impanga 2022/23

I will link convex geometry and homological algebra by associating a convex cone to an abelian category; a fan to a bounded heart in a triangulated category and a multifan to the triangulated category itself. These constructions are motivated by toric geometry; g-vector fans in representation theory; and stability conditions: a doubled-up convex-geometric construction gives Bridgeland’s stability space. (Joint work in progress with Nathan Broomhead, David Pauksztello, Jon Woolf.)

I will introduce the notion of a regular integrable connection on a smooth log scheme over $\mathbf{C}$ and construct an equivalence between the category of such connections and the category of integrable connections on its analytification, compatible with de Rham cohomology. This extends the work of Deligne (when the log structure is trivial), and combined with the work of Ogus yields a topological description of the category of regular connections in terms of certain constructible sheaves on the Kato--Nakayama space. The key ingredients are the notion of a canonical extension in this context and the existence of good compactifications of log schemes obtained recently by Włodarczyk.

In this talk, I will first recall the construction of Lagrangian fibrations by Prym varieties starting from a K3 surface with a non-symplectic involution. Then I will discuss a criterion to ensure that the normalization of such a fibration is an irreducible symplectic variety. This is joint work in progress with E. Brakkee, A. Grossi, L. Pertusi, G. Saccà and A. Viktorova.

We show that the cohomology of moduli spaces of Higgs bundles decomposes in elementary summands depending on the topology of the symplectic singularities on a (fixed!) master object and/or the combinatorics of certain posets and lattice polytopes. This is based on a joint work with Luca Migliorini and Roberto Pagaria.

Given a singularity with a crepant resolution, an associated symmetry of the derived category of coherent sheaves on the resolution may often be constructed, with applications to homological mirror symmetry and enumerative geometry. I relate such constructions to the derived category of a smooth ambient space for the given singularity. This builds on previous results with Segal, and is inspired by work of Bodzenta–Bondal.

The Tate–Oort group is a parametrised family of group schemes over a base scheme of mixed characteristic, providing a good reduction of the cyclic groups $\mathbf{Z}/p$ or $\mu_p$ in characteristic zero to its three characteristic $p$ degenerations.

I start with a colloquial treatment of $\mathrm{TO}_p$ and its representations. Our applications to Godeaux surfaces or Godeaux Calabi–Yau 3-folds contain computer algebra that can be seen as a slide-show, but not lectured in real time. If you want to study it, the code is available from my webpage + TOp.

The Cauchy–Liouville–Mirimanoff polynomials underlie the definition of $\mathrm{TO}_p$, and their arithmetic and Galois theory is a deep source of interesting conjectures that go back to A. L. Cauchy and J. Liouville [C. R. Acad. Sci. 9 (1839), 359].

The goal of this talk is to introduce the tempered fundamental group of an analytic space over a non-archimedean field, and to present its application to anabelian geometric questions. Introduced by Yves André, this group captures both the good topological nature of Berkovich spaces, but also all their finite étale behaviours. Its profinite completion is the usual profinite étale fundamental group, which is not so interesting over algebraically closed fields. However, we will see that the tempered group has interesting anabelian features for curves over algebraically closed fields. For a large class of curves this group captures a canonical subgraph called the skeleton, intimately related to the special fiber of the stable model, and in some special cases it even captures all the topology of the space.

In this talk, we consider the homogeneous coordinate rings $A(Y_{n,d}) \cong \mathbb{K}[\Omega_{n,d}]$ of monomial projections $Y_{n,d}$ of Veronese varieties, i. e. varieties parameterized by subsets $\Omega_{n,d}$ of monomials of degree $d$ in $n+1$ variable. Our goal is to study when $\mathbb{K}[\Omega_{n,d}]$ is a quadratic algebra and, if so, when $\mathbb{K}[\Omega_{n,d}]$ is Koszul or G-quadratic. We present new results and conjectures for two families of monomial projections: (1) $\Omega_{n,d}$ contains all monomials supported in at most $s$ variables, and (2) $\Omega_{n,d}$ is a set of monomial invariants of a finite diagonal abelian group $G \subset GL(n+1,\mathbb{K})$ of order $d$.

An Enriques surface is the quotient of a K3 surface by a fixed point-free involution. There is a beautiful conjectural formula predicted by Klemm and Marino through string theory which link counts of elliptic curves on the Enriques surface to an automorphic form on the moduli space of Enriques surface constructed famously by Borcherds. I will explain recent work in progress towards proving this formula.

The (weighted) Hurwitz numbers are one of the primary objects in the enumerative geometry. From the integrable hierarchies point of view one can approach them by studying the associated generating series that is encoded by Schur symmetric functions. The latter has a famous one-parameter deformation called Jack polynomials. It seems that when we replace the Schur functions by the Jack functions in this generating series, the positivity and integrality is mysteriously not affected.

We prove that this $b$-deformed generating series of (weighted) Hurwitz numbers can be understood as the generating series of some geometric objects that we construct. In particular we give an explicit interpretation of the conjectured positivity. These objects can be thought of as generalized branched coverings of the sphere, since the covering space is no longer required to be orientable. We describe our construction and explain how it provides topological expansion for various matrix models studied previously in the literature. Based on the work with Bonzom and Chapuy.

I will explain how differential operators coming from algebraic geometry produce interesting $p$-adic numbers. In a recent work with Frits Beukers we give examples of families of Calabi-Yau hypersurfaces in $n$ dimensions, for which one observes $p$-adic zeta values $\zeta_p(k)$ for $1 < k < n$. Appearance of $p$-adic zeta values for differential operators of Calabi-Yau type was conjectured by Candelas, de la Ossa and van Straten.

Singular contact varieties constitute a generalization of the notion of a complex contact manifold, which is well-known and studied in literature in the context of LeBrun-Salamon conjecture. There exists a classical construction that associates to a given contact manifold a symplectic one (the symplectization). Having that in mind, one can define singular contact varieties to correspond to (singular) symplectic varieties via an analogous construction. Then, one can study their properties in parallel with better-known symplectic varieties and (smooth) contact manifolds. In particular, one can show that singular contact varieties have negative Kodaira dimension, crepant resolutions of singularities produce classical contact manifolds and they admit stratification of the singular locus a la Kaledin. Moreover, there is a criterion that decides whether a finite quotient of a contact variety is another contact variety, that allows to construct new examples from known contact manifolds. All those gadgets allow in particular to provide a link between two distinct families of contact manifolds.

The algebraic structure of a complex algebraic variety cannot be determined by the underlying structure of a complex manifold. Perhaps the most-known example is due to Serre: he is credited with a non-affine algebraic structure on a manifold analytically isomorphic to $\mathbf{C}^* \times \mathbf{C}^*$. In some situations, Tate--Shafarevich twists of Lagrangian fibrations produce a countable number of non-isomorphic algebraic structures on the same complex manifold.

In our joint work in progress with Anna Abasheva, we construct a countable number of non-isomorphic algebraic structures on log-CY surfaces. Our construction is similar to the gluing construction of K3 surfaces from two log-CY surfaces developed by Koike and Uehara, and involves an equivalence relation of degree zero line bundles on elliptic curves, which we call "the analytic cobordism", and which has seemingly not been noticed before and is probably of interest on its own.

Dubrovin's conjecture, arising from physics, relates two different areas of algebraic geometry, quantum cohomology and derived categories. It states that, for any Fano variety, the derived category admits a full exceptional decomposition if and only if the quantum cohomology is generically semisimple. Test cases to show that the conjecture holds are homogeneous projective spaces: for some of them, among which (co)adjoint varieties, the conjecture has been shown to be true. In this talk we will focus on the cohomology side of the conjecture and we will study hyperplane sections of (co)adjoint varieties. These varieties, even though they are not homogeneous, admit an action of a torus with a finite number of fixed points. The aim of the talk is to show how to use this action to compute the cohomology and the quantum cohomology of such varieties; if time permits, we will summarize what is known with respect to Dubrovin’s conjecture in this context. This is a joint work with N. Perrin.

We study étale morphisms of adic spaces $f: Y \to X$. For a point $y \in Y$ with image $x=f(y)$ the corresponding residue field extension $k(y)/k(x)$ is a finite separable extension of valued fields. If this extension is wildly ramified, we say that $f$ is wild at $y$. In a joint project with Michael Temkin we show that if $f$ is wild at some point $y \in Y$, it has to be wild at a divisorial point, i.e. a point corresponding to an irreducible component of the special fiber of some formal model. A consequence of our result is that we can test tameness on curves. This was known previously thanks to an article by Kerz and Schmidt but using resolution of singularities.

I will show how to generalize various well-known theorems for Higgs sheaves on smooth projective varieties to normal projective varieties. I will start with a survey of some known results for singular varieties in the characteristic zero case due to Greb, Kebekus, Peternell, Taji et al. Much of the talk will be devoted to positive characteristic case that is easier to study due to existence of the Frobenius morphism. I will show various applications of these results including some variants of Simpson's correspondence on normal projective varieties.