Titles for the short talks on Monday
Daniele Angella - Canonical metrics on complex manifolds
Giuseppe Barbaro - Cohomology and pluriclosed metrics
Joana Cirici - Dolbeault cohomology for generalized almost complex manifolds
Pietro Ciusa - The constant scalar curvature equation with B-field on blow ups
Marie-Camille Delarue - Stable homology of Thompson groups
Colin Fourel - Morse theory and flow categories
Corrado Mazzoccoli - A Brief Overview of the Hull-Strominger System
Antonio Pio Contrò - A Recent Specialization of the Notion of Formality
Giovanni Placini - Holomorphic isometries of Kähler manifolds
Anna Sopena - A-infinity algebras and minimal models
Jonas Stelzig - The ddbar-equation and homotopy theory
Scott Wilson - Massey products for homotopy inner products
Titles and abstracts for the workshop
Daniele Angella
Emergence of canonical metrics in complex geometry
We investigate various tools and questions related to the emergence of canonical Hermitian metrics from complex structures, with particular attention to identifying and understanding possible obstructions.
__________________________________________
Giuseppe Barbaro
Aeppli cohomology, pluriclosed metrics, and deformations
We exploit the structure of toric fibration of compact semisimple Lie groups to construct a model for their cohomology. We compute the 1,1 Aeppli cohomology which is measuring the dimension of the cone of pluriclosed metrics. We thus pose two questions: one regards the non-existence of pluriclosed metrics deforming the complex structure within the family of left-invariant complex structures; the other about suitable deformations that guaranty the existence of pluriclosed metrics but are not invariant.
__________________________________________
Joana Cirici
On the real homotopy type of complex surfaces
On a compact complex surface, all Massey products of elements in degree 1 vanish beyond length three. Likewise, the real Malcev Lie algebra of its fundamental group is 3-step nilpotent. I will give a proof building an explicit model based on Bott-Chern cohomology.
__________________________________________
Pietro Ciusa
The constant scalar curvature equation with B-field on blow ups
The constant scalar curvature equation with B-field is a generalization of cscK equation for complexified Kähler classes. We want to prove an analogue of the Arezzo-Pacard theorem for these coupled equations. We present some results in this direction regarding the case when the coupled solution is in some sense trivial coming from an actual cscK metric.
__________________________________________
Marie-Camille Delarue
Scanning methods and building topological models for families of groups
The Higman-Thompson groups are groups of certain self-homeomorphisms of Cantor sets. They can be described using tree diagrams. We compute their homology in a stable range by building a topological model for these groups and constructing a scanning map following the work of Madsen, Weiss, Galatius and others. This scanning map allows us to express the stable homology of the Higman-Thompson groups as the homology of a certain spectrum.
__________________________________________
Colin Fourel
Flow categories as exit path categories
Given a Morse function on a closed smooth manifold and a Smale gradient-like vector field adapted to it, one can construct a topological category called the flow category associated with this data. Its objects are the critical points of the function, and its morphisms are the broken trajectories of the vector field connecting critical points. Like any topological category, a flow category models an ∞-category, and a theorem of Cohen–Jones–Segal states that the homotopy type underlying this ∞-category is that of the manifold itself. However, different choices of Morse–Smale pairs on the same manifold can give rise to non-equivalent ∞-categories. After making these ideas more precise, I will present a result I obtained, which asserts that the ∞-category associated with the flow category of a Morse–Smale pair is equivalent to the exit-path ∞-category associated with the stratification of the manifold by the ascending manifolds of the critical points.
__________________________________________
Corrado Mazzoccoli
Exploring Theory and Solutions to the Hull–Strominger System
Lying at a crossroads between modern Theoretical Physics and Complex Differential Geometry, the Hull-Strominger system presents a compelling and challenging problem. For superstring theorists, the system describes the geometric structure of space-time in the heterotic framework; for complex geometers, it provides a new and unexplored notion of special metric. Indeed, this system of partial differential equations over a hermitian manifold combines a conformally balanced condition--making the manifold a non-Kähler Calabi-Yau--with an "anomaly" equation that links the torsion and curvature of the metric. Moreover, the manifold is endowed with a connection satisfying the Hermitian-Yang-Mills equation. Directly solving this system remains a considerable analytical challenge; However, several explicit solutions to the system have been constructed, offering valuable insight into its intricate properties. After a brief introduction to the underlying concepts, this talk will survey the landscape of known solutions, with a focus on selected examples of interest.
__________________________________________
Antonio Pio Contrò
Are Hyperkähler Manifolds Strongly Formal?
Following J. Stelzig and A. Milivojević’s introduction of the notion of strong formality for complex manifolds (a bigraded extension of the already well-known concept of formality for manifolds), it has been natural to ask which classes of complex manifolds are strongly formal. This question has become even more relevant after G. Placini, J. Stelzig, and L. Zoller have recently shown the existence of compact Kähler manifolds that are not strongly formal, thus marking a notable difference from the classical case (indeed, in 1975 P. Deligne, P. Griffiths, J. Morgan, and D. Sullivan proved that all compact Kähler manifolds are formal). In this talk, I will give a general overview of the current state of the art regarding the possible strong formality of compact Hyperkähler manifolds, providing all the necessary background to understand the problem, and aiming to: 1) highlight the points that suggest a positive answer concerning their strong formality, and 2) underline the analogies with the case of compact Kähler manifolds and formality (it seems, in fact, that Hyperkähler manifolds may play, with respect to strong formality, a role analogous to that of Kähler manifolds with respect to formality). This talk is based on joint work/discussions with G. Placini and J. Stelzig.
__________________________________________
Giovanni Placini
Holomorphic isometries of Kähler manifolds
Kähler geometry is rather “rigid”. A particular instance in which we encounter rigidity phenomena is that of Kähler submanifolds. As an example, an interesting problem is to determine whether two Kähler manifolds share any submanifold, the trivial case being when one is embeddable in the other. We will discuss classical and recent results on local and global holomorphic isometric immersions of Kahler manifolds. We will introduce the essential work of Calabi on embedding of complex spaces forms into one another and present more recent result and open problems of the same flavour.
__________________________________________
Sönke Rollenske
Compact complex parallelisable nilmanifolds with unobstructed deformations
Compact quotients of non-abelian complex nilpotent Lie groups are non-Kähler manifolds with trivial tangent and hence trivial canonical bundle, the most prominent example being the Iwasawa manifold. We give a precise characterisation of the (rare) cases where such manifolds have unobstructed deformations in terms of so-called verbal ideals in free nilpotent Lie-algebras. (joint with M. Paulsen and K. Wehler)
__________________________________________
Anna Sopena
Pluripotential A-infinity structures on the cohomology of a complex manifold
I will introduce pluripotential A-infinity algebras and show how these structures naturally emerge in the cohomology of complex manifolds, providing a refined invariant. Notably, this framework recovers the triple Aeppli–Bott–Chern–Massey products studied by Angella and Tomassini, just as classical A-infinity algebras recover Massey products.
__________________________________________
Jonas Stelzig
Title 1: Strong formality over Q in actions.
Abstract 1: I'll show that compact homogeneous Kähler manifolds and toric manifolds are strongly formal. Then, I'll introduce strong formality over Q, which roughly means rational and strong formality in a compatible way, and show that it is stronger than strong formality and that the aforementioned results still hold over Q. I'll end with some open questions and ideas for future directions. Based on joint work with Giovanni Placini and Leopold Zoller.
Title 2 Pluripotential, Hodge, and arithmetic
Abstract 2: I will survey some work in progress and open questions relating pluripotential homotopy theory and ABC Massey products with invariants studied 'classically' in complex geometry and arithmetic. In particular: Mixed Hodge structures, archimedean height functions ('holomorphic linking numbers') and Massey products in Deligne cohomology.
__________________________________________
Scott Wilson
Massey products for homotopy inner products
I will describe the higher homotopical notion of inner products and define “Massey products” for such structures, as in recent work with Kate Poirier and Thomas Tradler. These are certain A-infinity type structures and give invariants which are natural with respect to morphisms. Explicit geometric examples will be given, coming from links, that show these contain information that is different than ordinary Massey products, defined for A-infinity algebras. It is expected (but not yet explicated in detail) that there are similar “homotopy inner products” and “Massey inner products” in the bi-graded context as well, complementary to ABC-Massey products for complex manifolds. This would be an excellent project pursue with workshop participants.