Broken lines for vector fields

• Mark Gross

Room: Aug 29, 30: Kjølv Egelands hus KE A-205; Aug 31 Kjell Arholmshus  KA U-138 Broken lines, in conjunction with scattering diagrams, have been a very powerful tool for constructing regular functions on schemes obtained by gluing standard charts  using the data of scattering diagrams. One could hope that broken lines could also be used to build other geometric objects on the  glued schemes, e.g., vector fields. At first glance, this seems to give very poorly behaved objects. However, new insights have shown that this seemingly poor behaviour in fact uncovers some very intricate enumerative structures. This is joint work with D. Pomerleano and B. Siebert.

Unobstructedness of Gorenstein Terminal Singularities

• Matej Filip

Room: Aug 29, 30: Kjølv Egelands hus KE A-205; Aug 31 Kjell Arholmshus  KA U-138 We prove that Gorenstein terminal toric fourfold singularities are unobstructed. We also establish vanishing results for the obstruction space for certain Gorenstein toric Fano varieties. Finally, we discuss connections with mirror symmetry and Laurent polynomials.

Smoothing Gorenstein toric affine 3-folds

• Alessio Corti

Room: Aug 29, 30: Kjølv Egelands hus KE A-205; Aug 31 Kjell Arholmshus  KA U-138 I discuss the structure of the deformation space of Gorenstein toric affine 3-folds following work of Filip. This is an introduction to the talk by Graefnitz.

Mutations, Log Resolutions and Smoothings

• Tim Gräfnitz

Room: Aug 29, 30: Kjølv Egelands hus KE A-205; Aug 31 Kjell Arholmshus  KA U-138 I explain how Fano smoothings of Gorenstein toric varieties can be constructed using logarithmic geometry. Maximally mutable Laurent polynomials induce singular log structures on reducible toroidal crossing spaces. Conjecturally, such a space admits a log crepant resolution if and only if the Laurent polynomial is rigid. For three-dimensional examples, I construct these log resolutions explicitly using sequences of blow-ups and flops. Once a log resolution has been obtained, a general smoothing theorem of Felten-Filip-Ruddat applies. The resulting construction also provides a framework for studying mirror symmetry and tropical geometry. This is joint work with Alessio Corti and Helge Ruddat.

K-moduli of Fano family 3.11 and degenerations of elliptic quartics

• Robert Śmiech

Room: Aug 29, 30: Kjølv Egelands hus KE A-205; Aug 31 Kjell Arholmshus  KA U-138 I will report on joint work in progress with Ivan Cheltsov, Anne-Sophie Kaloghiros, and Junyan Zhao, in which we construct the $K$-moduli space of Fano varieties obtained by first blowing up a point in $\mathbb{P}^3$ and then blowing up the strict transform of a $(2,2)$ complete intersection passing through that point. Our construction is based on the moduli continuity method, enhanced by the study of possible degenerations of elliptic quartics.

Towards a Thomas-Yau-Joyce result on K3 surfaces and products

• Jacopo Stoppa

Room: Aug 29, 30: Kjølv Egelands hus KE A-205; Aug 31 Kjell Arholmshus  KA U-138 Thomas, Yau and Joyce made the remarkable conjecture that the existence of a special Lagrangian submanifold representing a fixed Fukaya equivalence class on a Calabi-Yau manifold should be equivalent to the stability of that class with respect to a suitable Bridgeland stability condition on the Fukaya category. I will describe work in progress towards proving the implication from Bridgeland stability to the existence of a special Lagrangian for some nontrivial examples, namely certain Lagrangian spheres in suitable K3 surfaces or products of such K3 surfaces with an elliptic curve. The proposed approach uses mirror symmetry for K3 surfaces rather than more standard but difficult analytic methods such as the Lagrangian mean curvature flow or gluing, but I will discuss why it should be (conjecturally) equivalent to a gluing construction.

Enriques surface fibrations and the integral Hodge conjecture

• John Christian Ottem

Room: Aug 29, 30: Kjølv Egelands hus KE A-205; Aug 31 Kjell Arholmshus  KA U-138 I will explain a geometric construction of an Enriques surface fibration over $\mathbb{P}^1$ of even index. This answers a question of Colliot-Thélène and Voisin, and provides new counterexamples to the Integral Hodge conjecture. This is joint work with Fumiaki Suzuki.

Classification of terminal Fano 3-fold hypersurfaces in rank 2 toric varieties

• Thamarai Valli Venkatachalam

Room: Aug 29, 30: Kjølv Egelands hus KE A-205; Aug 31 Kjell Arholmshus  KA U-138 Fano 3-folds with terminal singularities are fundamental building blocks in birational geometry, but their classification remains an important open problem. In this talk, I will discuss my work on the classification of terminal Fano 3-fold hypersurfaces in rank 2 toric varieties, viewed as GIT quotients. By exploiting this toric structure and the Minimal Model Program, I provide a complete list of families in my setup and relate them to existing results on terminal Fano 3-folds.

The Logarithmic Derived Category

• Xianyu Hu

Room: Aug 29, 30: Kjølv Egelands hus KE A-205; Aug 31 Kjell Arholmshus  KA U-138 Let $X \to S$ be a proper semistable degeneration. For the singular central fibre, the usual choices $D^b_{\mathrm{coh}}(X_0)$ and $\operatorname{Perf}(X_0)$ fail, in complementary ways, to be simultaneously smooth and proper. Inspired by the fact that the natural log structures make the degeneration log smooth, we define logarithmic coherent sheaves and the associated category of log perfect complexes. This category is proper over $S$ and satisfies a logarithmic version of categorical smoothness. I will outline the motivation and construction, using a nodal degeneration as a guiding example. This is joint work with Hannah Dell, Patrick Kennedy-Hunt, Kabeer Manali Rahul, and Maximilian Schimpf.

Convex Fujita numbers

• Alex Küronya

Room: Aug 29, 30: Kjølv Egelands hus KE A-205; Aug 31 Kjell Arholmshus  KA U-138 This is an account of an ongoing research project with Yusuf Mustopa and Jakob Stix. We study effective global generation properties of adjoint line bundles, nevertheless in a somewhat more extended setting than considered traditionally. Originally motivated by Fujita's conjecture on global generation, we investigate when adjoint line bundles coming from sums of ample ones become globally generated. It turns out that there is a lot to say  about the resulting new invariants dubbed convex Fujita numbers. I will discuss a variety of cases when the  answer is known, and directions for future research.

Rank 2 sheaves on P^3, Quot schemes and wall-crossing

• Andrea Ricolfi

Room: Aug 29, 30: Kjølv Egelands hus KE A-205; Aug 31 Kjell Arholmshus  KA U-138 Moduli spaces of stable sheaves on threefolds are notoriously difficult to understand and describe geometrically, especially in rank $>1$. In this talk, I will give an explicit description, in terms of Quot schemes of reflexive sheaves, of an infinite family of moduli spaces of rank-$2$ sheaves on $\mathbb{P}^3$. Several properties of these families will be discussed. One of the key tools used to establish this description at the scheme-theoretic level is DT/PT wall-crossing in the framework of Bridgeland stability. This is joint work in progress with Charles Almeida, Nicolò Bignami, and Marcos Jardim.

The 4D ADHM quiver and framed sheaves on (P^1)^4

• Jørgen Vold Rennemo

Room: Aug 29, 30: Kjølv Egelands hus KE A-205; Aug 31 Kjell Arholmshus  KA U-138 In recent work, Arbesfeld, Kool, and Lim, following work in physics by Nekrasov, studied the moduli space of representations of a quiver with relations called the 4D ADHM quiver. In particular, they placed Nekrasov’s partition function on a rigorous algebro-geometric footing using the fourfold counting invariants of Oh–Thomas. In the same paper, they conjectured that the moduli space of stable framed representations of this quiver is isomorphic to a moduli space of two-dimensional framed sheaves on $(\mathbb{P}^1)^4$. This generalizes a result going back to Barth, which identifies representations of the ADHM quiver with torsion-free framed sheaves on $\mathbb{P}^2$. I will explain these results and present a proof of the conjecture.

Classification of simple 3-fold flops

• Gavin Brown

Room: Aug 29, 30: Kjølv Egelands hus KE A-205; Aug 31 Kjell Arholmshus  KA U-138 Simple flops (the contraction of a single irreducible canonically trivial rational curve on a smooth 3-fold) are in bijection with their Donovan-Wemyss contraction algebras, that is particular noncommutative deformation algebras. We give an intrinsic characterisation of such algebras and classify them, and explain how to recover the flopping curves directly. Joint with Michael Wemyss.

The Heisenberg algebra and Hochschild homology

• Timothy Logvinenko

Room: Aug 29, 30: Kjølv Egelands hus KE A-205; Aug 31 Kjell Arholmshus  KA U-138 In this talk, I will explain how for any noncommutative smooth and proper variety X we can construct three actions of the Heisenberg algebra on the total Hochschild homology of its symmetric quotient stacks $[X^n/S_n]$. One is defined algebraically using the orbifold decomposition of Hochschild homology. One decategorifies the 2-categorical Heisenberg action of arXiv:2105.13334. One is defined representation theoretically via explicit operators intrinsic to the symmetric quotient stacks. It turns out that these three actions coincide, and thus give alternative descriptions of one natural action. For ordinary commutative varities, we give a fourth, geometrical description by correspondences similar to those used by Grojnowski and Nakajima for the Hilbert schemes of points on surfaces.

Quantum Area

• Duco van Straten

Room: Aug 29, 30: Kjølv Egelands hus KE A-205; Aug 31 Kjell Arholmshus  KA U-138 I will report on work in progress joint with Mauricio Garay from Paris. Near a critical point of a Hamiltonian function on the symplectic $p-q$ plane, the level set $H(p,q)=c$ forms a simple closed loop that encloses an area and has the dimension of an action. In quantum theory the variables no longer commute: $pq-qp=h$. We introduce the notion of quantised area, which is a formal series in $h$. In examples we find strong integrality properties of the appearing coefficients that suggest looking for an enumerative meaning.

Normed matrix factorizations

• Jake Solomon

Room: Aug 29, 30: Kjølv Egelands hus KE A-205; Aug 31 Kjell Arholmshus  KA U-138 I will discuss a version of homological mirror symmetry for Fano manifolds that incorporates the non-Archimedean norms on the morphism spaces of the Fukaya category. A Fano manifold corresponds to a Banach Landau Ginzburg model, to which is associated a category of normed matrix factorizations. This correspondence gives rise to conjectural formulas for the singular cohomology with rational coefficients, the Floer cohomology over the Novikov ring, and the open Gromov-Witten invariants of the real locus of a toric Fano manifold that is relatively spin. Joint work with M. Sela.