Smooth 1-Dimensional Algebraic Quantum Field Theories

This paper proposes a refinement of the usual concept of algebraic quantum field theories (AQFTs) to theories that are smooth in the sense that they assign to every smooth family of spacetimes a smooth family of observable algebras. Using stacks of categories, this proposal is realized concretely for the simplest case of 1-dimensional spacetimes, leading to a stack of smooth 1-dimensional AQFTs. Concrete examples of smooth AQFTs, of smooth families of smooth AQFTs and of equivariant smooth AQFTs are constructed. The main open problems that arise in upgrading this approach to higher dimensions and gauge theories are identified and discussed.

Hyperbolicity for Log Smooth Families with Maximal Variation

We prove that the base space of a log smooth family of log canonical pairs of log general type is of log general type as well as algebraically degenerate, when the family admits a relative good minimal model over a Zariski open subset of the base and the relative log canonical model is of maximal variation.

Decomposition of degenerate Gromov–Witten invariants

We prove a decomposition formula of logarithmic Gromov–Witten invariants in a degeneration setting. A one-parameter log smooth family $X \longrightarrow B$ with singular fibre over $b_0\in B$ yields a family $\mathscr {M}(X/B,\beta ) \longrightarrow B$ of moduli stacks of stable logarithmic maps. We give a virtual decomposition of the fibre of this family over $b_0$ in terms of rigid tropical maps to the tropicalization of $X/B$. This generalizes one aspect of known results in the case that the fibre $X_{b_0}$ is a normal crossings union of two divisors. We exhibit our formulas in explicit examples.

The isotriviality of smooth families of canonically polarized manifolds over a special quasi-projective base

In this paper we prove that a smooth family of canonically polarized manifolds parametrized by a special (in the sense of Campana) quasi-projective variety is isotrivial.

Families of canonically polarized manifolds over log Fano varieties

Let $(X,D)$ be a dlt pair, where $X$ is a normal projective variety. We show that any smooth family of canonically polarized varieties over $X\setminus \,{\rm Supp}\lfloor D \rfloor $ is isotrivial if the divisor $-(K_X+D)$ is ample. This result extends results of Viehweg–Zuo and Kebekus–Kovács. To prove this result we show that any extremal ray of the moving cone is generated by a family of curves, and these curves are contracted after a certain run of the minimal model program. In the log Fano case, this generalizes a theorem by Araujo from the klt to the dlt case. In order to run the minimal model program, we have to switch to a $\mathbb Q$-factorialization of $X$. As $\mathbb Q$-factorializations are generally not unique, we use flops to pass from one $\mathbb Q$-factorialization to another, proving the existence of a $\mathbb Q$-factorialization suitable for our purposes.

Lines, circles, focal and symmetry sets

Let X be a surface in Euclidean 3-space, hereafter denoted by ℝ3. In the paper [13] Montaldi considered the contact of the surface X with circles, and obtained some very attractive results. In this piece of work we want to address some more detailed questions concerning such contact. In keeping with a general theme within singularity theory we shall bundle the circles up into fibres of certain maps and consider the restriction of these mappings to our surface X. In other words we shall be interested in the simultaneous contact of the surface X with special families of circles. The particular families we shall consider are parameterized by the set K of all lines in ℝ3; associated to such a line we have the family of all circles lying in planes orthogonal to the line, and centred on the line. The line will be referred to as the axis of the circle. Suppose, for example, the line in question is given by x1 = x2 = 0. We can consider the map ℝ3 → ℝ2 given by . The fibres of this mapping are clearly the set of circles with the properties described above together, of course, with single points on the line itself. So the family of oriented lines parameterizes a family of mappings ℝ3 → ℝ2, and by restriction a family of mappings X → ℝ2. It is of interest to relate the singularities of this mapping to the differential geometry of X. The key geometric invariant of any smooth family is its bifurcation set, that is the set of parameter values for which the corresponding map fails to be stable. We shall see that for the family of circle maps the bifurcation set is of some interest.

Elliptic ruled surfaces on Calabi–Yau threefolds

In [5], we studied the behaviour of the Kähler cone of Calabi–Yau threefolds under deformations. We saw that the Kähler cone is locally constant in a smooth family of Calabi–Yau threefolds, unless some of the threefolds Xb contain elliptic ruled surfaces. Moreover, if X is a Calabi–Yau threefold containing an elliptic ruled surface, then the Kähler cone is not invariant under a generic small deformation.