Giuseppe Barbaro (Università di Aarhus)

Title:  Non-Kähler Calabi-Yau metrics

Abstract:  Pluriclosed metrics with vanishing Bismut-Ricci form, called Bismut-Hermitian-Einstein (BHE), provide a crucial example of non-Kähler Calabi-Yau geometries and are therefore relevant to mathematical physics and uniformization problems in complex geometry. They are natural candidates for ‘canonical metrics’ in complex non-Kähler geometry, arising as fixed points of the pluriclosed flow and being related to Yau’s Problem concerning compact Hermitian manifolds with holonomy reduced to subgroups of U(n). A remarkable result of Gauduchon-Ivanov states that the only complex non-Kähler BHE surfaces, are quotients of the standard Bismut-flat Hopf surface. We further extend this rigidity to BHE 3-folds. Furthermore, using a stability result for the pluriclosed flow, we deduce the non-existence of non-trivial homogeneous BHE metrics. These results in low dimensions and in the homogeneous setting together with the examples we have motivate the hypothesis that all BHE manifolds have parallel Bismut torsion (PBT). We therefore provide a complete classification of BHE metrics with PBT by specializing our more general classification of pluriclosed manifolds with PBT. 

Jaroslaw Buczynski (IMPAN, Varsavia)

Title: Three stories of Riemannian and holomorphic manifolds 

Abstract: Compact holomorphic manifolds and Riemannian manifolds invite you all to participate in their three epic stories. In the first tale, the main character is going to be a compact holomorphic manifold, and as in every story, there will be some action going on. More specifically, the group of invertible complex numbers, or even better, several copies of those, act on the manifold. The spirit of late Andrzej Białynicki-Birula until this day helps us to comprehend what is going on. The second story is a tale of holonomies, it begins with "a long time ago,..." and concludes with "... and the last missing piece of this mystery is undiscovered til this day". The protagonist of this part is a quaternion-Kahler manifold, while the legacy of Marcel Berger is in the background all the time. In the third part we meet legendary distributions, which are subbundles in the tangent bundle of one of our main characters. Among others, distributions can be foliations, or contact distributions, which like yin and yang live on the opposite sides of the world, yet they strongly interact with one another. Ferdinand Georg Frobenius is supervising this third part. Finally, in the epilogue, all the threads and characters so far connect in an exquisite theorem on classification of low dimensional complex contact manifolds. In any dimension the analogous classification is conjectured by Claude LeBrun and Simon Salamon, while in low dimensions it is proved by Jarosław Wiśniewski, Andrzej Weber, in a joint work with the narrator. 

Gian Maria Dall’Ara (SNS Pisa)

Sara Angela Filippini (Università degli Studi del Salento)

Title: Residual intersections and Schubert varieties

Abstract: The notion of residual intersections was introduced by Artin and Nagata. Let I be an ideal in a local Cohen-Macaulay ring R, and A = (a_1, \ldots, a_s) \subsetneq I. Then J = A:I is called an s-residual intersection of I if ht(J) \geq s \geq ht(I). Residual intersections provide a generalization of linkage. Indeed, if J = A:I and I = A:J for A a regular sequence, I and J are said to be linked.

I will show how results of Huneke and of Kustin and Ulrich on residual intersections for standard determinantal ideals and Pfaffian ideals arise in the context of ideals of Schubert varieties in the big opposite cell of homogeneous spaces. This is joint work with X. Ni, J. Torres and J. Weyman.

Alice Garbagnati (Università degli Studi di Milano)

Title: Singular symplectic surfaces

Abstract: There are several possible generalizations of the definition of irreducible holomorphic symplectic manifold to the singular context (primitive symplectic variety, irreducible symplectic orbifold, irreducible symplectic variety). We review these definitions and we consider them in the lowest dimensional case, i.e. the case of the surfaces. We classify the surfaces which satisfy these definitions, showing that all of them are contractions of rational curves on K3 surfaces, but that not all the possible contractions of rational curves on K3 surfaces satisfy all these definitions. We observe that the Hilbert scheme of points of the singular surfaces considered are irreducible symplectic orbifolds, under certain conditions. The talk is based on a joint work with M. Penegini and A. Perego.

Simone Gutt (ULB, Bruxelles)

Title:  Connections adapted to symplectic frameworks

Abstract: This talk will present a partial overview on choices of special symplectic connections. 

A symplectic connection on a symplectic manifold (M, ω) is a torsion free linear connection such that the covariant derivative of the symplectic two form vanishes (∇ω = 0). Such connections always exist but are not unique. 

To select some of those connections, one way to proceed is to select identities for some components of the curvature.

The curvature R∇ of a symplectic connection ∇ splits under the action of the symplectic group into two components R∇ = E∇ + W∇ where E∇ is determined by the Ricci tensor of ∇. One calls Ricci-type a symplectic connection such that W∇ = 0. We shall briefly review the geometric interpretation of this condition in terms of a twistor space and descibe constructions of models for symplectic manifolds admitting a Ricci-type connection. A compatible almost complex structure J on a symplectic manifold (M, ω) is an almost complex structure such that gJ (X, Y ) = ω(X, JY ) is symmetric; such structures exist on any symplectic manifold. If there exists a symplectic connection which leaves J parallel, it is necessarily the Levi Civita connection for the pseudo-Riemannian metric gJ and the manifold (M, gJ , J) is pseudo-Kähler. The curvature of the Levi Civita connection of a pseudo-Kähler manifold splits under the action of the pseudo-unitary group into three components, R = S + E + B, where S is defined in terms of the scalar curvature and E is defined in terms of the traceless part of the Ricci tensor. The remaininig tensor B is called the Bochner tensor. If B = 0, the space is called a pseudo-Kähler Bochner space. There exists a construction of local models for pseudo-Kähler Bochner spaces which is analogous to the construction of Ricci-type spaces and also generalizes for symplectic connections with special holonomy, in particular for quaternionic symplectic manifolds.

Eva Miranda (UPC and CRM – Barcellona e Università di Colonia)

Title: Symplectic Geometry in Flow

Abstract: This lecture explores how contact and cosymplectic structures provide a geometric lens for understanding stationary solutions of the Euler and Navier–Stokes equations. We discuss applications to embedding theorems, the search for periodic orbits, and the construction of universal Turing machines encoded within the flows of Reeb vector fields.

Gabriele Mondello (Università degli Studi di Roma La Sapienza)

Title: Unique hyperbolization of relatively maximal PSL(2,R)-representations of punctured surface groups

Abstract: A hyperbolic metric on a punctured surface comes equipped with a PSL(2, ℝ)-valued monodromy representation. If such a metric has conical singularities at the punctures, then the conjugacy class of the monodromy around the i-th peripheral loop is uniquely determined by the angle at the i-th conical point.

In the opposite direction, it is not generally understood when a PSL(2, ℝ) representation of the fundamental group of a punctured surface (with prescribed conjugacy classes around the peripheral loops) arises as the monodromy of a hyperbolic metric with conical points of given angles. A necessary condition is that the Euler number of the representation be relatively maximal.

In joint work with Nicolas Tholozan (ENS Paris), we show that such relatively maximal representations arise as monodromies of a unique hyperbolic metric if and only if the angles satisfy a precise smallness condition.

After introducing the main objects, I will describe the key ingredients of the proof, which include Teichmüller space, the Weil–Petersson metric, and the gradient flow of the energy functional associated with an equivariant map into the hyperbolic plane. 

Francesco Polizzi (Università degli Studi di Napoli, Federico II)

Title: Groups of order 64 and non-homeomorphic double Kodaira fibrations with the same biregular invariants

Abstract: We investigate some finite, non-abelian quotients G of the pure braid group on two strands P2(Σb), where Σb is a closed Riemann surface of genus b. Building on our previous work on some special systems of generators on finite groups that we called  "diagonal double Kodaira structures", we prove that, if G has not order 32, then |G | ≥ 64, and we completely classify the cases where equality holds. In the last section, as a geometric application of our algebraic results, we construct two 3-dimensional families of double Kodaira fibrations having the same biregular invariants and the same Betti numbers but different fundamental groups. If time permits, we will also outline some potential future lines of investigation.

This is a joint work with Pietro Sabatino.

Silvia Sabatini (Università di Colonia)

Title: Positive monotone symplectic manifolds with symmetries and reflexive GKM graphs

Abstract: Positive monotone symplectic manifolds are the symplectic analogues of Fano varieties, namely they are compact symplectic manifolds for which the first Chern class equals the cohomology class of the symplectic form. In dimension 6, if the positive monotone symplectic manifold is acted on by a circle in a Hamiltonian way, a conjecture of Fine and Panov asserts that it is diffeomorphic to a Fano variety. In this talk I will report on recent classification results of positive monotone symplectic manifolds endowed with some special Hamiltonian actions of a torus, showing some evidence that they are indeed (homotopy equivalent/homeomorphic/diffeomorphic to) Fano varieties.  Moreover I will introduce reflexive GKM graphs, which are a generalization of reflexive polytopes, and illustrate their relation to some positive monotone symplectic manifolds as well as some of their properties.

Enrico Schlesinger (Politecnico di Milano)

Title:  Hilbert schemes of curves in projective space and the maximum genus problem.

Abstract: After a brief introduction to Hilbert schemes, I will present results on the the maximum genus problem. The classical maximum genus problem asks what is the maximum genus of a smooth curve of degree d in projective space that is not contained in a surface of degree < s (this is still open!). I will review what is known on the analogous question for the larger class of locally Cohen-Macaulay curves: these are curves that may have several irreducible components and a nonreduced algebra of regular functions, but do not have isolated or embedded points.  Finally, I will discuss results that settle the problem when d=s and when  d > 2s-2. These results were obtained in collaboration with V. Beorchia and P. Lella first, and then with A. Sammartano.