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Ugo Bruzzo

Publications and source records attributed to Ugo Bruzzo.

At least 37 records · Page 2Linked to original sources
arXiv2021

Moduli of rank 2 Higgs sheaves on elliptic surfaces

We study torsion-free, rank 2 Higgs sheaves on genus one fibered surfaces, (semi)stable with respect to suitable polarizations in the sense of Friedman and O'Grady. We prove that slope-semistability of a Higgs sheaf on the surface implies semistability on the generic fiber. In the case of Higgs sheaves of odd fiber degree on elliptic surfaces in characteristic $\neq 2$, we prove that any moduli space of Higgs sheaves with fixed numerical invariants splits canonically as the product of the moduli space of ordinary sheaves (with the same invariants), and the space of global regular $1$-forms on the surface. For elliptic surfaces with section in characteristic zero, and in the case arbitrary fiber degree, we prove that if a Higgs sheaf has reduced Friedman spectral curve, or is regular on a general fiber with non-reduced spectral cover, then its Higgs field takes values in the saturation of the pull-back of the canonical bundle of the base curve in the cotangent bundle of the surface.

math.AG↗
arXiv2021

The supermoduli of SUSY curves with Ramond punctures

We construct local and global moduli spaces of supersymmetric curves with Ramond-Ramond punctures. We assume that the underlying ordinary algebraic curves have a level n structure and build these supermoduli spaces as algebraic superspaces, i.e., quotients of étale equivalence relations between superschemes.

math.AG↗
arXiv2021

Resolution a la Kronheimer of $\mathbb{C}^3/Γ$ singularities and the Monge-Ampere equation for Ricci-flat Kaehler metrics in view of D3-brane solutions of supergravity

We analyze the relevance of the generalized Kronheimer construction for the gauge-gravity correspondence. We study the general structure of IIB supergravity D3-brane solutions on crepant resolutions $Y$ of singularities $\mathbb{C}^3/Γ$ with $Γ$ a finite subgroup of $SU(3)$. Next we concentrate on another essential item for the D3-brane construction, i.e., the existence of a Ricci-flat metric on $Y$, with particular attention to the case $Γ=\mathbb{Z}_4$. We conjecture that on the exceptional divisor the Kronheimer Kähler metric and the Ricci-flat one, that is locally flat at infinity, coincide. The conjecture is shown to be true in the case of the Ricci-flat metric on ${\rm tot} K_{{\mathbb WP}[112]}$ that we construct, which is a partial resolution of $\mathbb{C}^3/\mathbb{Z}_4$. For the full resolution we have $Y=\operatorname{tot} K_{\mathbb{F}_{2}}$, where $\mathbb{F}_2$ is the second Hizebruch surface. We try to extend the proof of the conjecture to this case using the one-parameter Kähler metric on $\mathbb{F}_2$ produced by the Kronheimer construction as initial datum in a Monge-Ampère (MA) equation. We exhibit three formulations of this MA equation, one in terms of the Kähler potential, the other two in terms of the symplectic potential; in all cases one can establish a series solution in powers of the fiber variable of the canonical bundle. The main property of the MA equation is that it does not impose any condition on the initial geometry of the exceptional divisor, but uniquely determines all the subsequent terms as local functionals of the initial datum. While a formal proof is still missing, numerical and analytical results support the conjecture. As a by-product of our investigation we have identified some new properties of this type of MA equations that we believe to be so far unknown.

math.DG↗
arXiv2021

On the Hodge conjecture for quasi-smooth intersections in toric varieties

We establish the Hodge conjecture for some subvarieties of a class of toric varieties. First we study quasi-smooth intersections in a projective simplicial toric variety, which is a suitable notion to generalize smooth complete intersection subvarieties in the toric environment, and in particular quasi-smooth hypersurfaces. We show that under appropriate conditions, the Hodge Conjecture holds for a very general quasi-smooth intersection subvariety, generalizing the work on quasi-smooth hypersurfaces of the first author and Grassi in [3]. We also show that the Hodge Conjecture holds asymptotically for suitable quasi-smooth hypersurface in the Noether-Lefschetz locus, where "asymptotically" means that the degree of the hypersurface is big enough. This extendes to toric varieties Otwinowska's result in [15].

math.AG↗
arXiv2021

Semistable Higgs bundles on elliptic surfaces

We analyze Higgs bundles $(V,ϕ)$ on a class of elliptic surfaces $π:X\to B$, whose underlying vector bundle $V$ has vertical determinant and is fiberwise semistable. We prove that if the spectral curve of $V$ is reduced, then $ϕ$ is vertical, while if $V$ is fiberwise regular with reduced (resp. integral) spectral curve, and if its rank and second Chern number satisfy an inequality involving the genus of $B$ and the degree of the fundamental line bundle of $π$ (resp., if the fundamental line bundle is sufficiently ample), then $ϕ$ is scalar. We apply these results to the problem of characterizing slope-semistable Higgs bundles with vanishing discriminant in terms of the semistability of their pull-backs via maps from arbitrary (smooth, irreducible, complete) curves to $X$.

math.AG↗
arXiv2020

On the Irreducibility of Some Quiver Varieties

We prove that certain quiver varieties are irreducible and therefore are isomorphic to Hilbert schemes of points of the total spaces of the bundles $\mathcal O_{\mathbb P^1}(-n)$ for $n \ge 1$.

math.AG↗
arXiv2020

Codimension bounds for the Noether-Lefschetz components for toric varieties

For a quasi-smooth hyper-surface $X$ in a projective simplicial toric variety $P$, the morphism $i:H^p(P) \to H^p(X)$ induced by the inclusion is injective for $p=d$ and an isomorphism for $p<d-1$, where $d=dim\ P$. This allows one to define the Noether-Lefschetz locus $NL_β$ as the locus of quasi-smooth hypersurfaces of degree $β$ such that $i$ acting on the middle algebraic cohomology is not an isomorphism. In this paper we prove that, under some assumptions, if $dim P =2k+1$ and $kβ-β_0=nη$ $(n\in\mathbb N)$, where $η$ is the class of a 0-regular ample divisor, and $β_0$ is the anticanonical class, then every irreducible component $V$ of the Noether-Lefschetz locus quasi-smooth hypersurfaces of degree $β$ satifies the bounds $n+1\leq codim\ V \leq h^{k-1,k+1}(X)$.

math.AG↗
arXiv2019

Crepant Resolutions of $\mathbb{C}^3/\mathbb{Z}_4$ and the Generalized Kronheimer Construction (in view of the Gauge/Gravity Correspondence)

As a continuation of a general program started in two previous publications, in the present paper we study the Kähler quotient resolution of the orbifold $\mathbb{C}^3/\mathbb{Z}_4$, comparing with the results of a toric description of the same. In this way we determine the algebraic structure of the exceptional divisor, whose compact component is the second Hirzebruch surface $\mathbb F_2$. We determine the explicit Kähler geometry of the smooth resolved manifold $Y$, which is the total space of the canonical bundle of $\mathbb F_2$. We study in detail the chamber structure of the space of stability parameters (corresponding in gauge theory to the Fayet-Iliopoulos parameters) that are involved in the construction of the desingularizations either by generalized Kronheimer quotient, or as algebro-geometric quotients. The walls of the chambers correspond to two degenerations; one is a partial desingularization of the quotient, which is the total space of the canonical bundle of the weighted projective space $\mathbb P[1,1,2]$, while the other is the product of the ALE space $A_1$ by a line, and is related to the full resolution in a subtler way. These geometrical results will be used to look for exact supergravity brane solutions and dual superconformal gauge theories.

hep-th↗
arXiv2019

Homology of twisted quiver bundles with relations

We study the Ext modules in the category of left modules over a twisted algebra of a finite quiver over a ringed space $(X,\mathcal O_X)$, allowing for the presence of relations. We introduce a spectral sequence which relates the Ext modules in that category with the Ext modules in the category of $\mathcal O_X$-modules. Contrary to what happens in the absence of relations, this spectral sequence in general does not degenerate at the second page. We also consider local Ext sheaves. Under suitable hypotheses, the Ext modules are represented as hypercohomology groups

math.RT↗
arXiv2018

The Kähler Quotient Resolution of $\mathbb{C}^3/Γ$ singularities, the McKay correspondence and D=3 $\mathcal{N}=2$ Chern-Simons gauge theories

We advocate that a generalized Kronheimer construction of the Kähler quotient crepant resolution $\mathcal{M}_ζ\longrightarrow \mathbb{C}^3/Γ$ of an orbifold singularity where $Γ\subset \mathrm{SU(3)}$ is a finite subgroup naturally defines the field content and interaction structure of a superconformal Chern-Simons Gauge Theory. This is supposedly the dual of an M2-brane solution of $D=11$ supergravity with $\mathbb{C}\times\mathcal{M}_ζ$ as transverse space. We illustrate and discuss many aspects of this of constructions emphasizing that the equation $\pmb{p}\wedge\pmb{p}=0$ which provides the Kähler analogue of the holomorphic sector in the hyperKähler moment map equations canonically defines the structure of a universal superpotential in the CS theory. The kernel of the above equation can be described as the orbit with respect to a quiver Lie group $\mathcal{G}_Γ$ of a locus $L_Γ\subset \mathrm{Hom}_Γ(\mathcal{Q}\otimes R,R)$ that has also a universal definition. We discuss the relation between the coset manifold $\mathcal{G}_Γ/\mathcal{F}_Γ$, the gauge group $\mathcal{F}_Γ$ being the maximal compact subgroup of the quiver group, the moment map equations and the first Chern classes of the tautological vector bundles that are in a one-to-one correspondence with the nontrivial irreps of $Γ$. These first Chern classes provide a basis for the cohomology group $H^2(\mathcal{M}_ζ)$. We discuss the relation with conjugacy classes of $Γ$ and provide the explicit construction of several examples emphasizing the role of a generalized McKay correspondence. The case of the ALE manifold resolution of $\mathbb{C}^2/Γ$ singularities is utilized as a comparison term and new formulae related with the complex presentation of Gibbons-Hawking metrics are exhibited.

hep-th↗
arXiv2017

Lie algebroid cohomology and Lie algebroid extensions

We consider the extension problem for Lie algebroids over schemes over a field. Given a locally free Lie algebroid Q over a scheme (X,O), and a sheaf of finitely generated Lie O-algebras L, we determine the obstruction to the existence of extensions 0 --> L --> E --> Q --> 0, and classify the extensions in terms of a suitable Lie algebroid hypercohomology group. In the preliminary sections we study free Lie algebroids and recall some basic facts about Lie algebroid hypercohomology.

math.RA↗
arXiv2017

Semistable Higgs bundles on Calabi-Yau manifolds

We provide a partial classification of semistable Higgs bundles over a simply connected Calabi-Yau manifolds. Applications to a conjecture about a special class of semistable Higgs bundles are given. In particular, the conjecture is proved for K3 and Enriques surfaces, and some related classes of surfaces.

math.AG↗
arXiv2017

On the Hodge conjecture for hypersurfaces in toric varieties

We show that very general hypersurfaces in odd-dimensional simplicial projective toric varieties verifying a certain combinatorial property satisfy the Hodge conjecture (these include projective spaces). This gives a connection between the Oda conjecture and Hodge conjecture. We also give an explicit criterion which depends on the degree for very general hypersurfaces for the combinatorial condition to be verified.

math.AG↗
arXiv2017

The Noether-Lefschetz locus of surfaces in toric threefolds

The Noether-Lefschetz theorem asserts that any curve in a very general surface $X$ in $\mathbb P^3$ of degree $d \geq 4$ is a restriction of a surface in the ambient space, that is, the Picard number of $X$ is $1$. We proved previously that under some conditions, which replace the condition $d \geq 4$, a very general surface in a simplicial toric threefold $\mathbb P_Σ$ (with orbifold singularities) has the same Picard number as $\mathbb P_Σ$. Here we define the Noether-Lefschetz loci of quasi-smooth surfaces in $\mathbb P_Σ$ in a linear system of a Cartier ample divisor with respect to a (-1)-regular, respectively 0-regular, ample Cartier divisor, and give bounds on their codimensions. We also study the components of the Noether-Lefschetz loci which contain a line, defined as a rational curve that is "minimal" in a suitable sense.

math.AG↗
arXiv2017

Lie algebroid cohomology as a derived functor

We show that the hypercohomology of the Chevalley-Eilenberg-de Rham complex of a Lie algebroid L over a scheme with coefficients in an L-module can be expressed as a derived functor. We use this fact to study a Hochschild-Serre type spectral sequence attached to an extension of Lie algebroids.

math.RA↗
arXiv2016

Hilbert schemes of points of $\mathcal O_{\mathbb P^1}(-n)$ as quiver varieties

In a previous paper, a realization of the moduli space of framed torsion-free sheaves on Hirzebruch surfaces in terms of monads was given. We build upon that result to construct ADHM data for the Hilbert scheme of points of the total space of the line bundles $\mathcal O(-n)$ on $\mathbb P^1$, for $n \ge 1$, i.e., the resolutions of the singularities of type $\frac1n(1,1)$. Basically by implementing a version of the special McKay correspondence, this ADHM description is in turn used to realize these Hilbert schemes as irreducible connected components of quiver varieties. We obtain in this way new examples of quiver varieties which are not of the Nakajima type.

math.AG↗