# Classification and Moduli Theory of Algebraic Varieties

The conference will take place from 10-17 September 2017 at Hotel Continental Ischia (NA), Italy.

Scientific Committee: Fabrizio Catanese (Bayreuth) and Yujiro Kawamata (Tokyo).

Organizers: Ingrid Bauer (Bayreuth), Fabrizio Catanese (Bayreuth) and Yujiro Kawamata (Tokyo).

Local Organizer: G. Ilardi (Napoli)

## Participants will include:

Barja Miguel Angel (Barcelona, Politecnica de Catalunya) Bauer Ingrid (Bayreuth) Campana Frederic(Nancy) Catanese Fabrizio (Bayreuth) Ciliberto Ciro (Roma, "Tor Vergata") Cho Yonghwa (Bayreuth) Corvaja Pietro (Udine) Coughlan Stephen (Bayreuth) De Poi Pietro (Udine) Demleitner Andreas (Bayreuth) Frapporti Davide (Bayreuth) Fujita Kento (Kyoto) Gleissner Christian (Bayreuth / Trento) Gongyo Yoshinori (Tokyo) Hwang Jun-Muk (Seoul, KIAS) Ilardi Giovanna (Napoli) Ishii Akira (Hiroshima) | Ito Atsushi (Kyoto / Nagoya) Katzarkov Ludmil (HSE Moscow/ Wien) Kawamata Yujiro (Tokyo) Keum Jong-Hae (Seoul, KIAS) Kondo Shigeyuki (Nagoya) Lee Yongnam (Daejeon, KAIST) Li Binru (Bayreuth) Lönne Michael (Bayreuth) Nakamura Yusuke (Tokyo) Okawa Shinnosuke (Osaka) Perroni Fabio (Trieste) Peternell Thomas (Bayreuth) Sernesi Edoardo (Roma III) Tian Gang (Princeton / Beijing) Van der Geer Gerard (Amsterdam / Beijing) Xu Chenyang (Beijing) |

## Schedule

### Monday, 13. September

- 09:30–10:15Peternell ThomasBayreuthDescent of numerically flat vector bundles, the Simpson correspondence and singular ball quotients10:15–10:50Coffee Break10:50–11:35Lee YongnamDaejeon, KAISTDeformation of a generically finite map to a hypersurface embedding
In my talk I will explain recent results with Greb, Kebekus and Taji concerning the Simpson correspondence on klt spaces and the uniformization of klt spaces whose (orbifold) Chern classes are extremal in the sense that they satisfy the Miyaoka-Yau equality. The key is a descent theorem for flat bundles on klt spaces.

17:00–17:45Ishii AkiraHiroshimaG-constellations and the maximal resolution of a quotient surface singularityIn this talk, we give a structure theorem for projective manifolds ${W}_{0}$ with the property of admitting a 1-parameter deformation where ${W}_{t}$ is a hypersurface in a projective manifold ${Z}_{t}$. There structure is the one of special iterated univariate coverings which we call of normal type, which essentially means that the line bundles where the univariate coverings live are tensor powers of the normal bundle to the image. We give applications to the case where ${Z}_{t}$ is a projective space, respectively an Abelian variety. This is a joint work with Fabrizio Catanese.

18:20–19:05Okawa ShinnosukeOsakaNoncommutative del Pezzo surfaces and their moduli spacesFor a finite subgroup G of GL(n,C), a G-constellation is a G-equivariant coherent sheaf with finite support on C^n whose global sections form the regular representation of G. The moduli space M_\ theta of G-constellations depends on the stability parameter \ theta and for some choice of \ theta it coincides with the Hilbert scheme of G-orbits. If G is in SL(3) then the moduli space is a crepant resolution of C^3/G. In a joint work with Alastair Craw in 2004, we proved that if G is a finite abelian subgroup of SL(3), then every projective crepant resolution of C^3/G is isomorphic to M_\ theta for some \ theta. In this talk, we consider the case when G is in GL(2) but not necessarily abelian. In this case, the moduli space is between the minimal and the maximal resolutions of C^2/G. When G is small, we prove every such resolution is isomorphic to M_\ theta for some \ theta.

19:10–20:00Welcome cocktailIt is well known that the derived category of coherent sheaves on a del Pezzo surface admits very nice generators, which can be regarded as an analogue of polarizations. In my talk I will introduce certain moduli spaces which contain the moduli spaces of marked del Pezzo surfaces as closed subschemes. Generic points of our moduli spaces, on the other hand, classify noncommutative del Pezzo surfaces in the sense of noncommutative algebraic geometry. If time permits, I will mention the action of the affine Weyl groups on the moduli spaces. This is a joint work with Tarig Abdelgadir and Kazushi Ueda.

### Tuesday, 14. September

- 09:30–10:15Kondo ShigeyukiNagoyaClassification of Enriques surfaces with finite automorphism group in characteristic 210:15–10:50Coffee Break10:50–11:35Xu ChenyangBeijingVolume and stability of singularities
We classify singular, supersingular and classical Enriques surfaces with finite automorphism group in characteristic 2 into 11 types according to their dual graphs of all nonsigular rational curves. It follows that the classification of all Enriques surfaces with finite automorphism group in any characteristics has been finished. This is a joint work with Toshiyuki Katsura and Gebhard Martin.

17:00–17:45Katzarkov LudmilHigher School of Economics Moscow / University of ViennaPDE Filtrations and CategoriesThe concept of KLT (Kawamata log terminal) singularities appears in the minimal model program (MMP) more than three decades ago, and became one of the most natural category of singularities to work on, for the study of MMP and many other reasons. One guiding principal is that the class of klt singularities is the local analogue of Fano varieties. In this talk, I will discuss our work (joint with Chi Li) on developing an algebraic stability theory of general klt singularities using Chi Li’s definition of normalised volumes. As an application, we prove that the intermediate semistable cone of the metric tangent cone of a klt singularity appearing on the GH limit of Kahler-Einstein Fano manifolds is only determined by the algebraic structure of the singularity but independent of the local metric, which confirms a conjecture by Donaldson-Sun.

18:20–19:05Perroni FabioTriesteOn homological stability for moduli spaces of curves with symmetryTBA

20:00Cena rustica (tradional food dinner)I will report on a joint project, in progress, with Fabrizio Catanese and Michael Lönne, where we investigate the homological stability of moduli spaces of (smooth) curves C with a given group of symmetry G, when the genus of the curves increases. Over the field of rational numbers, the homology of these moduli spaces can be interpreted as equivariant homology of the mapping class group Map_{g',d} acting on HV(G;g',d)/G, where HV(G;g',d) is the set of Hurwitz vectors corresponding to coverings C->C/G, g' is the genus of C/G, d is the number of branch points of the covering C -> C/G, and G acts on HV(G;g',d) by conjugation. In the case where d=0, we prove that the equivariant homology groups of Map_{g'}^1 (the mapping class group of a surface of genus g' with one boundary component) acting on HV(G;g',0) don't depend on g', for g' large enough. This result could be useful in the study of homological stability of Map_{g'} acting on HV(G;g',0)/G.

### Wednesday, 15. September

- 09:00–09:45Tian GangPrinceton / BeijingCounting holomorphic curves in P^209:45–10:20Coffee Break10:20–11:05Corvaja PietroUdineOn the Betti map associated to abelian logarithms
TBA

11:30–12:15Lönne MichaelBayreuthOn the branch stable homology of Hurwitz spacesGiven an abelian scheme provided with a section, one can locally express an abelian logarithm of the section in a local basis for the periods, obtaining the so-called Betti coordinates of the section. In a joint work with Y. André and U. Zannier, we studied the differential of this map, proving that under certain conditions it is a submersion. It turned out that these conditions are somewhat linked with the Kodaira-Spencer map of the abelian fibration. In the particular case of the universal hyper-elliptic scheme, we obtain explicit and general results.

14:45–19:00Boat tour of island IschiaHurwitz spaces of Galois branched covers of the disc are finite unramified covering spaces of configuration spaces of the disc. Their homology was studied by Ellenberg, Venkatesh, Westerland via certain stabilisation maps increasing the number of branch points in cases with rather restricted Galois groups. The case of arbitrary finite groups was addressed by F. Tietz in his thesis. I will present a slightly improved version of his results, in particular the identification of the branch stable rational homology with the homology of coloured braid groups.

### Thursday, 16. September

- 09:30–10:15Hwang Jun-MukSeoul, KIASRecognizing symplectic Grassmannians by their varieties of minimal rational tangents10:15–10:50Coffee Break10:50–11:35Van der Geer GerardAmsterdam / BeijingInvariant theory and modular forms for genus 2 and 3
Let G/P be a rational homogeneous space of Picard number 1 and let X be a Fano manifold of Picard number 1. The question we are interested in is: if the variety of minimal rational tangents at a general point of X is isomorphic to that of G/P, is X biholomorphic to G/P? Ngaiming Mok gave an affirmative answer when G/P is a Hermitian symmetric space or a homogeneous contact manifold. Mok's argument was generalized by Hong-Hwang to any G/P associated with a long root. In these cases, the underlying geometric structure modeled on G/P is a parabolic geometry, for which local differential geometric machinery is well-developed in the classical works of Tanaka. The question has been open for the other G/P, i.e., symplectic Grassmannians or two cases of F_4-homogeneous spaces. The main difficulty in the remaining cases is that the underlying geometric structure is no longer a parabolic geometry and certain degenerate structures may occur in a neighborhood of a minimal rational curve. Because of the potential degeneracy, standard Tanaka theory is not available. In a collaboration with Qifeng Li, we overcome this difficulty by constructing a Cartan connection associated to the geometric structure in a neighborhood of a minimal rational curve, under the assumption that certain vector bundles arising from Spencer complexes do not have nonzero sections. Using this construction, we settle the case of symplectic Grassmannians. Our argument also works for odd-symplectic Grassmannians.

17:00–17:45Campana FredericNancyA notion of rational connectedness for orbifold pairsThe best-known modular forms are elliptic modular forms; a natural generalization are vector-valued Siegel modular forms of genus g. For genus 2 and 3 these modular forms are intimately connected with the moduli of curves of genus 2 and 3. We give an explicit way to describe such vector-valued modular forms for genus 2 and 3 using invariant theory and give some applications. This is based on joint work with Fabien Clery and Carel Faber.

18:20–19:05Ito AtsushiKyoto / NagoyaOn reconstruction problem in algebraic visionTBA

20:00Social dinnerIn this talk, I will explain a reconstruction problem in computer vision from the view point of algebraic geometry. This is joint work with Makoto Miura and Kazushi Ueda.

### Friday, 17. September

- 09:30–10:15Barja Miguel AngelBarcelona, Politecnica de CatalunyaMaximal Albanese dimension Varieties on the Severi Lines10:15–10:50Coffee Break10:50–11:35Ciliberto CiroRoma, "Tor Vergata"Gaussian maps and extensions of canonical curves
In the first part of my talk I will review several properties of linear systems on irregular varieties and the so called Clifford-Severi Inequalities we can derive from them. Then problem of classifying maximal Albanese dimension varieties with small canonical volume (varieties on the Severi Lines) has been solved in dimension 2 by Lu and Zuo and, independently, by Pardini, Stoppino and myself in 2015. I will show how to study the higher dimensional cases using the so called continuous rank functions and the eventual maps associated to linear systems. This is a joint work with R. Pardini and L. Stoppino.

17:00–17:45Gongyo YoshinoriTokyoOn log CY structure of varieties admitting non-trivial polarized endomorphismIn this talk I will consider the ramification locus of the obvious map from the space of pairs $(S,C)$ where $S$ is a primitively polarised K3 surface and $C$ is a smooth curve in the primitive polarization with ${C}^{2}=2g-2$, to the moduli space of curves of genus $g$. The main result is that, if \geqslan 11$and$ \mathrm{ Cliff}(C)>2$,theabovemapramifiesat$(S,C)$ifandonlyif$ is the curve section of a (possibly fake) Fano variety $ of dimension at least 3. This follows form results by J. Wahl and by E. Arbarello, A. Bruno and E. Sernesi, is related with a theorem of L'vovsky and has some consequences which I intend to explain. This is joint work (in progress) with Thomas Dedieu and Edoardo Sernesi.

18:20–19:05Fujita KentoKyotoA valuative criterion for uniform K-stability of log Fano pairsThis is a joint work with Amael Broustet. I will discuss the log Calabi–Yau structure of varieties admitting polarized endomorphism. This is motivated in local theory for isolated singularities. I will talk from more global side.

We give a simple criterion for uniform K-stability of log Fano pairs (for the anti-log-canonical polarization). We also see several applications and recent progresses on this area.

### Saturday, 18. September

- 09:00–09:45Nakamura YusukeTokyoA vanishing theorem of Witt-vector cohomology for Fano threefolds09:45–10:20Coffee Break10:20–11:05Gleissner ChristianBayreuth /TrentoProduct Quotients in Dimension Three
In this talk, I will explain a vanishing theorem of Witt-vector cohomology of Nadel type. This is an analogy of Nadel vanishing theorem in positive characteristic. We also discuss its application to the connectedness lemma and the rational point formula. This is joint work with Hiromu Tanaka.

11:30–12:15Sernesi EdoardoRoma IIISyzygy schemes of a canonical curveTBA

It is possible to associate certain "syzygy schemes" to the minimal resolution of a projective curve. Not much is known about them. I will give an overview of known results in the case of canonical curves and report on recent work in collaboration with M. Aprodu and A. Bruno.

Research funded by

ERC Advanced Grant TADMICAMT