Geometry

Given an algebraic variety $X$ defined over a field $k$, one can associate to it various cohomology groups: coherent, étale, $p$-adic, or even motivic. These groups reflect the geometry, and in case $k$ is of arithmetic interest, the arithmetic of the variety $X$. Using cohomological methods we study, among other things:

• algebraic cycles
• algebraic fundamental groups
• local-global principles for rational points

Logarithmic geometry is an enhanced version of algebraic geometry, where spaces are equipped with an additional structure sheaf, which encodes information of a combinatorial nature (e.g. toric varieties). This recent theory has been fruitfully applied to questions regarding special kinds of degenerations of varieties or other more complicated objects, and compactifications of moduli spaces, for example in the context of mirror symmetry. There are also very interesting interactions with the field of tropical (and non-Archimedean) geometry. Our activity focuses for example on

• the study of moduli spaces of parabolic bundles (some notion of coherent sheaf, adapted to log schemes)
• sheaf-counting on log smooth varieties
• interactions between (log) algebraic and tropical moduli spaces (e.g. for curves with level structures)

The moduli space of surfaces of general type is well known to have an intricate structure. Its “geography” has been extensively studied. Furthermore, a modular compactification of it is the moduli
space of stable surfaces, i.e. semi-log-canonical surfaces with ample canonical divisor. The activity of the group is focused on the analysis of the compactified moduli space $\mathcal{M}(a,b)$ (where $a=K^2$ and $b$ is the holomorphic Euler characteristic), with particular attention to the case of surfaces with low numerical invariants. Such analysis is given by studying log-canonical pairs via a classical approach and analyzing the singularities, via Deformation Theory, and a detailed study of the canonical ring.
Related goals are to extend the knowledge of the moduli space by analyzing $\mathbb{Q}$-Gorenstein surfaces and to study the Hodge theoretic approach, by associating to a variety its cohomology and analyzing the induced variation of Hodge structures.

Quantum groups encode the ‘hidden symmetries’ of quantum physics through integrability. The geometric approach to these groups has proven successful in various areas, including representation theory (e.g. the theory of Maulik-Okounkov Yangians), the theory of moduli stacks and spaces (e.g. the proof of Beauville and Voisin’s conjectures of Maulik-Negut), and theoretical physics (e.g. the proof of the Alday-Gaiotto-Tachikawa conjecture).
Our group aims to investigate quantum groups through their geometric incarnations, focusing on Hall algebras and their refined versions (such as cohomological, K-theoretical, and categorified) associated with moduli stacks of coherent sheaves on curves or surfaces.

The interests of this group focus on moduli stacks of curves and surfaces, and on explicit computations of some of their invariants (e.g. Chow rings, unramified cohomology, Brauer groups). The stacks studied by the group are: moduli stacks of smooth/stable marked curves, moduli stacks of Weierstrass fibrations, moduli stacks of polarized K3 surfaces, moduli stacks of stable maps.

We studied complex Finsler manifolds, having in mind as a guiding example hyperbolic manifolds endowed with the Kobayashi metric, and in particular Kähler-Finsler manifolds with constant holomorphic curvature. Using techniques coming from both differential geometry and algebraic topology we studied the rich geometrical structure of analytic varieties that can be obtained as fixed point sets of a holomorphic self-map, proving a number of index theorems generalising to this setting classical Baum-Bott and Lehmann-Suwa theorems known for holomorphic foliations, and with applications to holomorphic dynamics.

We study the aspect of differential geometry that revolves around the explicit construction of Einstein or special metrics, with special emphasis on the case where the metric is either homogeneous or of cohomogeneity one.

Riemannian homogeneous Einstein metrics of negative scalar curvature can be identified with left-invariant metrics on a Lie group, and more precisely standard metrics on a solvable Lie algebra. In the pseudo-Riemannian case, homogeneous Einstein metric need not be standard, and the Lie algebra may be nilpotent. Our goal is to prove more general structure results for arbitrary signature, and obtain classifications under suitable extra assumptions.

Regardless of any assumption of invariance, among Einstein metrics one finds those of special holonomy and those that admit a Killing spinor; the special metrics that arise may carry natural contact, complex or (almost) hypercomplex structures, and their existence imposes restrictions on the topology. Of particular interest to us is the interplay between the intrinsic torsion and curvature of a special metric. We also aim at producing examples, both by direct inspection of left-invariant metrics on Lie groups and by exploiting the well-posedness of the Cauchy problem for hypersurfaces, which often holds in the context of special metrics.

A characteristic feature of complex analysis is the use of geometrical tools to study analytic phenomena. A typical example consists in using the behaviour of the natural invariant (under biholomorphisms) metrics and distances defined on complex manifolds to study the boundary behaviour of holomorphic functions or the action of integral operators on spaces of holomorphic functions. In particular, we are using the Kobayashi metric and distance in pseudoconvex and convex domains to study the boundary behaviour of the derivatives of a holomorphic function at a specific point in the boundary and, more recently, to study the mapping properties of Toeplitz operators on weighted Bergmann spaces using characterisations of Carleson measures expressed in terms of Kobayashi balls.

In the last forty years, the study of holomorphic dynamical systems has become one of the most important topics in complex analysis and complex geometry of one and several variables, at the forefront of contemporary mathematical research. In Pisa, we are particularly interested in studying:

• the global dynamics of holomorphic self-maps of hyperbolic manifolds and domains, and more generally of non-expanding self-maps of Gromov hyperbolic metric spaces;
• the local dynamics around a non-hyperbolic fixed point;
• the dynamics of meromorphic connections on hyperbolic Riemann surfaces;
• the global dynamics of complex dynamical systems in any dimension (endomorphisms of projective spaces, Hénon maps, automorphisms of Kahler manifolds), usually through pluripotential theory methods;
• Stability and bifurcation phenomena in one and several complex variables.

This area of research studies the geometric and dynamical properties of representations of the fundamental group of a surface $S$ (of negative Euler characteristic) into a Lie group $G$. For example, when $G=\mathbb{P}\mathrm{SL}(2, \mathbb{R})$, conjugacy classes of discrete and faithful representations are in bijection with the Teichmüller space of $S$, the space of marked hyperbolic (or complex) structures on $S$. More in general, and especially for Lie groups of rank $2$ (i.e., $G=\mathrm{SL}(3,\mathbb{R}), \mathrm{Sp}(4, \mathbb{R}, \mathrm{SO}(2,2), G_{2}$), researchers have identified special connected components of the character variety $\mathrm{Hom}(\pi_{1}(S), G)/G$ that parametrize geometric structures on $S$, or fiber bundles over $S$, and share a lot of similarities with the classical Teichmüller space.

The main goal of this research is to understand to which extent the classical Teichmüller theory generalizes to the higher rank. Some aspects include:

• the study of diverging sequences of representations and the definition of the analogue of Thurston’s boundary for higher rank Teichmüller spaces;
• the analysis of equivariant harmonic maps from the universal cover of $S$ into the symmetric space $G/K$ and the real Euclidean building modeled on $\mathfrak{g}$;
• the definition of natural (pseudo)-Riemannian metrics on these higher Teichmüller components and the study of their global geometry.

The uniformisation of surfaces of Koebe and Poincaré and the geometrisation of 3-manifolds of Thurston and Perelman have shown that every manifold of dimensions 2 and 3 admits a geometric structure (after cutting along some canonical spheres and tori in dimension 3). The prominent role among these geometric structures is played by hyperbolic geometry, that is by far the prevalent
structure. It is also the richest and most studied structure in dimensions 2 and 3.

The deformation spaces of hyperbolic 2- and 3-manifolds are the focus of a vast literature concerning Teichmueller spaces and hyperbolic fillings of open manifolds. Moreover, the topology of hyperbolic 3-manifolds is a central topic in low-dimensional topology. But hyperbolic manifolds are abundant in any dimension, and a major goal is to understand their topology as well as their deformation spaces. To this aim, the members of the research group rely on many techniques, from the decomposition of manifolds into hyperbolic polytopes to the study of the topology of fibrations over the circle, to the investigation of the variety of representations of discrete groups into the Lie group of the isometries of hyperbolic space.

The group investigates the combinatorial and topological properties of hyperplane arrangements. From such a point of view, we study the theory of Coxeter groups (seen as reflection groups), Artin groups (seen as fundamental groups of the complements of reflection arrangements), and the computation of cohomology groups of the complements of hyperplane arrangements, both in the linear and the toric cases.
To obtain an explicit characterization of the cohomology ring of the complement of a toric arrangement, the group is studying wonderful compactifications of the complements. This approach allows the definition of a certain differential graded algebra, which ‘governs’ the cohomology ring.
At the moment, the group is interested in the following topics:

• the $K(\pi, 1)$ conjecture for all possible Coxeter groups and the corresponding hyperplane arrangements (for example, the affine simplicial arrangements, which are a natural generalization of the affine reflection arrangements);
• the study of the so-called dual Coxeter groups, which depend on an element of the group and an interval formed by its divisors;
• the study of combinatorial properties (such as the shellability property) for intervals as above;
• the construction of an explicit basis of the integral ring cohomology of complements of toric arrangements and the explicit characterization of the corresponding differential graded algebra in specific examples.

This wide research area encompasses several subjects of interest for the research group.
In dimension 2, for example, we investigate the Hurwitz problem concerning the existence of branched covering between surfaces realizing a fixed combinatorial datum. To this aim, one may exploit Grothendieck’s dessins d’enfant, as well as the geometry of spherical, flat, and hyperbolic 2-orbifolds.

In dimension 3 some topics of interest for the group are the Heegaard-Floer homology of rational homology 3-spheres (with particular attention towards possible applications to the L-space conjecture) and the theory of knots and links in the sphere and in general 3-manifolds. A particular interest is devoted to Legendrian links, and to the study of Khovanov homology. 3-manifold topology is also involved in the study of apparent contours of surfaces in 3-space and in general 3-manifolds.

The topology of 4-manifolds is a very active research field, and the group is also interested in this area. Among the topics covered by the group, there are the study of handlebody decompositions of 4-manifolds, Heegaard-Floer homology, and  3-dimensional knot theory from a 4-dimensional viewpoint.

The simplicial volume is a homotopy invariant of manifolds defined by Gromov in 1982. Despite its purely topological definition, it is deeply related to the geometric structures that a manifold can carry.

Thanks to Thurston’s (now proved) Geometrization Conjecture, the simplicial volume of closed 3-manifolds is well understood. Much less is known in higher dimensions, or for open manifolds, and the group is interested in further investigating these research fields (with particular care devoted to aspherical manifolds).

A powerful tool for the computation of the simplicial volume is the so-called bounded cohomology (of groups and of spaces), which is itself a very active research field. Computing the bounded cohomology of groups is very challenging (for example, the problem of whether it vanishes or not for free groups in degrees bigger than 3 is still open), and the research group aims at achieving some progress in this direction, as well as at studying the relationship between bounded cohomology and other related areas like representation theory, group actions on the circle, ergodic theory of groups.