#### ALGEBRAIC GEOMETRY IN AUCKLAND

**16-20 December, 2019**

#### Schedule

All talks are in room MLT3. The address is 36 Princes Street. Go through the main entrance and take the lift to Floor 1. See the map

Wednesday is a free day.

On Thursday evening is the conference dinner, held at the Fale Pasifika between 6:30-9:00pm. See the map

**ABSTRACTS**

**Degenerations of K3 surfaces and 24 points on the sphere ** **(Valery Alexeev, University of Georgia) **I will discuss Kulikov and stable degenerations of K3 surfaces and describe explicit, geometric compactifications of their moduli spaces in several interesting cases. Based on joint work with Philip Engel and Alan Thompson.

**The Nielsen realisation problem for K3 surfaces (David Baraglia, The University of Adelaide)**

The Nielsen realisation problem, solved by Kerckhoff, says that any finite subgroup of the mapping class group of a smooth complex projective curve of negative Euler characteristic can be lifted to an action on the curve by diffeomorphisms. We consider the analogous problem for K3 surfaces. We show that there exists a finite subgroup of the mapping class group which can not be lifted to a corresponding action by diffeomorphisms. Our methods also show that there exists a topological family of K3 surfaces over the 2-sphere which can not be smoothed. Our proof of these results is based on Seiberg-Witten theory and the global Torelli theorem for K3 surfaces. This is joint work with Hokuto Konno.

**On several problems in complex geometry (Fedor Bogomolov, New York University)**

I will discuss several problems and constructions in complex geometry which I was confronted with in my research.

**Dense entire curves in rationally connected and some fibered Calabi-Yau manifolds (****Frederic Campana, Université de Lorraine)**

We (joint with J. Winkelmann) show among other things that any rationally connected manifold contains dense entire curves. Which implies the same property for Calabi-Yau (and Hyperk\\”ahler) manifolds fibered in Abelian varieties over a smooth rationally connected base. In dimension 3, one may remove the hypothesis over the base.

** Log canonical thresholds of surfaces of general type and threefold geography (Jungkai Chen, National Taiwan University)**

Log canonical thresholds (lct) plays very important role in the studies of Fano varieties and log Calabi-Yau varieties. In this talk, we are going to present some recent development of lct of surfaces of general type and its application to the investigation of Noether inequality of threefolds. This talk is base on the work jointly with Meng Chen, Chen Jiang and Janos Kollar.

**Stringy Kaehler moduli, mutation and monodromy (Will Donovan, Tsinghua University) **

The derived symmetries associated to a 3-fold admitting an Atiyah flop may be organised into an action of the fundamental group of a sphere with three punctures, thought of as a stringy Kaehler moduli space. I extend this to general flops of irreducible curves on 3-folds in joint work with M Wemyss. This uses a novel helix of sheaves supported on the flopping curve.

**Polynomial automorphisms (Eric Edo, University of New Caledonia)**

We present the group of polynomial automorphisms, ie automorphisms of the algebra of polynomials. We consider both algebraic (sub-groups, generators) and topological (limits, closure) aspects. The main results obtained over the last twenty years are discussed. We put forward the questions and conjectures that represent a challenge for future developments.

**On the uniform K-stability for some asymptotically log del Pezzo surfaces (Kento Fujita, Osaka University)**

We calculate the delta-invariant for some asymptotically log del Pezzo surfaces, motivated by the question of Cheltsov and Rubinstein. As a corollary, we give a complete answer for Cheltsov-Rubinstein’s question in dimension 2, which asserts about the existence of Kaehler-Einstein edge metrics for asymptotically log Fano varieties with small cone angles.

**Surface singularities in $R^4$: first steps towards Lipschitz knot theory (Andrei Gabrielov, Purdue University)**

A link of an isolated singularity of a two-dimensional semialgebraic surface in $R^4$ is a knot (or a link) in $S^3$. Thus the ambient Lipschitz classification of surface singularities in $R^4$ can be interpreted as a bi-Lipschitz refinement of the topological classification of knots (or links) in $S^3$. We show that, given a knot $K$ in $S^3$, there are infinitely many distinct ambient Lipschitz equivalence classes of outer metric Lipschitz equivalent singularities in $R^4$ with the links topologically equivalent to $K$.

**A generalization of Batyrev’s cone conjecture (Yoshinori Gongyo, University of Tokyo)**

We discuss a generalization of Batyrev’s cone conjecture for curves moving in codimension $l$. Thus we propose the cone conjecture integrating Mori’s cone theorem and the Batyrev’s cone conjecture. Moreover we prove a weak version of this conjecture by proving some rationality theorem. This is a work in progress with Sung Rak Choi.

**New constructions of Fano 3-folds from mirror symmetry (Liana Heuberger, Loughborough University)**

Mirror symmetry conjecturally associates to a Fano orbifold a (very special type of) Laurent polynomial. Laurent inversion is a method for reversing this process, obtaining a Fano variety from a candidate Laurent polynomial. We apply this to construct previously unknown Fano 3-folds with terminal quotient singularities.

A Laurent polynomial f determines, through its Newton polytope P, a toric variety X_P, which is in general highly singular. Laurent inversion constructs, from f and some auxiliary data, an embedding of X_P into an ambient toric variety Y. In many cases this embeds X_P as a complete intersection of line bundles on Y, and the general section of these line bundles is the Q-Fano 3-fold that we seek.

This is joint work with Tom Coates, Al Kasprzyk and Giuseppe Pitton.

**Manin’s conjecture for a log del Pezzo surface with A_4+K_5 singularities (DongSeong Hwang, Ajou University)**

Manin’s conjecture is a precise prediction on the asymptotic behavior on the number of rational points of bounded anticanonical height on Fano varieties. In this talk, I will explain how the geometry governs the arithmetic in the case of a log del Pezzo surface with A_4 and K_5 singularities using the torsor method. This talk is based on joint work in progress with Ulrich Derenthal.

**Infinitesimal neighborhoods of submanifolds (Jun-Muk Hwang, Korea Institute for Advanced Study)**

We discuss the rigidity problem of infinitesimal neighborhoods of compact submanifolds of complex manifolds, which goes back to a question of Nirenberg and Spencer asking whether a finite-order infinitesimal neighborhood determines the germ of a submanifold when the normal bundle is positive. Following Hirschowitz’s approach, we consider a geometric condition in terms of deformations of the submanifold, replacing the condition on the positivity of the normal bundle. This leads to a reformulation of the problem in terms of families of infinitesimal neighborhoods of submanifolds. We explain some results on this problem with examples.

**Planes in four-dimensional cubics (Ilia Itenberg, Sorbonne University)**

We discuss possible numbers of 2-planes in a smooth cubic hypersurface in the 5-dimensional projective space. We show that, in the complex case, the maximal number of planes is 405, the maximum being realized by the Fermat cubic. In the real case, the maximal number of planes is 357. The proofs are based on the global Torelli theorem for cubic hypersurfaces in the 5-dimensional complex projective space and the surjectivity of the period map for these hypersurfaces, as well as on Nikulin’s theory of discriminant forms.

Joint work with Alex Degtyarev and John Christian Ottem.

**ACC for minimal log discrepancies and divisorial contractions in dimension three (Masayuki Kawakita, Kyoto University)**

One of the remaining important problems in the birational geometry of threefolds is the ACC for minimal log discrepancies. When one works on a smooth threefold, it is equivalent to a certain boundedness of divisors computing the minimal log discrepancy. I will discuss an approach to it by the explicit study of threefold divisorial contractions.

**Non-commutative deformations of coherent sheaves (Yujiro Kawamata, University of Tokyo)**

We consider deformations of a coherent sheaf on an algebraic variety. The deformations are usually considered over commutative rings, and the parameter ring of the universal deformation is the local ring of the moduli space of the sheaf. When we allow non-commutativity of the parameter ring, then we obtain similar theory but with more deformations. For example, if we consider the structure sheaf of a line in a projective space, the moduli space of the usual deformations is the Grassmann variety. But there are more non-commutative deformations than the commutative deformations which are obstructed. We will explain how to describe the universal non-commutative deformations.

**Quantizations of conical symplectic singularities (Ivan Losev, Yale University)**

Quantizations of interesting Poisson algebras provide an important object of study for Representation theory — these are associative algebras whose representations we want to understand. It turns out that one can use recent advances in Birational geometry to classify the filtered quantizations of algebras of functions on conical symplectic singularities. In the talk we will explain this classification and discuss several aspects of the representation theory of resulting algebras.

**Topology of rank 1 linear systems on real curves (Grigory Mikhalkin, Université de Genève)**

Hilbert’s 16th problem includes the question of determining topology of a an algebraic curve of degree d in the real projective plane. This question is still open for d>=8. A related question asks for possible topology of a real linear system of degree d and rank 1 on a given real curve. The talk will review the corresponding classification problem (still open for g>=5).

**Collapsing K3 surfaces, hyperKahler metrics, and Moduli compactification (Yuji Odaka, Kyoto University)**

Moduli spaces of K3 surfaces and their generalization – holomorphic symplectic varieties – are well studied, largely thanks to the celebrated Torelli type theorems (Piatetski-Shapiro-Shafarevich, Burns-Rapoport, Verbitsky,..). Here, we propose to use a certain canonical and explicit compactification, which must correspond to considering/adding the limits of their canonical (hyperKahler) metrics with fixed diameters. The limits nor the compactifications are no longer varieties, but consequently, they can be explicitly described in a relatively algebraic way, and still has various interesting algebraic aspect.

This is a brief summary of part of the joint work with Yoshiki Oshima (Osaka university), the full paper is available at arXiv:1810.07685 (a short version: 1805.01724). For instance, it unconditionally follows that the limit along any maximal degeneration of polarized K3 surfaces is a sphere (“tropical K3″) with some additional structures (conjectured by Gross-Wilson, Todorov, Kontsevich-Soibelman). Some parts extend to higher dimensional hyperKahler case as well.

**Rational Elliptic Surfaces and Trigonometry of Non-Euclidean Tetrahedra (Daniil Rudenko, The University of Chicago)**

I will explain how to construct a rational elliptic surface out of every non-Euclidean tetrahedra. This surface “remembers” the trigonometry of the tetrahedron: the length of edges, dihedral angles and the volume can be naturally computed in terms of the surface. The main property of this construction is self-duality: the surfaces obtained from the tetrahedron and its dual coincide. This leads to some unexpected relations between angles and edges of the tetrahedron. For instance, the cross-ratio of the exponents of the spherical angles coincides with the cross-ratio of the exponents of the perimeters of its faces. The construction is based on relating mixed Hodge structures, associated to the tetrahedron and the corresponding surface.

**Lagrangian fibrations by Prym varieties (Justin Sawon, University of North Carolina at Chapel Hill)**

Lagrangian fibrations are fibrations on holomorphic symplectic manifolds/orbifolds whose general fibres are abelian varieties that are Lagrangian with respect to the symplectic structure. Beauville and Mukai considered examples coming from relative Jacobians of families of curves on K3 surfaces. We discuss the construction of examples whose fibres are Prym varieties, coming from K3 surfaces covering del Pezzo surfaces (following Markushevich-Tikhomirov, Menet, Matteini). Some interesting duality relations arise in these examples.

**Birational geometry and C* actions (Jaroslaw Wisniewski, University of Warsaw)**

A general lay out for relating C* actions and birational geometry is known by works of Reid, Thaddeus, Morelli and Wlodarczyk. I will explain how to use these ideas in order to understand complex contact manifolds (joint project with Romano, Occhetta, and Sola Conde) and to invert matrices via flips and flops (joint project with Michalek and Monin).