Schedule for: 22w5187 - Moduli, Motives and Bundles – New Trends in Algebraic Geometry

Classical enumerative geometry counts solutions to “geometric problems” in algebraic geometry that are expected to have a finite number of solutions, or more generally compute integer invariants of algebro-geometrical objects. Typical examples include: $$\begin{align*} &\text{• Bézout’s theorem: how many points of intersection are there among} \;n\; \text{hypersurfaces of degrees} \;d_1, . . . , d_n\; \text{in } \;\mathbb{P}^n\;\text{for example two curves} \;C_1, C_2\; \text{of degrees} \;d_1, d_2\; \text{in}\;\mathbb{P}^2?\\ &\text{• Find a formula for the Euler characteristic of a smooth hypersurface of degree}\;d\; \text{in}\;\mathbb{P}^n.\\ &\text{• How many lines are there on a (smooth) hypersurface of degree}\;2n − 3\; \text{in}\; \mathbb{P}^n\; \text{for example, how many lines are there on a smooth cubic surface in}\;\mathbb{P}^3?\\ &\text{• How many rational plane curves of degree d pass through}\;3d − 1\;\text{general points in }\;\mathbb{P}^2?\\ &\text{• How many conics in}\;\mathbb{P^2}\;\text{are tangent to}\;5\;\text{general lines}?\\ \end{align*}$$ Usually one looks for an answer to such questions over an algebraically closed field, where essentially discrete, topological invariants will give at least a first approximation to an answer. The goal of “quadratic” enumerative geometry is to refine the typically $\mathbb{Z}$-valued answer to an enumerative problem over an algebraically closed field to an element of the Grothendieck-Witt ring of non-degenerate quadratic forms over a field $k$ over which the problem makes sense, in the hope that this finer invariant will give additional information about the set of solutions over $k$. In this first lecture, we will concentrate on the example of the quadratic Euler characteristic, which has an abstract definition, but is also amenable to concrete computations.

Nonabelian Hodge theory can be seen as a vast generalization of the theory of harmonic forms on smooth and projective complex varieties. It is due to Hitchin, Simpson, Donaldson and Corlette. The fundamental result of the theory is the existence of a correspondence between the “Betti cohomology” and the “Dolbeault cohomology” of the variety. In his 1992 paper ‘Higgs bundles and local systems’, Carlos Simpson observed that the Betti side generalises to non-constant coefficients, and asked about the corresponding Dolbeault space in that case. The purpose of this course is to explain one possible answer to Simpson’s question. In the three lectures, we will cover the following topics: $$\begin{align*} &\text{1. Twisted local systems}\\ &\text{2. Twisted character varieties}\\ &\text{3. Higgs bundles for non-constant groups}\\ \end{align*}$$

Abstract: Lagrange interpolation is a classical and elementary tool with important implications in mathematics. One of its higher dimensional generalizations, involving higher-rank vector bundles, is a topic of current research with connections in birational geometry. In this talk I will discuss two compelling examples where interpolation problems have clear implications in the birational geometry of certain moduli spaces.

I want to explain some criteria for checking algebraicity of Hodge (and Tate) cycles on varieties fibered over lower dimensional ones. This will be applied to the checking the Hodge (and Tate) conjectures for some families of curves and abelian varieties over $M_2$ and certain Shimura curves respectively.

Ulrich bundles are vector bundles that were introduced in the 1980s in a commutative algebra context, by studying finitely generated modules over Cohen-Macaulay rings of positive dimension. Their first appearances in algebraic geometry date back to the works of Beauville (2000) and Eisenbud-Schreyer-Weyman (2003) where it is shown, for example, that the existence of an Ulrich bundle on a hypersurface is closely related to being able to express the equation that defines it as (a power of) a determinant of a matrix whose coefficients are linear forms. It is an open problem to prove the existence of Ulrich bundles for every (polarized) smooth projective algebraic variety, and there are many recent results in particular cases. In this talk, we will give a short introduction to Ulrich bundles and review some of the techniques used for constructing low-rank vector bundles. Finally, we will present a characterization of projective manifolds whose tangent bundle is Ulrich, by means of studying restrictions on the Chern classes of such bundles and by reducing the question to a Lie theoretic problem on rational homogeneous spaces. This is a joint work with Vladimiro Benedetti, Yulieth Prieto, and Sergio Troncoso.

We extend the concept of the Segre Invariant for vector bundles on a curve to vector bundles on a surface X. For a vector bundle $E$ of rank $2$ on $X$, the Segre invariant is defined as the minimum of the differences between the slope of $E$ and the slope of all line subbundles of $E$. This invariant defines a semicontinuous function on the families of vector bundles on $X$. Thus, the Segre invariant gives a stratification of the moduli space $M_{X, H}(2; c_1, c_2)$ of $H$-stable vector bundles of rank $2$and fixed Chern classes $c_1$ and $c_2$ on the surface $X$ into locally closed subvarieties $M_{X, H}(2; c_1, c_2; s)$ according to the value of $s$. For $X=\mathbb{P}^2$ we determine what numbers can appear as the Segre Invariant of a rank $2$ vector bundle with given Chern classes. The irreducibility of strata with fixed Segre invariant is proved and its dimensions are computed. Finally, we present applications to the Brill-Noether Theory for rank $2$ vector bundles on $\mathbb{P}^2$. This is joint work with H. Torres-López and A.G. Zamora (Universidad Autónoma de Zacatecas, Mexico) DOI:10.1515/advgeom-2021-0003.

Higher tangent spaces to moduli (in particular so called obstruction spaces) have been known since the beginning of infitesimal deformation theory in the fifties. They reflect the local structure of the moduli at a point. One way to globalize them is the introduction of obstruction theories; a more general geometric interpretation is based on higher geometry. In this series of lectures we will give a gentle introduction to this circle of ideas, focusing on selected examples from classical moduli problems in complex projective geometry rather than abstract foundations. In particular, we will review deformation theory of Quot schemes and of moduli of coherent sheaves and discuss their derived structures.

Nonabelian Hodge theory can be seen as a vast generalization of the theory of harmonic forms on smooth and projective complex varieties. It is due to Hitchin, Simpson, Donaldson and Corlette. The fundamental result of the theory is the existence of a correspondence between the “Betti cohomology” and the “Dolbeault cohomology” of the variety. In his 1992 paper ‘Higgs bundles and local systems’, Carlos Simpson observed that the Betti side generalises to non-constant coefficients, and asked about the corresponding Dolbeault space in that case. The purpose of this course is to explain one possible answer to Simpson’s question. In the three lectures, we will cover the following topics: $$\begin{align*} &\text{1. Twisted local systems}\\ &\text{2. Twisted character varieties}\\ &\text{3. Higgs bundles for non-constant groups}\\ \end{align*}$$

In this talk I will describe some moduli spaces of vector bundles using the algebra of endomorphisms of the bundles.

The Geometric Invariant Theory (GIT) is one of the most important tools to construct moduli spaces, although to obtain the explicit quotient varieties is a difficult problem and in most cases, the GIT quotient is very singular. This talk shows some techniques to compute the Betti numbers of GIT quotients, developed by Frances Kirwan, applied to the GIT quotient of quartic plane curves, where the GIT quotient variety is still unknown. As there is a birational morphism between the space of quartic plane curves and the space of holomorphic foliations of degree two on the complex projective plane, we prove that the quotient varieties have the same intersection Betti numbers.

Let $(L,V)$ be a generated linear system on a smooth complex algebraic curve $C$ and consider the following exact sequence induced by the evaluation map $$0\to M_{V,L}\to V\otimes O_C\to L\to 0.$$ The kernel bundle $M_{V,L}$ is called syzygy bundle associated to $(L,V)$. When $V=H^0(C,L)$ we write $M_L=M_{H^0(C,L), L}$. The syzygy bundle has been studied by many authors due to the applications it has within the theory of vector bundles on curves. Using the definition of linear stability given by Mumford we have that if $M_{V,L}$ is a slope-stable bundle, then the linear series $(L,V)$ is linearly stable. Consider a complete linear system, that is, let $V=H^0(C,L)$ on an algebraic curve $C$. In this talk I will give a proof of two results: $$\begin{align*} & \text{(i)}\;\text{Let}\; C\;\text{be a general curve in the sense of moduli, then linear stability of the complete and generated linear system}\; (L,H^0(C,L))\; \text{is equivalent to the slope-stability of}\; M_L.\; \text{(This result is a joint work wit Hugo Torres).}\\ & \text{(ii)}\; \text{On smooth plane curves of degree}\; 7\; \text{there exist a complete linear systems}\; (L,H^0(C,L))\; \text{which are linearly stable such that}\;M_L\; \text{is not slope-stable. (This result is a joint work with Hugo Torres and Ernesto Mistretta)} \end{align*}$$ I will give some consequences of these facts.

In the early 1980s, Bloch and Beilinson independently constructed height pairings on cycles over smooth projective varieties on global fields, generalising the Néron-Tate pairing. More recently, Rössler and Szamuely generalised one of Beilinson’s constructions to smooth projective varieties $X$ over the function field of a smooth variety $B$ over an algebraically closed field: their pairing takes values in $H^2(B,\mathbb{Q}_l(1))$ (for $l$ prime to the characteristic). I will explain the construction of a height pairing with values in $Pic(B)\otimes \mathbb{Q}$, for $X$ as above and $B$ smooth over any field.

Applying the intersection theory of the Chow ring to fundamental classes of varieties and Chern classes of vector bundles is the main tool used to compute classical enumerative invariants. More recently, this collection of objects has been enlarged by the introduction of virtual fundamental classes in Gromov-Witten theory. In this lecture we will introduce the framework needed to construct quadratic refinements of all of these objects. Here the Milnor-Witt K-sheaves and the sheaf of Witt groups will play an important role. We will illustrate with some examples, for instance, the quadratic Bézout theorem, quadratic counts of lines on hypersurfaces and complete intersections in a projective space, a quadratic Riemann-Hurwitz formula, and the quadratic Gauß-Bonnet theorem.

Higher tangent spaces to moduli (in particular so called obstruction spaces) have been known since the beginning of infitesimal deformation theory in the fifties. They reflect the local structure of the moduli at a point. One way to globalize them is the introduction of obstruction theories; a more general geometric interpretation is based on higher geometry. In this series of lectures we will give a gentle introduction to this circle of ideas, focusing on selected examples from classical moduli problems in complex projective geometry rather than abstract foundations. In particular, we will review deformation theory of Quot schemes and of moduli of coherent sheaves and discuss their derived structures.

Higher tangent spaces to moduli (in particular so called obstruction spaces) have been known since the beginning of infitesimal deformation theory in the fifties. They reflect the local structure of the moduli at a point. One way to globalize them is the introduction of obstruction theories; a more general geometric interpretation is based on higher geometry. In this series of lectures we will give a gentle introduction to this circle of ideas, focusing on selected examples from classical moduli problems in complex projective geometry rather than abstract foundations. In particular, we will review deformation theory of Quot schemes and of moduli of coherent sheaves and discuss their derived structures.

Nonabelian Hodge theory can be seen as a vast generalization of the theory of harmonic forms on smooth and projective complex varieties. It is due to Hitchin, Simpson, Donaldson and Corlette. The fundamental result of the theory is the existence of a correspondence between the “Betti cohomology” and the “Dolbeault cohomology” of the variety. In his 1992 paper ‘Higgs bundles and local systems’, Carlos Simpson observed that the Betti side generalises to non-constant coefficients, and asked about the corresponding Dolbeault space in that case. The purpose of this course is to explain one possible answer to Simpson’s question. In the three lectures, we will cover the following topics: $$\begin{align*} &\text{1. Twisted local systems}\\ &\text{2. Twisted character varieties}\\ &\text{3. Higgs bundles for non-constant groups}\\ \end{align*}$$

We will present the André-Oort conjecture and its proof under the generalised Riemann hypothesis. We will also present sone recent developments, namely the ideas of Jonathan Pila involving model theory that allow, in some cases to prove the conjecture unconditionally.

Zeta and L-functions are ubiquitous in modern number theory. While some work in the past has brought homotopical methods into the theory of zeta functions, there is in fact a lesser-known zeta function that is native to homotopy theory. Namely, every suitably finite decomposition space (aka $2$-Segal space) admits an abstract zeta function as an element of its incidence algebra. In this talk, I will show how many 'classical' zeta functions in number theory and algebraic geometry can be realized in this homotopical framework. I will also discuss work in progress towards a categorification of motivic zeta and L-functions.

We revisit Langton’s semistable reduction method to construct compactifications of stacks of $GL(r)$-shtukas with arbitrary modifications, which generalizes the work of V. Drinfeld and L. Lafforgue. We also apply our approach to obtain compactifications of stacks of D-shtukas with arbitrary modifications, which solves completely a problem studied by L. Lafforgue, E. Lau and Bao-Chau Ngo. This is a joint work with Y. Varshavsky.

In this talk, I will present a report of my thesis on the geometry of some families of curves in projective 3-space; in particular, curves of degree six and genus three. The main motivation to study these families stems from Liaison Theory. I will talk briefly about some aspects of this theory and its possible relation with the birational geometry of the Hilbert schemes that parametrize families of curves in the projective 3-space.

As they carry more information than the classical Z-valued invariants, the quadratic invariants are often more difficult to compute. In this lecture, we will go over some of the computational tools that have been developed to enable such computations. The methods include the development of a calculus of characteristic classes of vector bundles with values in Witt sheaf cohomology, algebraic computations of the quadratic Euler characteristics of smooth hypersurfaces in $\mathbb{P}^n$, and localization techniques for computing Euler classes and virtual fundamental classes. As a further example we look at a quadratic count of twisted cubic curves on hypersurfaces and complete intersections in a projective space.

We review the basic conjectures due to Simpson to the effect that rigid local systems should be of geometric origin and what is so far understood (based on joint work with Michael Groechenig and for one point with Johan de Jong as well).

Abstract: We will introduce different definition of semiampleness for vector bundles, and similar “generic” versions, then show how these can be used to give a birational, or biholomorphic, characterization of abelian varieties. In order to obtain a similar characterization for complex tori, or compact complex parallelizable manifolds, some basic questions on vector bundles need to be answered (Work In Progress).