# Autumn Term 2015

Balazs Szendroi (University of Oxford). Euler characteristics of Hilbert schemes of points on singular surfaces. Friday 16th Oct, 1.30-2.30pm. Huxley 139.

Abstract: Given a smooth surface, the generating series of Euler characteristics of its Hilbert schemes of points can be given in closed form by (a specialisation of) Goettsche’s formula. I will discuss a generalisation of this formula to surfaces with rational double points. A certain representation of the affine Lie algebra corresponding to the surface singularity (via the McKay correspondence), and its crystal basis theory, play an important role in our approach. (Joint work with Adam Gyenge and Andras Nemethi, Budapest)

Olivier Debarre (Ecole normale supérieure – Paris). Rational cohomology tori. Friday 23rd Oct, 1.30-2.30pm. Huxley 139.

Abstract: Hirzebruch and Kodaira proved in 1957 that when $n$ is odd, any compact K\”ahler manifold $X$ which is homeomorphic to ${\bf P}^n$ is isomorphic to ${\bf P}^n$. This holds for all $n$ by Aubin and Yau’s proofs of the Calabi conjecture. It is conjectured that it should be sufficient to assume that the integral cohomology rings $H^\bullet(X,{\bf Z})$ and $H^\bullet({\bf P}^n,{\bf Z})$ are isomorphic and $c_1(T_X)>0$.
Catanese recently observed that complex tori are characterized among compact K\”ahler manifolds by the fact that their integral cohomology rings are exterior algebras on $H^1$ and asked whether this remains true under the weaker assumptions that the rational cohomology ring is an exterior algebra on $H^1(X,{\bf Q})$ (we call the corresponding compact K\”ahler manifolds “rational cohomology tori”).
We give a negative answer to Catanese’s question by producing explicit examples. We also prove some structure theorems for rational cohomology tori. This is work in collaboration with Z. Jiang, M. Lahoz, and W. F. Sawin.

Renzo Cavalieri (Colorado State University). Open invariants and Crepant Transformations. Friday 30th Oct, 1.30-2.30pm. Huxley 139.

Abstract: The question that the Crepant Resolution Conjecture (CRC) wants to address is: given an orbifold X that admits a crepant resolution Y, can we systematically compare the Gromov-Witten theories of the two spaces? That this should happen was first observed by physicists and the question was imported into mathematics by Y.Ruan, who posited it as the search for an isomorphism in the quantum cohomologies of the two spaces. In the last fifteen years this question has evolved and found different formulations which various degree of generality and validity. Perhaps the most powerful approach to the CRC is through Givental’s formalism. In this case, Coates, Corti, Iritani and Tseng propose that the CRC should consist of the natural comparison of geometric objects constructed from the GW potential fo the space. We explore this approach in the setting of open GW invariants. We formulate an open version of the CRC using this formalism, and make some verifications. Our approach is well tuned with Iritani’s approach to the CRC via integral structures, and it seems to suggest that open invariants should play a prominent role in mirror symmetry.

Roberto Svaldi (University of Cambridge). A geometric characterization of toric varieties. Friday 6th Nov, 1.30-2.30pm. Huxley 139.

Abstract: Given a pair (X, D), where X is a proper variety and D a divisor with mild singularities, it is natural to ask how to bound the number of components of D. In general such bound does not exist. But when -(K_X+D) is positive, i.e. ample (or nef), then a conjecture of Shokurov says this bound should coincide with the sum of the dimension of X and its Picard number. We prove the conjecture and show that if the bound is achieved, or the number of components is close enough to said sum, then X is a toric variety and D is close to being the toric invariant divisor. This is joint work with M. Brown, J. McKernan, R. Zong.

Michel Van den Bergh (Hasselt University). Non-commutative resolutions. Friday 13th Nov, 1.30-2.30pm. Huxley 139.

Abstract: It has been observed in recent years that many singular varieties admit a resolution of singularities, in a suitable sense, given by a non-commutative sheaf of algebras. This sometimes happens in cases where no nice commutative resolution is known. Even if a good commutative resolution is known to exist the non-commutative resolution may still be smaller. In the talk I will give a general introduction to this phenomenon. In particular I will discuss some recent joint work with Spela Spenko about non-commutative resolutions of determinantal varieties for symmetric and skew-symmetric matrices.

Behrouz Taji (University of Freiburg). The Miyaoka-Yau inequality for minimal models of general type and
the uniformization by the ball.
Friday 20th Nov, 1.30-2.30pm. Huxley 139.

Abstract: By proving Calabi’s conjecture, Yau proved that the Chern classes of a compact manifold with ample canonical bundle encode the symmetries of the Kahler-Einstein metric via a simple inequality; the so-called Miyaoka-Yau inequality. Furthermore he showed that in the case of equality, the universal cover is the ball. Later Tsuji established the MY inequality for smooth minimal models of general type by constructing singular Kahler-Einstein metrics. The singularity of these metrics are usually a major obstacle towards uniformization, a problem that has not yet been resolved via analytic methods. In a joint project with Greb, Kebekus and Peternell we take a different approach, via Hermitian-Yang-Mills theory and Simpson’s groundbreaking work on complex variation of Hodge structures, and we prove the MY inequality for minimal models of general type and establish a uniformization for their canonical models.

Johannes Nicaise (Imperial College). Poles of maximal order of Igusa zeta functions. Friday 27th Nov, 1.30-2.30pm. Huxley 139.

Abstract: Igusa’s p-adic zeta function Z(s) attached to a polynomial f in N variables is a meromorphic function on the complex plane that encodes the numbers of solutions of the equation f=0 modulo powers of a prime p. It is expressed as a p-adic integral, and Igusa proved that it is rational in p^{-s} using resolution of singularities and the change of variables formula. From this computation it is immediately clear that the order of a pole of Z(s) is at most N, the number of variables in f. In 1999, Wim Veys conjectured that the only possible pole of order N of the so-called topological zeta function of f is minus the log canonical threshold of f. I will explain a proof of this conjecture, which also applies to the p-adic and motivic zeta functions. The proof is inspired by non-archimedean geometry and Mirror Symmetry, but the main technique that is used is the Minimal Model program in birational geometry. This talk is based on joint work with Chenyang Xu.

Spyros Alexakis (University of Toronto). The structure of integral invariants, in Riemannian and complex
geometries.
Friday 4th Dec, 1.30-2.30pm. Huxley 139.

Abstract: I will present older and newer work, on the following question:
Assume one has a natural geometric functional, which acts on metrics by forming scalar intrinsic invariants, and then integrating these invariants over a manifold. Assume that the resulting integral presents an invariance under certain deformations of the underlying metric. Then what information can one deduce on the integrand?
I will present answers to this question in two cases: When the class of metrics are Riemannian, or Kahler, and the invariance of the integral is under conformal, or Kahler deformations of the underlying metric. I will discuss some applications of this to renormalized volume and generalized Chern-Gauss Bonnet integrands in the Riemannian setting, and to the structure of the Bergman kernel of ample line bundles in the complex setting.

John Ottem (University of Cambridge). Positive cotangent bundles and their Chern classes. Friday 11th Dec, 1.30-2.30pm. Huxley 139.

Abstract: A central theme in algebraic geometry is that geometric properties of an algebraic variety are reflected in the positivity of its (co)tangent bundle. For instance, if the tangent bundle is ample, a result of Mori says that the variety is a projective space. On the other hand, if the cotangent bundle is ample, the variety contains no rational or elliptic curves at all. In this talk I’ll present a few results on varieties with positive (ample, nef, or pseudoeffective) cotangent bundle, and discuss the positivity of higher degree Chern classes of such varieties.

Luca Tasin (University of Bonn). On some unirational hypersurfaces that are not stably rational. Friday 18th Dec, 1.30-2.30pm. Huxley 139.

Abstract: Let n=6,7,8 or 9; in this talk I will show that a very general hypersurface of degree at least 4 in P^n is not stably rational (a fortiori not rational), whereas it is known that general quartics are unirational. This gives new counter-examples to the Lüroth problem. Moreover, it provides the first examples of hypersurfaces that are known to be unirational but not stably rational. Our approach is based on a new Chow theoretic method introduced by Voisin, which has recently allowed important advances in the subject. In particular, our result improves in low dimension a recent theorem of Totaro on the stable rationality of hypersurfaces. In the first part of the seminar I will introduce and explain this method.
The talk is based on a joint work with S. Schreieder.