We study Hilbert schemes of points on a smooth projective Calabi-Yau 4-fold X and define DT4 invariants by integrating the Euler class of a tautological vector bundle against the virtual class. We conjecture a formula for their generating series, which we prove in certain cases when L corresponds to a smooth divisor on X. A parallel equivariant conjecture for toric Calabi-Yau 4-folds is proposed. This conjecture is proved for smooth toric divisors and verified for more general toric divisors in many examples. Combining the equivariant conjecture with a vertex calculation, we find explicit positive rational weights, which can be assigned to solid partitions. The weighted generating function of solid partitions is given by exp(M(q) − 1), where M(q) denotes the MacMahon function. This is joint work with Martijn Kool.

# Past Algebraic Geometry Seminar

We develop a structure theory for del Pezzo surfaces that are regular but geometrically non-normal, based on work of Reid, but now independence on the p-degree of the ground field. This leads to existence results, as well as non-existence results for ground fields of p-degree one. In turn, we settle questions arising from Koll'ar's analysis on the structure of Mori fiber spaces in dimension three. This is joint work with Andrea Fanelli.

Following Grothendieck's vision that many cohomological invariants of of an algebraic variety should be captured by a common motive, Voevodsky introduced a triangulated category of mixed motives which partially realises this idea. After describing this category, I will explain how to define the motive of certain algebraic stacks in this context. I will then report on joint work in progress with Victoria Hoskins, in which we study the motive of the moduli stack of vector bundles on a smooth projective curve and show that this motive can be described in terms of the motive of this curve and its symmetric powers.

Starting from the seminal paper of Caporaso-Harris-Mazur, it has been proved that if Lang's Conjecture holds in arbitrary dimension, then it implies a uniform bound for the number of rational points in a curve of general type and analogue results in higher dimensions. In joint work with Kenny Ascher we prove analogue statements for integral points (or more specifically stably-integral points) on curves of log general type and we extend these to higher dimensions. The techniques rely on very recent developments in the theory of moduli spaces for stable pairs, a higher dimensional analogue of pointed stable curves.

If time permits we will discuss how very interesting problems arise in dimension 2 that are related to the geometry of the log-cotangent bundle.

Let $\mathbb K$ be a field, and $\mathcal M$ be the “projective linear" moduli stack of objects in a suitable $\mathbb K$-linear abelian category $\mathcal A$ (such as the coherent sheaves coh($X$) on a smooth projective $\mathbb K$-scheme $X$) or triangulated category $\mathcal T$ (such as the derived category $D^b$coh($X$)). I will explain how to define a Lie bracket [ , ] on the homology $H_*({\mathcal M})$ (with a nonstandard grading), making $H_*({\mathcal M})$ into a graded Lie algebra. This is a new variation on the idea of Ringel-Hall algebra.

There is also a differential-geometric version of this: if $X$ is a compact manifold with a geometric structure giving instanton-type equations (e.g. oriented Riemannian 4-manifold, $G_2$-manifold, Spin(7)-manifold) then we can define Lie brackets both on the homology of the moduli spaces of all $U(n)$ or $SU(n)$ connections on $X$ for all $n$, and on the homology of the moduli spaces of instanton $U(n)$ or $SU(n)$ connections on $X$ for all $n$.

All this is (at least conjecturally) related to enumerative invariants, virtual cycles, and wall-crossing formulae under change of stability condition.

Several important classes of invariants in algebraic and differential geometry — (higher rank) Donaldson invariants of 4-manifolds (in particular with $b^2_+=1$), Mochizuki invariants counting semistable coherent sheaves on surfaces, Donaldson-Thomas type invariants for Fano 3-folds and CY 4-folds — are defined by forming virtual classes for moduli spaces of “semistable” objects, and integrating some cohomology classes over them. The virtual classes live in the homology of the “projective linear" moduli stack. Yuuji Tanaka and I are working on a way to define virtual classes counting strictly semistables, as well as just stables / stable pairs.

I conjecture that in all these theories, the virtual classes transform under change of stability condition by a universal wall-crossing formula (from my previous work on motivic invariants) in the Lie algebra $(H_*({\mathcal M}), [ , ])$.

The specialization question for rationality is the following one: assume that very general fibers of a flat proper morphism are rational, does it imply that all fibers are rational? I will talk about recent solution of this question in characteristic zero due to myself and Nicaise, and Kontsevich-Tschinkel. The method relies on a construction of various specialization morphisms for the Grothendieck ring of varieties (stable rationality) and the Burnside ring of varieties (rationality), which in turn rely on the Weak Factorization and Semi-stable Reduction Theorems.

In mirror symmetry, the prepotential on the Kahler side has an expansion, the constant term of which is a rational multiple of zeta(3)/(2 pi i)^3 after an integral symplectic transformation. In this talk I will explain the connection between this constant term and the period of a mixed Hodge-Tate structure constructed from the limit MHS at large complex structure limit on the complex side. From Ayoub’s works on nearby cycle functor, there exists an object of Voevodsky’s category of mixed motives such that the mixed Hodge-Tate structure is expected to be a direct summand of the third cohomology of its Hodge realisation. I will present the connections between this constant term and conjecture about how mixed Tate motives sit inside Voevodsky’s category, which will also provide a motivic interpretation to the occurrence of zeta(3) in prepotential.

I will introduce a cohomology theory which combines topological and algebraic concepts. Interpretations of certain cohomology groups will be given. We also generalise the construction of the second Stiefel-Whitney class of a line bundle. As I will explain in my talk, the refined Stiefel-Whitney class of the canonical bundle on certain moduli stacks provides an obstruction for the construction of cohomological Hall algebras.

We investigate Néron models of Jacobians of singular curves over strictly Henselian discretely valued fields, and their behaviour under tame base change. For a semiabelian variety, this behaviour is governed by a finite sequence of (a priori) real numbers between 0 and 1, called "jumps". The jumps are conjectured to be rational, which is known in some cases. The purpose of this paper is to prove this conjecture in the case where the semiabelian variety is the Jacobian of a geometrically integral curve with a push-out singularity. Along the way, we prove the conjecture for algebraic tori which are induced along finite separable extensions, and generalize Raynaud's description of the identity component of the Néron model of the Jacobian of a smooth curve (in terms of the Picard functor of a proper, flat, and regular model) to our situation. The main technical result of this paper is that the exact sequence which decomposes the Jacobian of one of our singular curves into its toric and Abelian parts extends to an exact sequence of Néron models. Previously, only split semiabelian varieties were known to have this property.