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Main courses (6h each)
[1st week]
Cristina Manolache - Moduli Spaces of Sheaves (introductory course)
This course will be an introduction to Moduli Spaces of vector bundles. A moduli space of stable vector bundles on a smooth, algebraic variety X is a scheme whose points are in ”natural bijection” to isomorphic classes of stable vector bundles on X. Using Geometric Invariant Theory the moduli space can be constructed as a quotient of certain Quot-scheme by a natural group action. We introduce the crucial concept of stability of vector bundles over smooth projective varieties and we give a cohomological characterization of the (semi)stability. The notion of (semi)stability is needed to ensure that the set of vector bundles one wants to parameterize is small enough to be parameterized by a scheme of finite type. We introduce the formal definition of moduli functor, fine moduli space and coarse moduli space and we recall some generalities on moduli spaces of vector bundles. Then we focus on vector bundles on algebraic surfaces. Quite a lot is known in this case and we will review the main results, some of which will illustrate how the geometry of the surface is reflected in the geometry of the moduli space. Going beyond surfaces, we introduce the notion of monad, which allow the classification of vector bundles on, for instance, P^3. Monads appeared in a wide variety of contexts within algebraic geometry, and they are very useful when we want to construct vector bundles with prescribed invariants like rank, determinant, Chern classes, etc. Finally, we will study moduli spaces of vector bundles on higher dimensional varieties. As we will stress, the situation drastically differs and results like the smoothness and irreducibility of moduli spaces of stable vector bundles on algebraic surfaces turn to be false for moduli spaces of stable vector bundles on higher dimensional algebraic varieties.
REFERENCES:
D. Huybrechts, M. Lehn : The geometry of moduli spaces of sheaves
Notes by Rosa Maria Miro-Roig available at : Notes MR
Z Qin : Equivalence classes of polarizations and moduli spaces of sheaves (stability conditions lecture)
The material for this course is partially contained in:
Lyalya Guseva - Derived Categories of Sheaves (introductory course)
Derived category of coherent sheaves is a convenient environment of investigating algebraic geometry of a variety. It provides useful techniques and gives a perspective point of view. On the level of derived categories one can see unexpected connections which are not visible on the classical level. I will try to give an introduction into the techniques of derived categories and semiorthogonal decomposition and (hopefully) will present many examples of applications of derived categories in algebraic geometry.
REFERENCES:
D. Huybrechts : Fourier-Mukai Transforms in Algebraic Geometry
S. I. Gelfand , Y. I. Manin : Methods of Homological Algebra
Michel Brion - Geometric Invariant Theory (introductory course)
Geometric Invariant Theory (GIT) studies the actions of reductive algebraic groups on algebraic varieties and the constructions of quotients in algebraic geometry. One of the most relevant aspect of GIT is its application to moduli theory: we can indeed approach moduli problems trying to construct moduli spaces as GIT quotients. In this course we will present the basic notions of geometric invariant theory, we will illustrate its connections to moduli and provide concrete examples of moduli spaces obtained as GIT quotients.
The material for this course is contained in:
Abdelmoubine Amar Henni - Quiver Moduli Spaces (exercise session)
These exercise sessions will be divided in three main parts; the first part is about representations of a given quiver and morphisms between different representations. This allows one to introduce and check irreducibility of a given quiver representation. In the second part we introduce the notion of stability of a quiver representation and as an application we give some examples of the construction of the GIT quotient as done by A. King. The last part will be related to the construction and properties of moduli spaces of representations of quivers such as the Hilbert scheme of points on the affine plane C^2.
REFERENCES:
Schiffler, R. (2014). Quiver Representations.
Kirillov, A. Jr. (2016). Quiver Representations and Quiver Varieties.
King, A. D. (1994). Moduli of representations of finite-dimensional algebras.
Henni, A. A., & Jardim, M. (2018). Commuting matrices and the Hilbert scheme of points on affine spaces.
Nakajima, H. (1999). Lectures on Hilbert schemes of points on surfaces.
[2nd week]
Cristian Martinez - Bridgeland stability: the General Theory (introductory course)
Bridgeland stability is a powerful tool for extracting geometry from homological algebra. In particular, it gives a framework for studying moduli spaces of objects in a triangulated category, such as the derived category of an algebraic variety. The subject was born as a mathematical interpretation of work in string theory, but has since impacted many areas, including classical algebraic geometry, derived categories of coherent sheaves, enumerative geometry, homological mirror symmetry, and symplectic geometry. The goal of this course is to develop the foundations of Bridgeland stability, covering the following topics: 1) The theory of t-structures on triangulated categories, including examples via tilting. 2) The definition of stability conditions and the stability manifold, as well as Bridgeland's deformation theorem. 3) Constructions of stability conditions. 4) Moduli spaces of stable objects.
Slides from the third lecture: Notes Martinez Floripa 3
EXERCISES:
Marcos Jardim - Instanton Objects and Bridgeland Stability Conditions (introductory course)
Instantons, which emerged in particle physics, have been intensely studied since the 1970's and had an enormous impact on mathematics since then. In this paper, we focus on one particular way in which ideas originating in mathematical physics have guided the development of algebraic geometry in the past 40+ years. To be precise, we examine how the notion of mathematical instanton bundles in algebraic geometry has evolved from a class of vector bundles over the 3-dimensional projective space both to a class of torsion-free sheaves on projective varieties of arbitrary dimension, and to a class of objects in the derived category of Fano threefolds. The final goal of the course is to determine suitable chambers in the space of Bridgeland stability conditions that contain these instanton objects, which include the classical rank 2 instanton bundles over the projective space, as Bridgeland stable objects, and describe the corresponding Bridgeland moduli space.
The material for this course can be found at:
Xiaolei Zhao - Noncommutative abelian surfaces and Kummer type kyperkähler varieties (advanced course)
Examples of noncommutative K3 surfaces arise from semiorthogonal decompositions of the bounded derived category of certain Fano varieties. The most interesting cases are those of cubic fourfolds and Gushel-Mukai varieties of even dimension. Using the deep theory of families of stability conditions, locally complete families of hyperkähler manifolds deformation equivalent to Hilbert schemes of points on a K3 surface have been constructed from moduli spaces of stable objects in these noncommutative K3 surfaces. On the other hand, an explicit description of a locally complete family of hyperkähler manifolds deformation equivalent to a generalized Kummer variety is not available from classical geometry.
In this lecture series, we will construct families of noncommutative abelian surfaces as equivariant categories of the derived category of K3 surfaces which specialize to Kummer K3 surfaces. Then we will explain how to induce stability conditions on them and produce examples of locally complete families of hyperkähler manifolds of Kummer type. Based on joint work with Arend Bayer, Alex Perry and Laura Pertusi.
Hannah Dell - Stability conditions and group actions (advanced course)
Group actions on categories arise naturally from symmetries of varieties and quivers, but how does this interact with Bridgeland stability? In the first half of this course we will introduce equivariant categories, which generalises the category of equivariant sheaves. Then we will show there is a correspondence between stability conditions on a category with a finite group action, and stability conditions on the equivariant category -- this will also play a role in Xiaolei Zhao's course. We will use this to produce stability conditions on quotient varieties (and stacks).
In the second half of the course, we will apply this to study open questions about the geometry of the stability manifold: in particular, we will discuss "geometric stability conditions" -- those for which all skyscraper sheaves of points are stable. In practice, these are constructed using slope stability for sheaves. Some varieties have only geometric stability conditions, whereas in other cases, there are more (for example if there is an equivalence with quiver representations). Lie Fu, Chunyi Li, and Xiaolei Zhao were the first to provide a general result explaining this phenomenon. In particular, they showed that if a variety has a finite map to an abelian variety, then all stability conditions are geometric. We will test the converse on free quotients of abelian varieties by finite groups, including Beauville-type and bielliptic surfaces. This is based on joint work with Edmund Heng and Anthony Licata.
The material for this course can be found at:
Research talks (30min each)
Crislaine Kuster (1st week) - Codimension one foliations on adjoint varieties
In this talk, I will present the classification of codimension one foliations on adjoint varieties with the most positive anti-canonical class. As these varieties are uniruled, we define the degree of a foliation with respect to a family of rational curves on an uniruled variety. Let X be an adjoint variety. When X has Picard number one, such foliations are precisely those of degree zero with respect to a minimal family of rational curves on X. We show that all such foliations are induced by pencils of hyperplane sections under the minimal embedding. When X has Picard number two, it carries two distinct families of minimal dominating rational curves. I will present the classification of codimension one foliations of degree zero with respect to both families.
Hipolito Treffinger (1st week) - Stability conditions applied to representation theory of finite-dimensional algebras
In this we will show how the set of all possible (King's) stability conditions over the module category of a finite-dimensional algebra encodes plenty of the homological information of the category, and in particular of its torsion classes. Time permitting, we will show how one can use this information to study how many isomorphism classes of bricks a finite-dimensional algebra has in its module category. This talk is based on joint work with Thomas Brüstle, David Smith and Sibylle Schroll.
Leonardo Roa Leguizamon (2nd week) - Stability conditions for coherent systems on integral curves
In this talk, we define Bridgeland stability conditions on the derived category of coherent systems on an integral curve. By studying semistable objects, we get some results concerning bounds for the dimension of the space of global sections of torsion-free sheaves on C. It’s a joint work with Marcos Jardim and Renato Vidal Martins.
Lucas Mioranci (2nd week) - Normal and tangent bundles of rational curves on projective hypersurfaces.
Let C ⊂ P^n be a rational normal curve of degree e ≤ n. For any hypersurface X containing C, the normal bundle NC/X splits as a direct sum of line bundles called its splitting type. We determine all possible splitting types of NC/X and find explicit examples of degree d hypersurfaces that induce each one of them. Additionally, for d ≥ 3, we compute the dimension of the space of hypersurfaces X such that NC/X has a given splitting type.
We classify all triples (e,d,n) such that a general degree d hypersurface X ⊂ P^n contains a rational curve C of degree e whose restricted tangent bundle TX|C is balanced. In particular, we compute the maximum number t of general points of X that can be interpolated by morphisms from general t-pointed P^1. As a notable case, we show that odd-degree rational curves do not interpolate the expected number of points on quadric hypersurfaces.
Leonardo Soares Moço (2nd week) - Hecke Modifications and Parabolic Bundles: A Stack-Theoretic Correspondence
This work investigates the relationship between Hecke modifications and parabolic bundles on algebraic curves. While Hecke modifications alter vector bundles at specified points, parabolic bundles incorporate flags and weights at marked points. In the rank 2 case, a well-known correspondence exists between Hecke modifications and quasi-parabolic structures, extending naturally to moduli spaces.
We aim to generalize this correspondence to vector bundles of arbitrary rank and an arbitrary number of marked points. This requires analyzing sequences of Hecke modifications and formulating the problem in the language of algebraic stacks.
Our main result, based on recent work by Alvarenga, Kaur, and Moço, establishes a natural transformation between the Hecke stack and the parabolic stack, revealing a deep structural connection between these two moduli problems. This correspondence offers new tools for understanding Hecke modifications in higher rank and contributes to open questions in the theory of parabolic bundles.
Arpan Saha (2nd week) - Real analogues of the Bridgeland–Strachan construction
In recent work, Tom Bridgeland and Ian Strachan have interpreted differential equations satisfied by generating functions encoding Donaldson–Thomas invariants of a triangulated Calabi–Yau threefold category in terms of natural geometric structures on the space of stability conditions of the category. In particular, we have a one-parameter family of flat holomorphic symplectic Ehresmann connections on the tangent bundle giving rise to a complex hyperkähler structure on the total space. In this talk, I discuss two related real versions of this construction, both of which produce real hyperkähler structures as output. In the first case, this real hyperkähler structure is defined on the space of stability conditions itself equipped with some extra data, and in the second case, on the total space of the tangent bundle of a manifold which has half the dimension as that of the space of stability conditions and is equipped with a two-parameter family of flat symplectic Ehresmann connections. The second construction turns out to generalise the c-map in supergravity literature. This is based on joint work in progress with Vicente Cortés, Alejandro Gil García, and Iván Tulli.
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