Claudio Bartocci

• Ph.D.
• University of Genoa

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

Over the past two decades several different approaches to defining a geometry over ${\mathbb F}_1$ have been proposed. In this paper, relying on To\"en and Vaqui\'e's formalism (2009), we investigate the category ${\mathsf{Sch}}_{\mathsf B}$ of schemes relative to the category of blueprints introduced by Lorscheid (2012). A blueprint, that may be t...

The primary role played by analogy in Henri Poincaré’s work, and in particular in his “analysis situs” papers, is emphasized. Poincaré’s “sixth example” (showing that Betti numbers do not suffice to classify 3-manifolds) and his construction of the homology sphere are discussed in detail.

We study the Ext modules in the category of representations of 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 modulesin the category of $\mathcal O_X$-modules. Contrary to what happens in...

This paper is an erratum to our paper Moduli spaces of framed sheaves and quiver varieties (J. Geom. Phys., 2016). As a byproduct, we prove a result (Prop. 2.5) providing a description of the fibre TV∨Gr(a,r)⊕n−1, for each V∈Gr(a,r), as the space of isomorphism classes of certain extensions of sheaves on Hirzebruch surfaces.

We introduce a notion of noncommutative Poisson-Nijenhuis structure on the path algebra of a quiver. In particular, we focus on the case when the Poisson bracket arises from a noncommutative symplectic form. The formalism is then applied to the study of the Calogero-Moser and Gibbons-Hermsen integrable systems. In the former case, we give a new int...

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 on , for , i.e., the resolutions of the singularities of type . Basically by implementing...

In the first part of this paper we provide a survey of some fundamental results about moduli spaces of framed sheaves on smooth projective surfaces. In particular, we outline a result by Bruzzo and Markushevich, and discuss a few significant examples. The moduli spaces of framed sheaves on $\mathbb{P}^2$, on multiple blowup of $\mathbb{P}^2$ are de...

Relying on a monadic description of the moduli space of framed sheaves on Hirzebruch surfaces, we construct ADHM data for the Hilbert scheme of points of the total space of the line bundle $\mathcal O(-n)$ on $\mathbb P^1$. This ADHM description is then used to realize these Hilbert schemes as quiver varieties.

Relying on a monadic description of the moduli space of framed sheaves on Hirzebruch surfaces, we construct ADHM data for the Hilbert scheme of points of the total space of the line bundle $\mathcal O(-n)$ on $\mathbb P^1$.

In this paper the emergence of Poincaré’s “analysis situs” is described by means of an overview of the original memoir and its supplements. In particular, the genesis of the celebrated “Poincaré conjecture” is discussed.

Henri Poincaré was close to his brother-in-law Émile Boutroux, with whom he shared an interest in epistemology. Ultimately, however, Boutroux’s spiritualism was irreconcilable with Poincaré’s empiricist scepticism.

The French poet and essayist Paul Valéry was fascinated by Henri Poincaré, who he never met but often observed. His description of Poincaré is an unusual and captivating portrait of the great mathematician.

We define monads for framed torsion-free sheaves on Hirzebruch surfaces and use them to construct moduli spaces for these objects. These moduli spaces are smooth algebraic varieties, and we show that they are fine by constructing a universal monad.

This volume focuses on the interactions between mathematics, physics, biology and neuroscience by exploring new geometrical and topological modelling in these fields. Among the highlights are the central roles played by multilevel and scale-change approaches in these disciplines. The integration of mathematics with physics, as well as molecular and...

Of all my childhood companions, one figure still remains shadowy, a figure that I have always tried to grasp among the many recollections that have surrendered themselves so sweetly to being entrapped in my pages.

Mathematics is the majestic structure conceived by man to grant him comprehension of the universe. It holds both the absolute and the infinite, the understandable and the forever elusive. It has walls before which one may pace up and down without result; sometimes there is a door: one opens it – enters – one is in another realm, the realm of the go...

Something invincible kept me from ever making Verlaine’s acquaintance. I lived very near the Jardin du Luxembourg; it would have taken me only a few steps to reach the marble table where he would sit from eleven to midday, in the back room of a café which took the form, I don’t know why, of a rocky grotto.

Enrico Bombieri was born in Milan in 1940. His precocious talent for mathematics was supported by his family, and while still a boy he came into contact with some eminent mathematical scholars. One of these was Giovanni Ricci, who worked in analysis and number theory; his influence would be a determining factor in Bombieri’s development. During the...

Andrew Wiles, born in Cambridge in 1953, has been interested in number theory since he was a child, and in particular, in Fermat’s Last Theorem.Pierre de Fermat – born in Toulouse, a lawyer by profession, and a great “mathematical dilettante” – making brief notes in the margins of his copy of Arithmetica by Diophantus, Observations sur Diophante, p...

There are prizes that are awarded periodically to honour the best mathematicians, both the most promising and those who have already enjoyed a remarkable career. Here we note the two most important of such prizes.

Night sets on us its magic task:

“Kakania was, after all, a country for geniuses; which is probably what brought it to its ruin”. R. Musil, The Man without Qualities, transl. by Sophie Wilkins and Burton Pike, 2 vols., Vintage Books, New York, 1996, vol. I, p. 31. From this point on the work will be cited with the abbreviation MWQ followed by the volume and page number (the second...

How will the mathematics of the last 50 years appear to the eyes of future historians? As difficult as it is to make predictions that depend in large part on developments to come (Lakatos docet), we can hazard a guess that the second half of the twentieth century will probably be considered a period of extraordinary proliferation of new ideas, and...

The notion of stability condition on a triangulated category has been introduced by Bridgeland in [65], following ideas from physics by Douglas [104] on π-stability for D-branes. A stability condition on a triangulated category ℑ is given by abstracting the usual properties of μ-stability for sheaves on complex projective varieties; one introduces...

In looking for examples of Fourier-Mukai transforms on varieties other than the Abelian ones, it is natural to consider K3 surfaces, especially in view of Theorem 2.38 and the subsequent discussion. A forerunner of a Fourier-Mukai functor for K3 surfaces (which in our notation is a morphism of the type f Q: H •(X; Z) → H•(Y,Z), cf. Eq. (1.12)) was...

In this chapter we offer some applications of Fourier-Mukai transforms, namely, a classification of the Fourier-Mukai partners of complex projective surfaces, some issues in birational geometry, and an approach to the McKay correspondence via Fourier-Mukai transform.

Mukai’s 1981 paper [224] contains, in one way or another, in a more or less explicit form, many of the ideas that have been introduced and developed in subsequent years in connection with Fourier-Mukai transforms. Thus these ideas are often at the core of the theory of integral functors that we have quite systematically developed in the first chapt...

The original Nahm transform, i.e., a mechanism that starting from an instanton on a 4-dimensional at torus produces an instanton on the dual torus, was introduced by Nahm in 1983 [230]. This construction was formalized by Schenk [263] and Braam and van Baal [57] in later years. Their descriptions show that the Nahm transform is essentially an index...

According to a fundamental theorem due to D. Orlov, any equivalence between derived categories of coherent sheaves of two smooth projective varieties is an integral functor. This “representability” result — which lies at the heart of the current chapter — opens the way to the investigation of the geometric consequences of the equivalence between th...

The first instance of an integral functor is to be found in Mukai’s 1981 paper on the duality between the derived categories of an Abelian variety and of its dual variety [224]. Integral functors have also been called “Fourier-Mukai functors” or “Fourier-Mukai transforms.” However, we shall give these terms specific meanings that we shall introduce...

For the reader’s convenience, in this appendix we collect several standard results that are used throughout this book. These concern relative duality, Simpson’s notion of stability for pure sheaves, and Fitting ideals.

In this chapter we offer a quite comprehensive study of the relative Fourier-Mukai functors. We consider (proper) morphisms of algebraic schemes X → B, Y → B, and use an element in the derived category of the fibered product X ×B Y as a kernel to define an integral functor from the derived category of X to the derived category of Y . This generaliz...

We show that the bi-Hamiltonian structure of the rational n-particle (attractive) Calogero–Moser system can be obtained by means of a double projection from a very simple Poisson pair on the cotangent bundle of . The relation with the Lax formalism is also discussed.

Given a special Kahler manifold M, we give a new, direct proof of the relationship between the quaternionic structure on its cotangent bundle and the variation of Hodge structures on the complexification of TM.

“La Kakania era forse un paese di geni, e fu probabilmente per questo che andò in rovina”1. Nelle pagine dell’Uomo senza qualità, Kakania (neologismo coniato da Musil a partire dall’abbreviazione k.k., vale a dire kaiserlich-königlich,“imperial-regio”) è l’ironico e scatologico appellativo che designa la monarchia austroungarica al suo tramonto.

Andrew Wiles, nato a Cambridge nel 1953, manifestò fin da bambino un forte interesse per la teoria dei numeri e, in particolare, per l’ultimo teorema di Fermat.

Come apparirà la matematica degli ultimi cinquant’anni agli occhi degli storici futuri? Per quanto sia difficile fare previsioni che dipendono in buona parte da quelli che saranno gli sviluppi avvenire (Lakatos docet), possiamo azzardare che la seconda metà del XX secolo sarà probabilmente considerata un periodo di straordinaria proliferazione di n...

We introduce the basic elements of some cohomology theories that arise naturally in the context of supermanifold theory.

We study the separability of the Neumann–Rosochatius system on the n-dimensional sphere using the geometry of bi-Hamiltonian manifolds. Its well-known separation variables are recovered by means of a separability condition relating the Hamiltonian with a suitable (1,1) tensor field on the sphere. This also allows us to iteratively construct the int...

See original Math. Nachr. 238, 23–36 (2002)

We study complex projective surfaces admitting a Poisson structure. We prove a classification theorem and count how many independent Poisson structures there are on a given Poisson surface.

Given two hyperk\"ahler manifolds $M$ and $N$ and a quaternionic instanton on their product, a hyperk\"ahler Nahm transform can be defined, which maps quaternionic instantons on $M$ to quaternionic instantons on $N$. This construction includes the case of Nahm transform for periodic instantons on $\bR^4$, the Fourier-Mukai transform for instantons...

We define a Nahm transform for instantons over hyperkähler ALE 4-manifolds, and explore some of its basic properties.

We prove that an integrable system over a symplectic manifold, whose symplectic form is covariantly constant w.r.t. the Gauss-Manin connection, carries a natural hyper-symplectic structure. Moreover, a special Kaehler structure is induced on the base manifold. Comment: LaTeX file, 7 pages; to be published in Journal of Geometry and Physics

We consider a relative Fourier-Mukai transform defined on elliptic fibrations over an arbitrary normal base scheme. This is used to construct relative Atiyah sheaves and generalize Atiyah's and Tu's results about semistable sheaves over elliptic curves to the case of elliptic fibrations. Moreover we show that this transform preserves relative (semi...

The Fourier-Mukai transform is extended to the context of Higgs bundles under certain conditions. Some results valid for sheaves on abelian varieties and K3 surfaces are extended to the situation of Higgs bundles. An application to the relative setting is given.

We study the structure of a modified Fukaya category ?(X) associated with a K3 surface X, and prove that whenever XX is an elliptic K3 surface with a section, the derived category of ?(X) is equivalent to a subcategory of the derived category of coherent sheaves on the mirror K3 surface .

We study the structure of a modified Fukaya category ${\frak F}(X)$ associated with a K3 surface $X$, and prove that whenever $X$ is an elliptic K3 surface with a section, the derived category of $\fF(X)$ is equivalent to a subcategory of the derived category ${\bold D}(\hat X)$ of coherent sheaves on the mirror K3 surface $\hat X$.

A necessary and sufficient condition for the existence of a supermanifold structure on a quotient defined by equivalence relation is established. Furthermore, we show that an equivalence relation R on a Berezin–Le˘ ıtes–Kostant supermanifold X determines a quotient supermanifold X/R if and only if the restriction R 0 of R to the underlying smooth m...

We use a relative Fourier–Mukai transform on elliptic K3 surfaces X to describe mirror symmetry. The action of this Fourier–Mukai transform on the cohomology ring of X reproduces relative T-duality and provides an infinitesimal isometry of the moduli space of algebraic structures on X which, in view of the triviality of the quantum cohomology of K3...

In a previous paper a new category of supermanifolds, called -supermanifolds, was introduced. The objects of that category are pairs (M, ), with M a topological space and a suitably defined sheaf of 2 -graded commutative BL - algebras, BL being a Grassmann algebra with L generators. In this note we complete the analysis of that category by showing...

Given two hyperkähler manifolds X and Y and a quaternionic instanton on X × Y, one can define a Fourier-Mukai transform, which maps quaternionic instantons on X to quaternionic instantons on Y. This construction encompasses the cases of two-dimensional algebraic tori and K3 surfaces already treated elsewhere. We also sketch some higher dimensional...

Considering a generic polarized complex Kummer surface X, we prove that for a particular choice of the Chern classes, the moduli space b X of stable vector bundles on X is not empty. The bundles we find do not satisfy a sufficient condition for the existence of semistable bundles recently established by Sorger. Actually, b X is a generic Kummer sur...

We define a Fourier-Mukai transform for sheaves on K3 surfaces over $\C$, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli space of stable sheaves on $X$. For a wide class of K3 surfaces $\hat X$ can be chosen to be iso...

We discuss the counting of minimal geodesic ball coverings of n-dimensional (n ≥ 3) riemannian manifolds of bounded geometry, fixed Euler characteristic, and Reidemeister torsion in a given representation of the fundamental group. This counting bears relevance to the analysis of the continuum limit of discrete models of quantum gravity. We establis...

Given a submersive morphism of complex manifoldsf: X→Y, and a complex vector bundleE onX, there is a relationship between the higher direct images of ε (the sheaf of holomorphic sections ofE) and the index of the relative Dolbeault complex twisted byE. This relationship allows one to yield a global and simple proof of the equivalence between the Mu...

Given two compact hyperk\"ahler surfaces $X$ and $Y$ and a holomorphic vector bundle $Q$ on $X\times Y$, which is a generalized instanton, one can define a Fourier-Mukai transform, which, under suitable assumptions, maps vector bundles on $X$ to vector bundles on $Y$. If $X$ and $Y$ are dual complex tori, this transform maps instantons on $X$ to in...

On the analogy of the so-called Nahm-Mukai transform on four-tori, we define a functor from instantons over a K3 surface X to instantons over a surface component of the moduli space of stable sheaves on X, which is again a K3 surface.

We discuss an axiomatic approach to supermanifolds valid for arbitrary ground graded- commutative Banach algebras B. Rothstein's axiomatics is revisited and completed by a further requirement which calls for the completeness of the rings of sections of the structure sheaves, and allows one to dispose of some undesirable features of Rothstein superm...

It is usually thought that any approach to superstrings à la Polyakov should involve a suitable generalization of the notion of Riemann surface.

This last Chapter is devoted to developing the rudiments of a theory of Lie supergroups within the category of G-supermanifolds, together with the basic definitions related to principal superfibre bundles and associated superbundles. Since a G-supermanifold structure is not determined by the underlying topological space, the group axioms must be ex...

The aim of this Chapter is to unfold a basic cohomological theory for supermanifolds, which will be exploited in the next Chapter to study the structure of superbundles; in particular to build a theory of characteristic classes. This cohomology theory does not embody only trivial extensions of results valid for differentiable manifolds. For instanc...

The category of G-supermanifolds [BB1,BBH] provides a consistent and concrete model for the development of supergeometry. In order to supply proper motivations for the introduction of these objects, and also for historical reasons, we shall start with a brief description of graded manifolds; these were originally introduced by Berezin and Leĭtes [B...

Sheaves provide a powerful tool for studying graded manifolds and supermanifolds. Sheaf theoretic techniques convey the elegant flavour of algebraic and analytic geometry. However, the use of sheaves is not merely a matter of aesthetics, but is unavoidable in the graded setting. Indeed, graded manifolds are intrinsically defined in terms of sheaves...

Our purpose in this Chapter is to study the main features of the theory of vector bundles in the category of G-supermanifolds.

This introductory Chapter aims at establishing, together with the basic notation and terminology, some elementary results about Z2-graded algebra that we shall constantly use in the sequel. The topics covered include Z2-graded rings and modules, Z2-graded tensor algebra, Lie superalgebras, and matrices with entries in a Z2-graded commutative ring.

The first five Sections of this Chapter, the technical core of this book, are dedicated to set down the basic differential geometry of G-supermanifolds, by introducing the fundamental objects one needs: morphisms, products, supervector bundles, and differential forms. It should be pointed out that the relevant definitions are quite different from t...

A graded Weil homomorphism is defined for principal superfiber bundles and the related transgression (or Chern–Simons) forms are introduced. As an example of the application of these concepts, a ‘‘superextension’’ of the Dirac monopole is discussed.

By using global geometric constructions on superfibre bundles, we provide a geometric interpretation of the standard constraints in supersymmetric gauge theories together with a proof of Weil triviality that holds for arbitrary superspace topologies.

We show that the existence of a connection on a super vector bundle or on a principal super fibre bundle is equivalent to the vanishing of a cohomological invariant of the superbundle. This invariant is proved to vanish in the case of a De Witt base supermanifold. Finally, some examples are discussed.

Super line bundles over supermanifolds are introduced as natural generalizations of line bundles over smooth manifolds. Their classification in terms of their obstruction class and the representation of their Chern class in terms of a connection on the super line bundle are discussed. The case where the base supermanifold is De Witt is analyzed in...

This paper defines the obstruction class of a complex super-line bundle over a supermanifold and shows that nontrivial bundles may have vanishing obstruction class. The relationship between the Picard group of a superanalytic De Witt supermanifold and the Picard group of its body is defined.

In this paper, a theory of characteristic classes for super vector bundles over is studied. Some properties of even and odd Chern classes, constructed in this theory, are established. At last, the relevance of the theory to supergeometry is discussed.

The cohomology of the structure sheaf of real and complex supermanifolds is studied. It is found to be nontrivial (also in the real case), unless the supermanifold is De Witt, i.e., it is a fiber bundle on an ordinary manifold with a vector fiber. As a consequence, the Dolbeault theorem can be extended only to complex De Witt supermanifolds. The re...

We analyze the category of GH∞ supermanifolds recently introduced by Rogers and show that these supermanifolds do not have a good graded tangent bundle, and that a natural definition of super vector bundle is not possible within that category. However, any GH∞ supermanifold can be turned into a supermanifold of a new category (that we call a -super...