Home » Lectures » Lorenzo Foscolo: Lectures

Lorenzo Foscolo: Lectures

September 11, 2017
TITLE: ALC manifolds with special holonomy

ABSTRACT: At the meeting in January, I described a new analytic construction, obtained in joint work with Mark Haskins and Johannes Nordström, of complete non-compact G_2-manifolds. The construction starts with an asymptotically conical (AC) Calabi-Yau 3-fold B and produces a 1-parameter family of complete G_2-metrics on a suitable circle bundle over B. These G_2-metrics have asymptotically locally conical (ALC) geometry, the natural generalisation of the asymptotic geometry of ALF hyperkähler 4-manifolds to higher dimensions. In this talk I will describe some initial steps in two different directions related to that result. Firstly, I will show how in our construction of ALC G_2-manifolds rigid compact holomorphic curves in the AC Calabi-Yau 3-fold lift to rigid associative submanifolds in the ALC G_2-manifold. Secondly, I will describe the analogous construction of ALC Spin(7)-holonomy metrics on suitable circle bundles over AC G_2-manifolds.

January 10, 2017 (jointly with Mark Haskins)
TITLE: Non-compact G2 manifolds collapsing to asymptotically conical Calabi-Yau 3-folds

ABSTRACT:We will present a new analytic construction of complete non-compact G_2 holonomy metrics, that yields infinitely many families of examples. The underlying 7-manifolds are all circle bundles over asymptotically conical (AC) Calabi-Yau 3-folds endowed with circle-invariant metrics. Their geometry at infinity is that of a circle bundle over a Calabi-Yau cone with circle fibres of fixed finite length. The G_2 manifolds we construct are therefore 7-dimensional analogues of 4-dimensional ALF hyperkähler metrics. Physicists have termed metrics with such asymptotics ALC (asymptotically locally conical). Within the Melrose school, ALC metrics form a special subclass of so-called fibred boundary metrics.

The dimensional reduction of the equations for G_2 holonomy in the presence of a Killing field was considered by Apostolov-Salamon and by several groups of physicists. We reinterpret the dimensionally-reduced equations in terms of a pair consisting of an SU(3) structure (with tightly constrained torsion) on the 6-dimensional orbit space coupled to an abelian Calabi-Yau monopole on this 6-manifold. We solve this nonlinear coupled system of PDEs by considering the geometric limit in which the circle fibres of the associated circle-invariant G_2 holonomy metrics collapse. In this collapsed limit, given the asymptotically conical Calabi-Yau 3-fold to which the 7-dimensional metrics collapse, the problem comes close to linearising and the crux of the matter is therefore to develop a sufficiently good understanding of solutions to these linearised equations in suitable spaces of functions. For this we need the well-known Fredholm theory for elliptic operators acting on weighted Sobolev spaces on asymptotically conical Calabi-Yau 3-folds and an extension of this theory to the setting of weighted Sobolev spaces on ALC manifolds.

The G_2 holonomy metrics we construct should be thought of as arising from (a suitable perturbation of) abelian Hermitian-Yang-Mills connections on AC Calabi-Yau 3-folds, especially those that arise as crepant resolutions of Calabi-Yau cones. From the physics viewpoint our solutions correspond to M theory uplifts of type IIA solutions in the absence of D6-branes. The existence of such a rich spectrum of “no brane” solutions does not seem to have been anticipated on the physics side, and is a new feature of G_2 holonomy metrics compared to the 4-dimensional hyperkähler setting. In particular all our examples provide instances of families of G_2 holonomy metrics that collapse with bounded curvature to Calabi-Yau 3-folds.

Time permitting we will outline extensions of the construction described here and how it fits into a broader programme.

The talk is closely related to the talks by Carron, Collins and Acharya and is based on joint work with Mark Haskins and Johannes Nordström.


September 8, 2016
TITLE: ALF spaces and collapsing Ricci-flat metrics on the K3 surface

ABSTRACT: The Kummer construction of Kähler Ricci-flat metrics on the (smooth 4-manifold underlying a complex) K3 surface provides the prototypical example of the formation of orbifold singularities in non-collapsing sequences of Einstein 4-manifolds. Much less is known about the structure of the singularities forming along sequences of collapsing Einstein metrics.

I will describe the construction of large families of Ricci-flat metrics on the K3 surface collapsing to the quotient of a flat 3-torus by an involution. The collapse occurs with bounded curvature away from finitely many points. The geometry around these points is modelled by ALF gravitational instantons (complete non-compact hyperkähler 4-manifolds with decaying curvature and cubic volume growth).

This 4-dimensional construction serves as a good warm-up for the much more involved case of collapsing families of G_2 holonomy metrics described in the talk of Mark Haskins.

Slides of lecture