Geometria in Bicocca 2023
Geometria in Bicocca 2023 took place in Milan, Italy, on September 11-12, 2023 at the University of Milano-Bicocca. The workshop brought together experts in geometry, covering a wide range of topics such as algebraic geometry, differential and symplectic geometry.
The following was the list of confirmed speakers.
Giovanni Bazzoni, Università degli studi dell’Insubria, Como
Michele Bolognesi, Université de Montpellier, Montpellier
Gianfranco Casnati, Politecnico di Torino, Torino
Lorenzo Foscolo, University College London, London
Federica Galluzzi, Università di Torino, Torino
Antonella Grassi, Università di Bologna, Bologna
João Pimentel Nunes, Instituto Superior Técnico, Lisboa
Federico Rossi, Università degli Studi di Perugia, Perugia
Paolo Stellari, Università degli Studi di Milano, Milano
Organizers: Sonia Brivio, Alberto Della Vedova, Gianluca Faraco, Andrea Galasso, Samuele Mongodi, Roberto Paoletti, Stefano Pigola, Michele Rossi
Titles and abstracts
Giovanni Bazzoni
Titolo: Homotopy Invariants and almost non-negative curvature
Abstract: Almost non-negative sectional curvature (ANSC) is a curvature condition on a Riemannian manifold, which encompasses both the almost flat and the non-negatively curved case. It was shown in a remarkable paper by Kapovitch, Petrunin and Tuschmann that, modulo some technical details, a compact ANSC manifold is a fiber bundle over a nilmanifold, and that the fiber satisfies a curvature condition only slightly more general than ANSC. In this talk, based on joint work with G. Lupton and J. Oprea, I will discuss such manifolds from the point of view of Rational Homotopy Theory, presenting various invariants of bundles of such type, and proving a (rational) Bochner-type theorem.
Michele Bolognesi
Titolo: Cubic fourfolds and quartic scrolls
Abstract: I will start by giving a general overview of the birational geometry of cubic hypersurfaces, and then will focus on the case of cubic fourfolds in P5. I will describe their moduli space and in particular will concentrate on the divisor C14, whose general element is a smooth cubic containing a smooth quartic rational normal scroll surface. The general element of C14 also has a Pfaffian equation. By showing that all degenerations of quartic scrolls in P5 contained in a smooth cubic hypersurface are surfaces with one apparent double point, I will prove that every cubic hypersurface contained in C14 is rational. As an application of these results and of the construction of some explicit examples, I will show that the locus of cubic fourfold having a Pfaffian equation is not open in C14, contrary to common belief. This was a joint work with F.Russo and G.Stagliano'.
Gianfranco Casnati
Titolo: Instanton sheaves on smooth hypersurfaces
Abstract: Let X be a non–degenerate variety of dimension n ≥ 2 in P^N(C) and denote by h its hyperplane class. An instanton sheaf E with quantum number k on X is a non–zero sheaf such that h^1(E(−h))=h^{n−1}(E(−nh))=k and h^q(E(ph))=0 elsewhere in the range −n ≤ p ≤ −1, 0 ≤ q ≤ n.
Instanton sheaves with quantum number k have many interesting properties. E.g. they are k–regular in the sense of Castelnuovo–Mumford and are related to the cohomology of certain monads. Moreover, instanton sheaves with k = 0 are exactly the Ulrich sheaves.
In the talk we focus our attention on instanton sheaves on an integral hypersurface X, showing that their existence is related to the problem of expressing a power of the polynomial defining X as the determinant of a suitable twisted morphism between Steiner bundles. We also briefly discuss some examples in P^3(C), P^4(C), P^5(C).
Lorenzo Foscolo
Titolo: Hypertoric varieties, W-Hilbert schemes and Coulomb branches
Abstract: In an influential paper motivated by IIA/M-theory duality, Sen discussed a construction of complete non-compact 4-dimensional hyperkähler metrics with ALF (asymptotically locally flat) asymptotics of dihedral type in terms of simpler building blocks: Z2-invariant multi-Taub-NUT metrics, i.e. complete 4-dimensional hyperkähler metrics with a circle symmetry, and the Atiyah-Hitchin metric, the simplest ALF space of dihedral type. In this talk I will discuss joint work with R. Bielawski about a higher dimensional version of this story. We study transverse equivariant Hilbert schemes of hypertoric varieties (the higher dimensional analogues of multi-Taub-NUT metrics) invariant under the action of a Weyl group W. In particular, we show that the Coulomb branches of 3-dimensional N=4 supersymmetric gauge theories can be given a concrete geometric realisation as (symplectic quotients of) such Hilbert schemes. These Coulomb branches are certain conjectural hyperkähler spaces arising in theoretical physics. They have been recently given a rigorous mathematical definition as holomorphic symplectic varieties by Braverman, Finkelberg and Nakajima. Building on our geometric description of the holomorphic symplectic structure, we also investigate the conjectural hyperkähler metric on these spaces in terms of twistor theory and Nahm's equations.
Federica Galluzzi
Titolo: Double planes of Picard Rank two and Cubic Fourfolds
Abstract: We discuss some examples of cubic hypersurfaces in P^5(C) containing a plane. The plane defines a quadric surface fibration with base P^2 and the discriminant curve is a plane sextic. The Hodge structure of such a fourfold is strictly related to the Hodge structure of the K3 surface obtained as the double cover of P^2 ramified over the sextic and parameterizing the rulings of the quadrics in the fibration. We describe some of these examples when the Picard rank of the K3 surface is two. This is a joint work with Bert van Geemen.
Antonella Grassi
Titolo: The spectrum of elliptic Calabi-Yau threefolds
Abstract: I will discuss a dictionary between the quantities in the spectrum of the string theory compactified on the Calabi-Yau and objects in the Calabi-Yau. I will also discuss bounds and some properties.
João Pimentel Nunes
Titolo: A (quantum) geometric interpretation of the Peter-Weyl theorem
Abstract: "For K a compact Lie group, we will review the construction of Mabuchi geodesic rays of K–invariant Kähler structures on T*K, via Hamiltonian flows in imaginary time, and the corresponding geometric quantizations. At infinite geodesic time, one obtains a very interesting mixed polarization of T*K, the Kirwin-Wu polarization. The geometric quantization of T*K along this family of polarizations is described by a generalized coherent state transform that, as geodesic time goes to infinity, describes the convergence of holomorphic sections to distributional sections supported on Bohr-Sommerfeld cycles, corresponding to coadjoint orbits. One then obtains a concrete quantum geometric interpretation of the Peter-Weyl theorem. This is joint work with T. Baier, J. Hilgert, O. Kaya and J. Mourão."
Federico Rossi
Titolo: Exploring Indefinite Einstein Solvmanifolds and Compatible Geometric Structures
Abstract: This seminar explores indefinite Einstein solvmanifolds and their connections to various geometric structures, including Sasaki, pseudo-Kähler, and para-Kähler.
We introduce Einstein nilmanifolds, addressing obstructions to achieving the Einstein condition and briefly presenting the rôle of "nice" nilpotent Lie algebras. Subsequently, we delve into the construction of Einstein nilpotent Lie algebras and their relationship with nilsolitons, with a specific focus on the pseudo-Iwasawa condition. This examination underscores the unique challenges posed by indefinite metrics.
Our discussion then extends to pseudo-Kähler geometries and para-Kähler structures and their connections with Einstein condition. We will present explicit constructions and examples. However, we demonstrate that the pseudo-Iwasawa condition, whilst instrumental in constructing Einstein solvmanifolds, is not compatible with Sasaki-Einstein structures, thereby suggesting the need for exploring new constructions.
This is a joint work with D. Conti and R. Segnan Dalmasso.
Paolo Stellari
Titolo: Deformations of stability conditions with applications to Hilbert schemes of points and very general abelian varieties
Abstract: The construction of stability conditions on the bounded derived category of coherent sheaves on smooth projective varieties is notoriously a difficult problem, especially when the canonical bundle is trivial. In this talk, I will illustrate a new and very effective technique based on deformations. A key ingredient is a general result about deformations of bounded t-structures (and with some additional and mild assumptions). Two remarkable applications are the construction of stability conditions for very general abelian varieties in any dimension and for some irreducible holomorphic symplectic manifolds, again in all possible dimensions. This is joint work with C. Li, E. Macri' and X. Zhao.