I will review the notion in the title (which, is not a group) and show how to construct in certain cases an "exponential" in the complexification. The construction is motivated by quantum mechanics.

If $M$ is a real-analytic Riemannian manifold, then there exists a canonical "adapted complex structure" on a neighborhood of the zero-section in the tangent bundle $T M$. These structures were introduced independently by Guillemin and Stenzel and by Lempert and Szoke. It is possible to understand the adapted complex structure in terms of the "imaginary time geodesic flow." I will describe a new family of complex structures in which one modifies the problem by adding a "magnetic field" on $M$, described by a closed 2-form. These new complex structures are described in terms of the Hamiltonian flow for a particle in a magnetic field, evaluated (in a suitably interpreted way) at an imaginary time. For a constant magnetic field on $\mathbb{R}^2$ or $S^2$, the magnetic complex structure can be computed explicitly and is defined on the entire tangent bundle.

Fano surfaces parametrize the lines on smooth cubic threefolds. In this talk, we will explain the remarkable geometric properties of these surfaces and we will explain how to use these properties in order to compute the invariants of the surfaces obtained as quotients of Fano surfaces by an automorphism group.

In a real vector space addition is settled by scalar multiplication. This lead us to the following description of smooth vector bundles: they are manifolds endowed with a nice action of the multiplicative monoid of the real numbers. In this talk I will deal with vector bundles over Lie groupoids and algebroids. Lie groupoids are presentations for singular smooth spaces, constitute a concrete unified framework to deal with manifolds, Lie groups, actions, foliations and others. Lie algebroids are their infinitesimal counterpart, together they play a rich theory in actual development. Vector bundles over groupoids and algebroids are generalized representations and help us to understand the geometry of our singular spaces. I will present definitions and examples, provide a characterization by using the multiplicative monoid, and discuss the integration of these structures. This is part of a joint work with H. Bursztyn and A. Cabrera.

The Arnold conjecture in Symplectic Topology states existence of many fixed points for Hamiltonian symplectomorphisms of a compact symplectic manifold. In my talk I will discuss an analogue of this conjecture in Contact Topology, based on the notion of translated points.

I will explain why each integral convex compact polytope is the moment polytope of a Kähler-Einstein orbifold (unique up to a dilatation). More generally, I will explain how to prove that each convex compact polytope admits a unique ray of symplectic potentials solving the toric Kähler-Einstein equation. Then, I'll bring out some geometric applications.

This is a report on joint work in collaboration with A. Knutsen (University of Bergen, Norway).

The pairs $(S,H)$ with $S$ a complex K3 surface (i.e. $S$ is simply connected and ${\Omega }_S^2$ is trivial), and $H$ is an ample line bundle of genus $p$ (i.e. $H^2=2p-2$), have an irreducible moduli space ${𝒦}_p$ of dimension 19. Let $(S,H)\in {𝒦}_p$ be general. The linear system $\mid H\mid$ has dimension $p$ and its general element is a smooth curve of genus $p$. I consider $V_{\mid H\mid ,\delta }$, the Severi variety of $\delta$-nodal curves in $\mid H\mid$, which is locally closed of codimension $\delta$ in $\mid H\mid$ and its general element is an irreducible curve with exactly $\delta$ nodes and no other singularities, so that its geometric genus (i.e. the genus of its desingularization) is $g=p-\delta$. The image of $V_{\mid H\mid ,\delta }$ in ${ℳ}_g$, the moduli space of curves of genus $g$, has dimension $g$ and its importance in moduli problems is well known. For any integer $k\ge 2$, consider the subscheme $V_{\mid H\mid ,\delta }^k\subset V_{\mid H\mid ,\delta }$ given by $$\{C\in V_{\mid H\mid ,\delta }:\text{the normalization of}C\text{is a}k\text{-tuple cover of}{\mathbb{P}}^1\}.$$

The first question is: when is $V_k\subset V_{\mid H\mid ,\delta }$ not empty? Using a (by now classical) vector bundle technique due to Lazarsfeld, we show that a necessary condition is $\delta \ge \alpha (g-(k-1)(\alpha +1))$ where $\alpha :=\lfloor \frac{g}{2(k-1)}\rfloor$. A dimension count shows that, if $V_{\mid H\mid ,\delta }^k$ is not empty, its expected dimension should be $2k-2$. As a matter of fact, we prove that for all even $p$ and all $\delta ,k$ verifying the above relation, $V_{\mid H\mid ,\delta }^k$ is actually not empty, of dimension $2k-2$. For $p$ odd we have a slightly weaker result. Finally, I will talk about some interesting relations of these results with a conjecture by Hassett and Tschinkel on the Mori cone of the hyperkhähler manifold ${\text{Hilb}}^k(S)$ (which parametrizes subschemes of $S$ of finite lenght $k$).

In this talk we will present Steiner and Schwarzenberger bundles on the Grassmannians and show the theory that relates them. In the first part we will define the two families of bundles mentioned before and study their properties. We will introduce then the concept of jumping pair for a Steiner bundle and we will study the dimension of the jumping locus of the bundle. Finally, we will give a complete classification of Steiner bundles on the Grassmannian whose jumping locus has maximal dimension and will describe them as Schwarzenberger bundles.

We intend to give an elementary talk to explain why (smooth) subvarieties of small codimension are expected to be quite special. We will concentrate on a theorem by Barth stating that a subvariety of dimension $n$ in a projective space of dimension $N$ inherits much of the topology of the projective space, namely the integral cohomology up to order $2n-N$ must be the same. We will give a new geometrical approach to this theorem, which will allow us to extend Barth's theorem to other ambient spaces different from the projective space. We will put all this in relation with the famous Hartshorne's conjecture about subvarieties of small codimension in the projective space

Gauged vortices are configurations of fields for certain gauge theories on fibre bundles over a surface $S$. Their moduli spaces support natural $L^2$-metrics, which are Kaehler, and whose geodesic flow approximates vortex dynamics at low speed. My talk will focus on vortices in line bundles, for which the moduli spaces are modelled on the spaces of effective divisors on $S$ with a fixed degree $k$; I shall describe the behaviour of the underlying $L^2$-metrics in a "dissolving limit" where the $L^2$-geometry simplifies and can be related to the geometry of the Jacobian variety of the surface. Some intuition about multivortex dynamics in this limit will be provided by analysing the simplest nontrivial example (two dissolving vortices moving on a hyperelliptic curve of genus three). This is joint work with N. Manton.

We consider moduli spaces of polygons in Minkowski 3-space with a fixed number of sides lying in the future time-like cone and the others in the past (where each side has a fixed Minkowski length) and study their relation with the fixed-point set of a natural involution on the space of rank-2 parabolic Higgs bundles over ${\mathrm{CP}}^1$.

This is joint work with I. Biswas, C. Florentino and A. Mandini.

Brill-Noether theory seeks to answer the basic question of when an abstract complex curve $C$ of genus $g$ comes equipped with a nondegenerate degree-$d$ morphism to $P^r$.

When $C$ is general in the moduli space $M_g$, an answer is provided by a celebrated theorem of Griffiths and Harris, which establishes that $C$'s behavior is as "expected" whenever $\rho =g-(r+1)(g-d+r)$ is positive, and that there are no morphisms when $\rho$ is negative.

Payne, et al. recently gave a proof of the nonexistence statement using a combinatorial analysis of metric graphs of a particular combinatorial type. These belong to a moduli space $M_g^{\mathrm{trop}}$ for stable metric graphs, which admits a stratification that is dual to that of the Deligne-Mumford space of stable curves.

We will discuss work in progress with Melo, Neves, and Viviani aimed at understanding the Brill-Noether-type behavior of metric graphs of other combinatorial types, focusing on a remarkable infinite family of these.

We present gluing formulas for zeta regularized determinants of Dolbeault laplacians on Riemann surfaces. These are expressed in terms of determinants of associated operators on surfaces with boundary satisfying local elliptic boundary conditions. The conditions are defined using the additional structure of a framing, or trivialization of the bundle near the boundary. An application to the computation of bosonization constants follows directly from these formulas.

Let $K$ be a compact Lie group, $G$ be its complexication, and $F$ be any finitely generated Abelian group. We prove that the conjugation orbit space $\operatorname{Hom}(F,K)/K$ is a strong deformation retract of the GIT conjugation orbit space $\operatorname{Hom}(F,G)/G$. As a corollary, we determine necessary and sufficient conditions for $\operatorname{Hom}(F,G)/G$ to be irreducible when $G$ is connected and semisimple, and $F$ is free Abelian. This is joint work with C. Florentino.

A rigidity result on the coisotropic Maslov index states that there exists a non-trivial loop (tangent to the characteristic foliation of a stable coisotropic submanifold) with certain bounds on its symplectic area and its Maslov index. This was proved by Ginzburg for symplectically aspherical ambient manifolds. The result also holds for some symplectic manifolds not necessarily aspherical. We shall state the theorem for the “rational case” and sketch its proof.

It is known that a compact complex manifold of dimension $2$ with the same Betti numbers as the complex projective plane is projective. Such a manifold is called "a fake projective plane" if it is not isomorphic to the complex projective plane. So a fake projective plane is exactly a smooth surface $X$ of general type with $p_g(X)=0$ and $c_1(X)^2=3c_2(X)=9$. By a result of Aubin and Yau, its universal cover is the unit $2$-ball, hence its fundamental group $\pi_1(X)$ is a discrete torsion-free cocompact subgroup of $PU(2,1)$ satisfying certain conditions. The classification of such subgroups has been done by G. Parasad and S.-K. Yeung, with the computer based computation by D. Cartwright and J. Steger. This has settled the arithmetic side of the classification problem. I will go over this, and then report recent progress on the other side of the problem-geometric construction.

The 1st lecture will review the singularity of class $T$ and $Q$-Gorenstein deformation theory. The notion of singularity of class $T$, which is defined as a quotient surface singularity admitting a $Q$-Gorenstein smoothing, was introduced by Kollár and Shepherd-Barron. They also gave an explicit description of the singularity of class $T$. The notion of $Q$-Gorenstein deformation is popular in the study of degenerations of normal algebraic varieties in characteristic zero related to the minimal model theory and the moduli theory since the paper by Kollár and Shepherd-Barron. A typical example of $Q$-Gorenstein deformation appears as a deformation of the weighted projective plane ${P}(1, 1, 4)$: Its versal deformation space has two irreducible components, in which the one-dimensional component corresponds to the $Q$-Gorenstein deformation and its general fibers are projective planes. By developing the theory of $Q$-Gorenstein deformation functor, we can generalize their results to surfaces in positive characteristics.

References

• J. Kollár and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), 299-338.
• Y. Lee and J. Park, A simply connected surface of general type with $p_g=0$ and $K^2=2$, Invent. Math. 170 (2007), 483-505.
• Y. Lee and N. Nakayama, Simply connected surfaces of general type in positive characteristic via deformation theory, preprint 2011 (arXiv:1103.5185, to appear in PLMS).

The relationship between the algebra of symmetric differentials (sections of the symmetric powers of the holomorphic cotangent bundle) and the topology of a projective manifold is still considered quite mysterious. In general this relationship will be quite loose, since for example it is known that there are manifolds with the same topology but diametrically distinct spaces of symmetric differentials (e.g. no symmetric differentials and asymptotically as many as possible). On the other hand, it is expected that properties of the fundamental group to be reflected on the space of symmetric differentials.

The goal of these lectures is to show that there is a class of symmetric differentials that is quite topological in nature. This class constitutes an extension to all degrees of the class of closed symmetric differentials of degree 1 (i.e. closed holomorphic 1-forms) which are well known to reflect topological properties. We will: discuss primarily the case of degree 2; describe examples; connect to the theory of foliations and fibrations; and show that the presence of closed symmetric differentials imply that the fundamental group has to be infinite.

It is known that a compact complex manifold of dimension $2$ with the same Betti numbers as the complex projective plane is projective. Such a manifold is called "a fake projective plane" if it is not isomorphic to the complex projective plane. So a fake projective plane is exactly a smooth surface $X$ of general type with $p_g(X)=0$ and $c_1(X)^2=3c_2(X)=9$. By a result of Aubin and Yau, its universal cover is the unit $2$-ball, hence its fundamental group $\pi_1(X)$ is a discrete torsion-free cocompact subgroup of $PU(2,1)$ satisfying certain conditions. The classification of such subgroups has been done by G. Parasad and S.-K. Yeung, with the computer based computation by D. Cartwright and J. Steger. This has settled the arithmetic side of the classification problem. I will go over this, and then report recent progress on the other side of the problem-geometric construction.

The 2nd lecture will explain several methods for constructions of surfaces of general type via $Q$-Gorenstein smoothings. After the paper by Lee and Park, which constructs a simply connected Campedelli surface, several interesting examples of surfaces of general type with $p_g=0$ were constructed via $Q$-Gorenstein smoothings. Now, these $Q$-Gorenstein smoothing methods are extended to some other type of surfaces and to surfaces in positive characteristics.

References

• J. Kollár and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), 299-338.
• Y. Lee and J. Park, A simply connected surface of general type with $p_g=0$ and $K^2=2$, Invent. Math. 170 (2007), 483-505.
• Y. Lee and N. Nakayama, Simply connected surfaces of general type in positive characteristic via deformation theory, preprint 2011 (arXiv:1103.5185, to appear in PLMS).

The relationship between the algebra of symmetric differentials (sections of the symmetric powers of the holomorphic cotangent bundle) and the topology of a projective manifold is still considered quite mysterious. In general this relationship will be quite loose, since for example it is known that there are manifolds with the same topology but diametrically distinct spaces of symmetric differentials (e.g. no symmetric differentials and asymptotically as many as possible). On the other hand, it is expected that properties of the fundamental group to be reflected on the space of symmetric differentials.

The goal of these lectures is to show that there is a class of symmetric differentials that is quite topological in nature. This class constitutes an extension to all degrees of the class of closed symmetric differentials of degree 1 (i.e. closed holomorphic 1-forms) which are well known to reflect topological properties. We will: discuss primarily the case of degree 2; describe examples; connect to the theory of foliations and fibrations; and show that the presence of closed symmetric differentials imply that the fundamental group has to be infinite.

There is a natural holomorphic Poisson structure on any flag manifold $G/P$ with infinitely many symplectic leaves, but only finitely many $T$-orbits of symplectic leaves. I'll describe these in detail for the case of the Grassmannian, where the leaves are naturally indexed by juggling patterns (this work is joint with Thomas Lam and David Speyer). Then I'll talk about manifolds with atlases consisting of Kac-Moody Bruhat cells, which conjecturally includes wonderful compactifications of groups.

In this talk, I will give a survey in convex geometries associated with Teichmuller space. In particular, an emphasis is put on the Weil-Petersson geometry, where deformations of hyperbolic metrics on a given topological surface is measured by the $L^2$ norm of the deformation tensors. Also mentioned is the several convex geometries naturally occurring in the investigation of Teichmuller space, including the geometries of the Teichmuller metric and the Thurston metric.

After recalling the main geometric properties of 3-quasi-Sasakian manifolds, I will present our preliminary results on the topology of this class of Riemannian manifolds. They include both 3-cosymplectic and 3-Sasakian manifolds as special cases of minimum and maximum rank, respectively. Recently, there has been an increasing interest toward these manifolds, especially the 3-Sasakian case, due to their relevance in theoretical physics. It is known that any 3-quasi-Sasakian manifold is locally the product of a Hyper-Kaehler manifold and a 3-Sasakian manifold. Nevertheless, we find examples of compact 3-quasi-Sasakian manifolds which are not the global product of a Hyper-Kaehler manifold and a 3-Sasakian manifold.

Let $G$ be a connected reductive group. A $G$-variety is called spherical if it is normal and has a dense orbit under some Borel subgroup of $G$. Examples include flag varieties, symmetric varieties, toric varieties and, more generally, group embeddings. The latter are closely related to algebraic monoids. An important topological property of spherical varieties is that they have only a finite number of $G$-orbits and, consequently, also a finite number of $T$-fixed points, for the induced action of a maximal torus $T$ of $G$. Furthermore, under certain assumptions on the singularity type and the number of $T$-invariant curves, the equivariant cohomology of spherical varieties can be understood via GKM-theory, a technique named after Goresky-Kottwitz-MacPherson. In this talk, we give an overview of these notions and, based on previous work by the author, provide a complete description of the equivariant cohomology of rationally smooth group embeddings. Finally, we report on current work extending these results to the study of equivariant Chow rings and K-theory.

I propose to discuss the main conjecture (the Green-Griffiths conjecture) on the complex hyperbolicity of algebraic surfaces, its consequences in the spirit of the dictum of Bloch: “nihil est in infinito quod non fuerit prius in finito”, and how one proves such things. Various further details may be found at http://www.mat.uniroma2.it/~mcquilla/

It is known that semistable sheaves V admit a filtration whose quotients are stable and have the same slope of V, named the Jordan-Hölder filtrations. We give the analogous result for principal Higgs bundles on curves.

We prove a sufficient criterion for a very general hypersurface in simplicial toric 3-fold to have the same Picard number as the ambient variety. We also give some preliminary results about the estimate of the codimension of the locus where the Picard number is bigger.

We have constructed a leafwise symplectic structure for a foliation on the 5 sphere which is associated to the Milnor fibration of a simple elliptic hypersurface singularity in 3 complex variables. The foliation associated with $x^3+y^3+z^3=0$ is nothing but Lawson’s foliation which was constructed as the 1^st codimension one foliation on the 5 sphere in 1970. In this talk we abstract the essence of the construction as a framework to produce a leafwise symplectic foliation or a family of almost contact structure converging to a symplectic foliation from an exact symplectic open book decomposition supporting a contact structure.

As a by-product of the framework, we can show that a similar construction works also for the cusp singularities in 3 variables. If the time allows we discuss about the possibility of further applicability of the framework as well as the impossibility.

All dynamical system, including non-hamiltonian and non-integrable ones, admits a natural intrinsic commutative symmetry group (torus action) which preserves not only the system but also any tensor field which is invariant with respect to the system. I will discuss the role played by these torus actions in the normalization problems, including action-angle variables, Poincare-Birkhoff normalization, and renormalization group method.

I will explain the construction of a bi-invariant metric on the universal cover of the contactomorphism group of any contact manifold, and discuss its relations to other contact rigidity phenomena such as the contact non-squeezing theorem, orderability of contact manifolds, and the existence of translated points for contactomorphisms. This is joint work with Vincent Colin.