hugh Thomas.
In recent joint work with Benjamin Nill, we introduced a natural generalisation of the notion of a reflexive polytope. These "l-reflexive polytopes" also appear as dual pairs, and in two-dimensions they arise from reflexive polygons via a change of the underlying lattice. Furthermore, any reflexive polygon of arbitrary index satisfies the famous “number 12” property. This is a new, infinite class of lattice polygons possessing this property, and extends the previously known sixteen instances.
In the spirit of Arcara & Bertram, we investigate wall-crossing phenomena in the stability manifold of an irreducible principally polarized abelian surface for objects with the same invariants as (twists of) ideal sheaves of points. In particular, we construct a sequence of fine moduli spaces which are related by Mukai flops and observe that the stability of these objects is completely determined by the configuration of points. Finally, we use Fourier-Mukai theory to show that these moduli are projective.
Let G be a finite subgroup of SL(3,C) and N be a normal subgroup of G. Then G-Hilb and G/N-Hilb(N-Hilb) are both crepant resolutions of C^3/G. In the talk I will explain how can we construct them as moduli spaces of representations of the McKay quiver, also known as moduli of G-constellations, and how can we calculate them explicitly in several cases. I will also address the problem of whether they are isomorphic (or not) as moduli spaces and as algebraic varieties. This is a joint work with A. Ishii and Y. Ito.
We study the stability spaces Stab(Q) and Stab(Γ_N Q), in the sense of Bridgeland, for the bounded derived category of a Dynkin quiver Q and the finite-dimensional derived category of the Calabi-Yau-N Ginzburg algebra Γ_N Q associated to Q. We will review the result of King-Qiu about the exchange graphs of hearts in such derived categories and cluster exchange graphs. Then we prove the simply connectedness of such spaces (which provides a topological realization of almost complete cluster tilting objects). A point of view is that, the quotient space of Stab(Γ_N Q) by the Seidel-Thomas braid group should be the stability space for the higher cluster category of Q.
This talk is on joint work with Robert Marsh. We describe a Landau-Ginzburg model for an arbitrary Grassmannian X in terms of Pluecker coordinates on a dual Grassmannian. This LG-model gives rise to a vector bundle with Gauss-Manin connection, which we relate explicitly with a bundle on the A-model side of X, the trivial bundle with fibre H*(X) and Dubrovin-Givental connection. The latter connection is defined in terms of the quantum cohomology of X. Our work makes use of the cluster structure of the homogeneous coordinate ring of the dual Grassmannian and involves some beautiful Postnikov diagrams.
I shall describe the polarised moduli of one of the four known types of compact irreducible symplectic variety, the 10-dimensional examples of O'Grady. The moduli space has dimension 21 and its birational geometry may be studied using modular forms, along the lines previously used for moduli of K3 surfaces and of deformations of their Hilbert schemes. This is joint work with V. Gritsenko and K. Hulek.