Forthcoming events in this series
(HoRSe seminar) Realizations of motives
Abstract
A categorification of cycle class maps consists to define
realization functors from constructible motivic sheaves to other
categories of coefficients (e.g. constructible $l$-adic sheaves), which are compatible with the six operations. Given a field $k$, we
will describe a systematic construction, which associates,
to any cohomology theory $E$, represented in $DM(k)$, a
triangulated category of constructible $E$-modules $D(X,E)$, for $X$
of finite type over $k$, endowed with a realization functor from
the triangulated category of constructible motivic sheaves over $X$.
In the case $E$ is either algebraic de Rham cohomology (with $char(k)=0$), or $E$ is $l$-adic cohomology, one recovers in this way the triangulated categories of $D$-modules or of $l$-adic sheaves. In the case $E$ is rigid cohomology (with $char(k)=p>0$), this construction provides a nice system of $p$-adic coefficients which is closed under the six operations.
(HoRSe seminar) Motivic sheaves over excellent schemes
Abstract
Starting from Morel and Voevodsky's stable homotopy theory of schemes, one defines, for each noetherian scheme of finite dimension $X$, the triangulated category $DM(X)$ of motives over $X$ (with rational coefficients). These categories satisfy all the the expected functorialities (Grothendieck's six operations), from
which one deduces that $DM$ also satisfies cohomological proper
descent. Together with Gabber's weak local uniformisation theorem,
this allows to prove other expected properties (e.g. finiteness
theorems, duality theorems), at least for motivic sheaves over
excellent schemes.
(HoRSe seminar) ''Stability conditions on the local projective plane and $\Gamma_1(3)$-action II'
Abstract
We report on joint work with Arend Bayer on the space of stability conditions for the canonical bundle on the projective plane.
We will describe a connected component of this space, generalizing and completing a previous construction of Bridgeland.
In particular, we will see how this space is related to classical results of Drezet-Le Potier on stable vector bundles on the projective plane. Using this, we can determine the group of autoequivalences of the derived category. As a consequence, we can identify a $\Gamma_1(3)$-action on the space of stability conditions, which will give a global picture of mirror symmetry for this example.
In the second hour we will give some details on the proof of the main theorem.
(HoRSe seminar) 'Stability conditions on the local projective plane and $\Gamma_1(3)$-action I'
Abstract
We report on joint work with Arend Bayer on the space of stability conditions for the canonical bundle on the projective plane.
We will describe a connected component of this space, generalizing and completing a previous construction of Bridgeland.
In particular, we will see how this space is related to classical results of Drezet-Le Potier on stable vector bundles on the projective plane. Using this, we can determine the group of autoequivalences of the derived category. As a consequence, we can identify a $\Gamma_1(3)$-action on the space of stability conditions, which will give a global picture of mirror symmetry for this example.
In the second hour we will give some details on the proof of the main theorem.
Symplectic homology of 4-dimensional Weinstein manifolds and Legendrian homology of links
Abstract
We show how to compute the symplectic homology of a 4-dimensional Weinstein manifold from a diagram of the Legendrian link which is the attaching locus of its 2-handles. The computation uses a combination of a generalization of Chekanov's description of the Legendrian homology of links in standard contact 3-space, where the ambient contact manifold is replaced by a connected sum of $S^1\times S^2$'s, and recent results on the behaviour of holomorphic curve invariants under Legendrian surgery.
Isotopy of Lagrangian submanifolds
Abstract
Lagrangian submanifolds are an important class of objects in symplectic geometry. They arise in diverse settings: as vanishing cycles in complex algebraic geometry, as invariant sets in integrable systems, as Heegaard tori in Heegaard-Floer theory and of course as "branes" in the A-model of mirror symmetry. We ask the difficult question: when are two Lagrangian submanifolds isotopic? Restricting to the simplest case of Lagrangian spheres in rational surfaces we will give examples where this question has a complete answer. We will also give some very pictorial examples (due to Seidel) illustrating how two Lagrangians can fail to be isotopic.
Normal forms for lattice polarized K3 surfaces and the Kuga-Satake Hodge Conjecture II (HoRSe seminar)
Normal forms for lattice polarized K3 surfaces and the Kuga-Satake Hodge Conjecture (HoRSe seminar)
Thom polynomials and the Green-Griffiths conjecture
Abstract
The Green-Griffiths conjecture from 1979 says that every projective algebraic variety $X$ of general type contains a certain proper algebraic subvariety $Y$ such that all nonconstant entire holomorphic curves in $X$ must lie inside $Y$. In this talk we explain that for projective hypersurfaces of degree $d>dim(X)^6$ this is the consequence of a positivity conjecture in global singularity theory.
(HoRSe seminar) Cluster category and applications
Abstract
I will introduce the theory of cluster categories after Amiot and Plamondon. For a quiver with a potential, the cluster category is defined as the quotient of the category of perfect dg-modules by the category of dg-modules with finite dimensional cohomologies. We can show the existence of the equivalence in the first talk as an application of the cluster category. I will also propose a definition of a counting invariant for each element in the cluster category.
(HoRSe seminar) Quiver mutations and stability conditions
Abstract
Let $(Q',w')$ be a quiver with a potential given by successive mutations from a quiver with a potential $(Q,w)$. Then we have an equivalence of the derived categories of dg-modules over the Ginzburg dg-algebras satisfying the following condition: a simple module over the dg-algebra for $(Q',w')$ is either concentrated on degree 0 or concentrated on degree 1 as a dg-module over the
dg-algebra for $(Q,w)$. As an application of this equivalence, I will give a description of the space of stability conditions.
Moduli Spaces of Sheaves on Toric Varieties
Abstract
Extending work of Klyachko, we give a combinatorial description of pure equivariant sheaves on a nonsingular projective toric variety X and use this description to construct moduli spaces of such sheaves. These moduli spaces are explicit and combinatorial in nature. Subsequently, we consider the moduli space M of all Gieseker stable sheaves on X and describe its fixed point locus in terms of the moduli spaces of pure equivariant sheaves on X. As an application, we compute generating functions of Euler characteristics of M in case X is a toric surface. In the torsion free case, one finds examples of new as well as known generating functions. In the pure dimension 1 case using a conjecture of Sheldon Katz, one obtains examples of genus zero Gopakumar-Vafa invariants of the canonical bundle of X.
(HoRSe seminar) Gromov-Witten Invariants and Modular Forms II
Abstract
I will show that generating functions for certain non-compact Calabi-Yau 3-folds are modular forms. This is joint work with Hiroshi Iritani.
(HoRSe seminar) Gromov-Witten Invariants and Modular Forms I
Abstract
I will show that generating functions for certain non-compact
Calabi-Yau 3-folds are modular forms. This is joint work with Hiroshi
Iritani.
Mutations of Quivers in the Minimal Model Programme
Abstract
Following work of Bridgeland in the smooth case and Chen in the terminal singularities case, I will explain a proposal that extends the existence of flops for threefolds (and the required derived equivalences) to also cover canonical singularities. Moreover this technique conjecturally says much more than just the existence of the flop, as it shows how the dual graph changes under the flop and also which curves in the flopped variety contract to points without contracting divisors. This allows us to continue the Minimal Model Programme on the flopped variety in an easy way, thus producing many varieties birational to the given input.
(HoRSe seminar) Symmetric and reduced obstruction theories
Abstract
I will describe some more of the deformation theory necessary for the first talk. This leads to a number of natural questions and counterexamples. This talk requires a strong stomach, or a fanatical devotion to symmetric obstruction theories.
(HoRSe seminar) GW/stable pairs on K3 surfaces
Abstract
(This is joint work with Davesh Maulik and Rahul Pandharipande. Only they understand the actual formulae. People who like modular forms are not encouraged to come to this talk.)
Big rational surfaces
Abstract
The Cox ring of a variety is an analogue of the homogeneous coordinate ring of projective space. Cox rings are not defined for every variety and even when they are defined, they need not be finitely generated. Varieties for which the Cox ring is finitely generated are called Mori dream spaces and, as the name suggests, they are particularly well-suited for the Minimal Model Program. Such varieties include toric varieties and del Pezzo surfaces.
I will report on joint work with T. Várilly and M. Velasco where we introduce a class of smooth projective surfaces having finitely generated Cox ring. This class of surfaces contains toric surfaces and (log) del Pezzo surfaces.
(HoRSe seminar at Imperial College) Moduli of Calabi-Yau 3-folds and instantons on $G_2$ manifolds
Abstract
This talk will be largely speculative. First we consider the formal properties that could be expected of a "topological field theory" in 6+1 dimensions defined by $G_2$ instantons. We explain that this could lead to holomorphic bundles over moduli spaces of Calabi-Yau 3-folds whose ranks are the DT-invariants. We also discuss in more detail the compactness problem for $G_2$ instantons and associative submanifolds.
The talk will be held in Room 408, Imperial College Maths Department, Huxley Building, 180 Queen’s Gate, London.
(HoRSe seminar at Imperial college) Gauge theory and exceptional holonomy
Abstract
This talk will review material, well-known to specialists, on calibrated geometry and Yang-Mills theory over manifolds with holonomy $SU(3)$, $G_2$ or $Spin(7)$. We will also describe extensions of the standard set-up, modelled on Gromov's "taming forms" for almost-complex structures.
The talk will be held in Room 408, Imperial College Maths Department, Huxley Building, 180 Queen’s Gate, London.
Constant scalar curvature orbifold metrics and stability of orbifolds through embeddings in weighted projective spaces
Abstract
There is a conjectural relationship due to Yau-Tian-Donaldson between stability of projective manifolds and the existence of canonical Kahler metrics (e.g. Kahler-Einstein metrics). Embedding the projective manifold in a large projective space gives, on one hand, a Geometric Invariant Theory stability problem (by changing coordinates on the projective space) and, on the other, a notion of balanced metric which can be used to approximate the canonical Kahler metric in question. I shall discuss joint work with Richard Thomas that extends this framework to orbifolds with cyclic quotient singularities using embeddings in weighted projective space, and examples that show how several obstructions to constant scalar curvature orbifold metrics can be interpreted in terms of stability.