GTAG

We study algebro-geometric aspects of the singular schemes of holomorphic foliations. We establish when a foliation is determined by their singular schemes(work joint with Carolina Araujo).
We prove that the tangent sheaf of a codimension one locally free distribution splits as a sum of line bundles if and only if its singular scheme is arithmetically Cohen-Macaulay. Moreover, we show that a foliation by curves is given by an intersection of generically transversal holomorphic distributions of codimension one if and only if its singular scheme is arithmetically Buchsbaum. (joint work with Marcos Jardim and Renato Vidal Martins).

Following the paper "Perverse coherent sheaves on blow-up. I. A quiver description" by Nakajima and Yoshioka, the chamber structure for a certain quiver stability (generalizing the one given by King for framed locally free sheaves on the blow-up of the projective plane) will be described in its entirety.

Modular invariance of characters of affine Lie algebras is one of the most beautiful connections between representation theory and number theory, with lots of applications in physics. This connection is by now quite well understood. Recently this relationship has been further extended: certain characters of affine Lie superalgebras are proved to be mock theta functions. Mock theta functions were introduced by Ramanujan in his last letter to Hardy, it is quite interesting these functions made their appearance almost one hundred years later in the representation theory of superalgebras. This talk will be quite general; we will skip the details in order to stress the fundamental ideas.

Last week, Henrique showed us the usefulness of the following key lemma: Let S be an anticanonical K3 divisor in a semi-Fano 3-fold; then given an incidence condition (a point and a tangent line) in S, there is an anticanonical pencil through S whose base locus curve has the prescribed incidence. This week, via basic manipulations of stable bundles on a K3 surface, I will show a concrete use of this key lemma providing an example of G_2-instanton on a G-2-manifold. Joint work with Henrique Sa Earp and Johannes Nordström.

Here is an apparently uninteresting result: Let S be an anticanonical K3 divisor in a semi-Fano 3-fold; then given an incidence condition (a point and a tangent line) in S, there is an anticanonical pencil through S whose base locus curve has the prescribed incidence. I will explain how this naive fact allows one to construct solutions to two sophisticated problems:
(1) to obtain a G2-instanton over a (compact) TCS manifold, hence obtaining a nontrivial 7-dimensional gauge theory;
(2) to produce a topological associative submanifold, which is in some sense a *minimal* 3-submanifold whose homology class however *does not* depend on the metric.
Both had been open problems, solved in collaboration resp. with Grégoire and Lazaro (and J. Nordstöm).

A codimension one foliation of degree $d\geq0$, is defined by an exact sequence $ 0\to F\to TP^3 \to I_Z(d+2)\to 0$ where $Z$ is a closed scheme of codimension $\geq2$ (the singular set), and $F$ a reflexive subsheaf of the tangent bundle of $P^3$, such that $F$ is closed under the Lie bracket of vector fields. We prove that the tangent sheaf of generic foliations with a reduced integrating factor, have stable tangent sheaf. We also calculate some cohomology groups of these foliations of low degrees that are relevant in the study of their deformations.

We report on quantum many-party correlations and the problem of their discontinuity. This problem exemplifies a demand for new methods to study quantum state spaces and their linear images, which are fundamental objects in quantum information theory. One method, Kippenhahn's theorem, shows that a planar linear image is the convex hull of a plane real affine curve. We present this theorem and some problems with its higher dimensional analogues.

We prove a residue formula for Morita-Futaki-Bott invariant with respect any holomorphic vector fields with isolated (possibly degenerated) singularities in terms of Grothendieck's residues. As an application we give a simple proof of the non-existence of Kahler-Einstein metric on singular weighted projective planes. This is a joint work with A. Miguel Rodríguez.

We present a family of monads whose cohomology are vector bundles of small rank on projective 3-spaces over the complex numbers, then we study the (Mumfurd-Takemoto) stability and a cohomological characterization of these vector bundles. We also study the dimension of this family on the moduli space of stable vector bundles over projective 3-spaces.

From a differential-geometric point of view, degenerations of ASD connections on a 4-manifold can be described by bubbling phenomena. The Kobayashi–Hitchin correspondence suggests that an analogue for vector bundles on algebraic surfaces should exists. After recalling some basic material, the bubbling of vector bundles is presented via a notion of stability on particular reducible varieties. I will conclude comparing these results with the example of bubble tree compactification constructed by Markushevich, Tikhomirov, and Trautmann.

Relying on a monadic representation of the moduli space of framed torsion-free sheaves on Hirzebruch surfaces, we get an ADHM description for the Hilbert scheme of points of the total space of the line bundle $\mathcal{O}(-n)$ on $\mathbb{P}^1$. It turns out that that these Hilbert schemes can be realised as Poisson quiver varieties.

In the paper "Monads for framed sheaves on Hirzebruch surfaces" (Adv. Geom. 15 (2015) 55–76), Bartocci, Bruzzo, and Rava provide a description of moduli spaces of framed sheaves on Hirzebruch surfaces in terms of monads. Specializing this description to the rank one case, we obtain ADHM data for Hilbert schemes of points of the total space of $\mathcal{O}_{\mathbb{P}^1}(−n)$. These linear data can be interpreted also within the theory of quiver representations, so that our Hilbert schemes turn out to be isomorphic to suitable moduli spaces of semistable representations of the ”framed” quiver.

Let $I(n)$ be the moduli space of rank 2 instanton bundles of charge n on $\mathbb{P}^3$. It is known f that $I(n)$ is an irreducible, nonsingular and affine variety of dimension $8n-3$. Since every rank 2 instanton bundle on $\mathbb{P}^3$ is stable, we may regard $I(n)$ as an open subset of the projective Gieseker--Maruyama moduli scheme M(n) of rank 2 semistable torsion free sheaves $F$ on $\mathbb{P}^3$ with Chern classes $c_1=c_3=0$ and $c_2=n$, and consider the closure $\overline{I(n)}$ of $I(n)$ in $M(n)$. We construct some of the irreducible components of dimension $8n-4$ of the boundary $\partial I(n):=\overline{I(n)}\setminus I(n)$. These components generically lie in the smooth locus of $\calm(n)$ and consist of rank 2 torsion free instanton sheaves with singularities along rational curves.

We introduce the problem of the existence of approximate Hermitian-Yang-Mills structure and its relation with the algebro-geometric notion of semistability for a Higgs bundle over a smooth projective variety. Eventually, using a modified version of the Donaldson heat flow, we show that the semistability of a twist bundle implies the existence of an approximate Hermitian-Yang-Mills structure. As a consequence of this we deduce some results about semistable twisted bundles.

The notion of symplectic instanton is a natural generalization to higher rank of the notion of rank-2 mathematical instanton vector bundle on projective space. In this talk we describe symplectic instantons via monads and use this description to study the problem of irreducibility of the moduli space of symplectic instantons. We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus of tame symplectic instantons is irreducible and has the expected dimension.

We present a modular compactification of the moduli space of framed symplectic bundles on the projective plane, and provide an ADHM-type description. This is a work in progress towards my PhD thesis.

We consider the structure of codimension one singular holomorphic foliations on $(\mathbb{C}^3,0)$ without invariant germs of analytic surface. We focus on the so-called CH-foliations, that is, foliations without saddle nodes in two dimensional sections. Considering a reduction of singularities, we detect the possible existence of “nodal components”, which are a higher dimensional version of the nodal separators in dimension two. If the foliation is without nodal components, we prove that all the leaves in a neighborhood of the origin contain at least one germ of analytic curve at the origin. We also study the structure of nodal components for the case of “Relatively Isolated CH-foliations” and we show that they cut the dicritical components or they exit the origin through a non compact invariant curve. This allows us to give a precise statement of a local version of Brunella’s alternative: if we do not have an invariant surface, all the leaves contain a germ of analytic curve or it is possible to detect the nodal components in the generic points of the singular curves before doing the reduction of singularities. Joint work with Felipe Cano (UVA) and Mariana Ravara-Vago

The Barth--Van de Ven--Tyurin--Sato Theorem claims that any finite rank vector bundle on the infinite complex projective space $\mathbf{P}^\infty$ is isomorphic to a direct sum of line bundles. We establish sufficient conditions on a locally complete linear ind-variety $\mathbf{X}$ which ensure that the same result holds on $\mathbf{X}$. We then exhibit natural classes of locally complete linear ind-varieties which satisfy these sufficient conditions. This is a joint work with Ivan Penkov (Jacobs Univ. Bremen).

A quiver $K$ is a finite oriented graph which may have double arrows and loops. If we assign vector spaces to all vertices of a quiver and some linear maps along arrows, then we a obtain a representation of $K$. The set of all representations of fixed dimensions of vertex vector spaces is called the representation space of $K$. The product of general linear groups acts naturally on the representation space of $K$. In the talk we consider several generalizations of this classical construction and describe its polynomial invariants.

In this seminar we give a closer look at semistability condition for a certain class of vector bundles, called tensors. Tensors are very general objects which give examples of principal bundles, Higgs bundles, framed sheaves and many other "decorated" bundles. The semistability notion for these objects is very hard to check, however we simplify this notion reducing the number of inequalities one has to check.

Joint work with M. Corrêa and Arturo Fernández (UFMG).
A codimension one holomorphic foliation on $\mathbb{P}^n$ is given by a class of sections $\omega\in \mathbb{P}(H^0(\mathbb{P}^n,\Omega^1(c))$, Where $cod(S(\omega))\geq2$ and $\omega\wedge d\omega=0$. $S(\omega)=\{ p \: | \: \omega(p)=0\}$. We are going to study the subset of pure codimension $S_2(\omega)\subset S(\omega)$. We will see that in some cases, the set $S_2(\omega)$ determines the global structure of the foliation. Also, we are going to count the number of degenerate points on $S_2(\omega)$.

Codimension 1 distributions on the projective space are rank 2 subsheaves of the tangent space whose quotient is torsion free. I will provide a classification of codimension 1 distributions of degree 0, 1 and 2.

The well-known Barth-Larsen Theorem implies in particular that, if X is a smooth n-dimensional subvariety of an N-dimensional projective space, then the Picard group of X is also generated by the hyperplane line bundle as long as N<=2n-2. In this talk we will provide the recipe to get a similar result when you replace the projective space with any other N-dimensional variety Y. We will illustrate the procedure doing in detail the case in which Y is a smooth quadric hypersurface of dimension 2n-2. More precisely, we will prove that, for any smooth n-dimensional subvariety X, the Picard group of X has rank one if n is odd and has rank at most two if n is even. This is part of the PhD thesis of my former student Jorge Caravantes.

The goal of the minimal model program is to construct a birational model of any complex projective variety which is as simple as possible in a suitable sense. This subject has its origins in the classical birational geometry of surfaces studied by the Italian school. In 1988 S. Mori extended the concept of minimal model to 3-folds by allowing suitable singularities on them. In 2010 there was a great breakthrough in the minimal model theory when C. Birkar, P. Cascini, C. Hacon and J. McKernan proved the existence of minimal models for varieties of log general type. Mori Dream Spaces, introduced by Y. Hu and S. Keel in 2002, form a class of algebraic varieties that behave very well from the point of view of Mori's minimal model program. They can be algebraically characterized as varieties whose total coordinate ring, called the Cox ring, is finitely generated. In addition to this algebraic characterization there are several algebraic varieties characterized by some positivity property of the anti-canonical divisor, such as weak Fano and log Fano varieties, that tourn out to be Mori Dream Spaces. In this talk, I will show how to obtain log Fano varieties and Mori Dream Spaces by blowing-up projective spaces in a certain number of general points.

The first examples of complete manifolds of holonomy G2 were constructed by Bryant and Salamon in the 1980's. More recently, gauge theory on manifolds on reduced holonomy has been introduced and greatly developed. I will explain this theory in the case of G2 holonomy and give a new construction of instantons on the manifolds on Bryant and Salamon.