Autumn Term 2016

János Kollár (Princeton University). Existence of Conic bundles that are not birational to numerical Calabi–Yau pairs. Friday 23rd Sep., 1:30-2:30pm. Huxley 503.

Abstract: Let X be a general conic bundle over the projective plane with branch curve of degree at least 19. We prove that there is no normal projective variety Y that is birational to X and such that some multiple of its anticanonical divisor is effective. We also give such examples for 2-dimensional conic bundles defined over a number field.

Jacob Rasmussen (Cambridge University). L-spaces, Left-orderings, and Lagrangians. Friday 14th Oct., 1:30-2:30pm. Huxley 341.

Abstract: Following Lekili, Perutz, and Auroux, we know that the Floer homology of a 3-manifold with torus boundary should be viewed as an element in the Fukaya category of the punctured torus. I’ll give a concrete description of how to do this and explain how it can be applied to study the relationship between L-spaces (3-manifolds with the simplest Heegaard Floer homology) and left orderings of their fundamental group.

Gavril Farkas (Humboldt Universität). Compact moduli of abelian differentials. Friday 21st Oct., 1:30-2:30pm. Huxley 341.

Abstract: The moduli space of holomorphic differentials (with prescribed zeros and poles) on nonsingular curves is not compact since the curve may degenerate. I will discuss a compactification of these strata in the moduli space of Deligne-Mumford stable pointed curves, which includes the space of canonical divisors as an open subset. The theory leads to geometric/combinatorial constraints on the closures of the strata of holomorphic differentials and as a consequence, one can determine the cohomology classes of the strata. This is joint work with Rahul Pandharipande.

Jenia Tevelev (University of Massachusetts at Amherst). The Craighero-Gattazzo surface is simply-connected. Friday 28th Oct., 1:30-2:30pm. Huxley 341.

Abstract: We show that the Craighero–Gattazzo surface, the minimal resolution of an explicit complex quintic surface with four elliptic singularities, is simply-connected. This was conjectured by Dolgachev and Werner, who proved that its fundamental group has a trivial profinite completion. The Craighero–Gattazzo surface is the only explicit example of a smooth simply-connected complex surface of geometric genus zero with ample canonical class. We hope that our method will find other applications: to prove a topological fact about a complex surface, we use an algebraic reduction mod p technique and deformation theory. Joint work with Julie Rana and Giancarlo Urzua.

Travis Schedler (Imperial College). Quantized symplectic singularities and Poisson homology. Friday 4th Nov., 1:30-2:30pm. Huxley 341.

Abstract: I will recall a unifying paradigm which incorporates representation theory of semisimple Lie algebras, Weyl algebras, D-modules, Cherednik algebras, and many more. In the case of semisimple Lie algebras, one considers the geometry of the cone of ad-nilpotent elements and its quantization (noncommutative deformation). I will explain how one can study these by a basic algebraic invariant, Poisson homology, and deduce powerful statements such as bounds on the number of finite-dimensional irreducible representations. I will end with some open problems and conjectures, such as on symplectic resolutions.

Johannes Nordstrom (University of Bath). Complete and conically singular G₂-manifolds of cohomogeneity​ one. Friday 11th Nov., 1:30-2:30pm. Huxley 341.

Abstract: Bryant and Salmon’s cohomogeneity 1 examples of complete, asymptotically conical G_2-manifolds provide a model for desingularising compact G_2-manifolds with conical singularities; however no examples of the latter are yet known, and there are also no further known examples of asymptotically conical G_2-manifolds. Theoretical physicists such as Cvetic-Gibbons-Lu-Pope and Brandhuber-Gomis-Gubser-Gukov have considered complete cohomogeneity 1 G_2-manifolds that are “asymptotically locally conical”–the model at infinity is a circle bundle over a cone–and which in a 1-parameter family converge to an asymptotically conical manifold. However, only some of these families have been studied rigorously (Bazaikin-Bogoyavlenskaya).
I will discuss joint work in progress with Foscolo and Haskins on these families, and some of their limits, which include a new asymptotically conical G_2-manifold and a conically singular G_2-manifold with locally conical asymptotics. The latter may provide an avenue to construction of compact G_2-manifolds with conical singularities.

Misha Verbitsky (Université libre de Bruxelles). Proof of Morrison-Kawamata cone conjecture for hyperkahler manifolds. Friday 18th Nov., 1:30-2:30pm. Huxley 341.

Abstract: Let M be a compact holomorphically symplectic manifold and K its Kahler cone. Morrison-Kawamata cone conjecture says that the automorphism group of M acts on polyhedral faces of K with finitely many orbits. I would explain the proof of this result (obtained jointly with Ekaterina Amerik), based on ergodic theory and hyperbolic geometry.
It turns out that the Morrison-Kawamata cone conjecture can be interpreted as a result of hyperbolic geometry: the quotient of the projectivization of rational positive cone of M by the group of Hodge isometries is a hyperbolic manifold H of finite volume, and the ample cone of M corresponds to a finite polyhedron in H with piecewisely geodesic boundary. As an application, we obtain that M has only finitely many holomorphically symplectic birational models.

Arnaud Beauville (Université de Nice). Recent developments in the Lüroth problem. Friday 25th Nov., 1:30-2:30pm. Huxley 341.

Abstract: The Lüroth problem asks whether every field K with CKC(x1,…, xn) is of the form C(y1,…, yp). In geometric terms, if an algebraic variety can be parametrized by rational functions, can one find a one-to-one such parametrization?

This holds for curves (Lüroth, 1875) and for surfaces (Castelnuovo, 1894); after various unsuccessful attempts, it was shown in 1971 that the answer is quite negative in dimension 3: there are many examples of unirational varieties which are not rational. Up to last year the known examples in dimension >3 were quite particular, but a new idea of Claire Voisin has significantly improved the situation.

I will survey the colorful history of the problem, then explain Voisin’s idea, and how it leads to a number of new results.

Eleonora Di Nezza (Imperial College). The space of Kähler metrics on singular varieties. Friday 2nd Dec., 1:30-2:30pm. Huxley 341.

Abstract: The geometry and topology of the space of Kähler metrics on a compact Kähler manifold is a classical subject, first systematically studied by Calabi in relation with the existence of extremal Kähler metrics. Then, Mabuchi proposed a Riemannian structure on the space of Kähler metrics under which it (formally) becomes a non-positive curved infinite dimensional space. Chen later proved that this is a metric space of non-positive curvature in the sense of Alexandrov and its metric completion was characterized only recently by Darvas.
In this talk we will talk about the extension of such a theory to the setting where the compact Kähler manifold is replaced by a compact singular normal Kähler space. As one application we give an analytical criterion for the existence of Kähler-Einstein metrics on certain mildly singular Fano varieties, an analogous to a criterion in the smooth case due to Darvas and Rubinstein.
This is based on a joint work with Vincent Guedj.

Dmitri Panov (King’s College). Real line arrangements with Hirzebruch property. Friday 9th Dec., 1:30-2:30pm. Huxley 341.

Abstract: A line arrangement of 3n lines in CP^2 satisfies Hirzebruch property if each line intersect others in n+1 points. Hirzebruch asked if all such arrangements are related to finite complex reflection groups. We give a positive answer to this question in the case when the line arrangement in CP^2 is real, confirming that there exist exactly four such arrangements.

Please note the unusual location:
Alessandra Sarti (Université de Poitiers). On the moduli space of cubic threefolds and irreducible holomorphic symplectic manifolds. Friday 16th Dec., 1:30-2:30pm. Huxley 144.

Abstract: In a famous paper Allcock, Carlson and Toledo describe the moduli space of smooth cubic threefolds as a ball quotient. Here we give an interpretation of this moduli space as moduli space of some irreducible holomorphic symplectic fourfolds with a special non-symplectic automorphism of order three. This is part of a more general construction, that I will explain in the talk. It is a joint work with S. Boissière and C. Camere.