Summer Term 2024

Paul Feehan (Rutgers University). Morse theory on moduli spaces of pairs and the Bogomolov-Miyaoka-Yau inequality. Friday 3rd May, 1:30-2:30pm. Huxley 140 (NOTE this is a different room to last term).

Abstract: We describe an approach to Bialynicki-Birula theory for holomorphic C^∗ actions on complex analytic spaces and Morse-Bott theory for Hamiltonian functions for the induced circle actions. A key principle is that positivity of a suitably defined “virtual Morse-Bott index” at a critical point of the Hamiltonian function implies that the critical point cannot be a local minimum even when it is a singular point in the moduli space. Inspired by Hitchin’s 1987 study of the moduli space of Higgs monopoles over Riemann surfaces, we apply our method in the context of the moduli space of non-Abelian monopoles or, equivalently, stable holomorphic pairs over a closed, complex, Kaehler surface. We use the Hirzebruch-Riemann-Roch Theorem to compute virtual Morse-Bott indices of all critical strata (Seiberg-Witten moduli subspaces) and show that these indices are positive in a setting motivated by a conjecture that all closed, smooth four-manifolds of Seiberg-Witten simple type (including symplectic four-manifolds) obey the Bogomolov-Miyaoka-Yau inequality.


Paul Norbury (University of Melbourne). Measures on the moduli space of curves and super volumes. Friday 10th May, 1:30-2:30pm. Huxley 140.

Abstract: I will define a family of finite measures on the moduli space of smooth curves with marked points. The measures are defined via a construction analogous to that of the Weil-Petersson metric using the extra data of a spin structure. In fact, the measures arise naturally out of the super Weil-Petersson metric defined over the moduli space of super curves. The total measure can be identified with the volume of the moduli space of super curves. It can be calculated in many examples, and conjecturally satisfies a recursion analogous to Mirzakhani’s recursion relations between Weil-Petersson volumes of moduli spaces of hyperbolic surfaces. This conjecture has been verified in many cases, including the so-called Neveu-Schwarz case where it coincides with the recursion of Stanford and Witten. The general case produces deformations of the Neveu-Schwarz volume polynomials, satisfying the same Mirzakhani-like recursion relations.


Jonny Evans (University of Lancaster). Tropical methods for stable Horikawa surfaces. Friday 17th May, 1:30-2:30pm. Huxley 140.

Abstract: The moduli space of surfaces of general type is a vast and largely uncharted mess. It admits a compactification, the Kollar–Shepherd-Barron–Alexeev compactification, whose points correspond to “stable” singular surfaces. Our understanding of even the coarse features of this compactification is even more fragmentary.

Some corners of the moduli space of smooth surfaces are well-understood, for example the Horikawa surfaces (on or close to the Noether line) are known to be branched double covers of certain rational surfaces. I will explain how one can use this fact, together with the toric and almost toric degenerations of rational surfaces, to get some control over the KSBA-boundary of the moduli space of Horikawa surfaces. For example, one can show that the component of surfaces with K^2 = 2 and p_g = 3 has precisely three KSBA-boundary strata corresponding to normal surfaces with at worst quotient singularities.

Abigail Ward (University of Cambridge). Weinstein manifolds without arboreal skeleta. Friday 24th May, 1:30-2:30pm. Huxley 140.

Abstract: The interplay between the topological or homotopy-invariant properties of a symplectic manifold X and the set of possible immersed or embedded Lagrangian submanifolds of X is rich and mostly mysterious. In 2020, D. Alvarez-Gavela, Y. Eliashberg, and D. Nadler proved that any Weinstein manifold (e.g. an affine variety) admitting a Lagrangian plane field retracts onto a Lagrangian submanifold with arboreal singularities (a certain class of singularities which can be described combinatorially). I will discuss work in progress with D. Alvarez-Gavela and T. Large investigating the other direction, in which we prove a partial converse to the AGEN result and show that most Weinstein manifolds do not admit such skeleta. This suggests that the Floer-theoretic invariants of some well-known open symplectic manifolds may be more complicated than expected..

Sam Johnston (Imperial College London).  Intrinsic mirror symmetry via Gromov-Witten theory of root stacks. Friday 31st May, 1:30-2:30pm. Huxley 140.

Abstract: Logarithmic Gromov-Witten theory and orbifold Gromov-Witten theory offer two distinct paths to probing the enumerative geometry of a normal crossing pair (X,D), and possess complementary strengths and weaknesses. Recent work of Battistella-Nabijou-Ranganathan demonstrates how these two theories are related after a sequence of strata blowups of (X,D). By using known relations in orbifold Gromov-Witten theory, this can be used to produce relations depending on discrete data of maps to (X,D) in the log Gromov-Witten theory of sufficiently blown-up targets. In the setting where (X,D) is log Calabi-Yau, we show how to use relations in the log Gromov-Witten theory of the blow-up to deduce relations in the log Gromov-Witten theory of (X,D) which give another proof of associativity of the intrinsic mirror algebra of Gross and Siebert, as well as a proof of the weak Frobenius structure property conjectured by Gross and Siebert.

Iacopo Brivio (Harvard University). Anti-Iitaka theorem for tame fibrations in characteristic p. Friday 7th June, 1:30-2:30pm. Huxley 140.

Abstract: Given a fibration $f\colon X\to Y$ of complex projective varieties with general fiber $F$, the Iitaka conjecture predicts the inequality $\kappa(K_X)\geq \kappa(K_F)+\kappa(K_Y)$. Recently Chang has shown that if one further assumes that the stable base locus $\mathbb{B}(-K_X)$ is vertical over $Y$, then we have a similar inequality for the anticanonical divisor $\kappa(-K_X)\leq \kappa(-K_F)+\kappa(-K_Y)$ . Both Iitaka’s conjecture and Chang’s theorem are known to fail in positive characteristic, but they are expected to hold when the action of the Frobenius on $F$ is sufficiently well behaved. In this talk I will show that this is indeed the case for Chang’s theorem. This is based on joint work with M. Benozzo and C.-K. Chang.

Yuji Odaka (Kyoto University). Compact algebraic moduli of Calabi-Yau cones, Sasaki-Einstein manifolds and affine Calabi-Yau varieties. Friday 14th June, 1:30-2:30pm. Huxley 341 (NOTE this is not the usual room).

Abstract: We construct proper moduli algebraic space of Sasaki-Einstein spaces, which in particular generalizes the compact moduli of (KE) Fano varieties with a different proof. Further we discuss certain general construction of affine Calabi-Yau varieties with (hopefully) Ricci-flat K\”ahler metrics of Euclidean volume growths. These use a general new method of canonical modifications of various limiting algebraic objects, along toric varieties or Novikov type rings. Partly available at arXiv:2405.07939, arXiv:2406.02489.

Ana Caraiani (Imperial College London). On the cohomology of Shimura varieties with torsion coefficients. Friday 21st June, 1:30-2:30pm. Huxley 140.

Abstract: Shimura varieties are certain highly symmetric algebraic varieties that generalise modular curves and that play an important role in the Langlands program. In this talk, I will survey a recent class of results about the vanishing of cohomology of Shimura varieties with torsion coefficients. I will focus on the most modern approach to proving these kinds of results, which involves a connection to the geometrisation of local Langlands over p-adic fields.

CANCELED. Friday 28th June, 10:30-11:30am. Huxley 140.



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