Ingrid Bauer, Fabrizio Catanese, Roberto Pignatelli
Chapters : Old and new inequalities; Surfaces with $χ=1$ and the bicanonical map; Surfaces with $p_g=4$; Surfaces isogeneous to a product, Beauville surfaces and the absolute Galois group;Lefschetz pencils and braid monodromies;DEF, DIFF and other equivalence relations.
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Recent developments on Gromov-Witten theory of Hilbert schemes of points on certain surfaces
本文综述了椭圆曲面和单连通极小一般型曲面上点的Hilbert概形的Gromov-Witten不变量理论的最新进展，重点探讨了低亏格情形的计算成果及重要猜想的解答进展。
We give a survey of results on the geometry of complex algebraic Q-acyclic surfaces, so-called 'Q-homology planes', including some recent results.
We survey the authors recent works, joint with A. Jacob, on Strominger-Yau-Zaslow mirror symmetry for rational elliptic surfaces and del Pezzo surfaces. We discuss some applications, including the Torelli theorem for $ALH^*$ gravitational instantons and explain how our results can be used to prove that all $ALH^*$ gravitational instantons can be compactified to a weak del Pezzo surface, recovering a recent result of Hein-Sun-Viaclovsky-Zhang.
Recent progress on the Minimal Model Program for foliations
本文综述了复杂变分上叶结构有理几何的最新进展，重点探讨了最小模型程序视角下的奇点、邻接及在曲面和三维翻转存在性中的应用。
In the last years there have been several new constructions of surfaces of general type with $p_g=0$, and important progress on their classification. The present paper presents the status of the art on surfaces of general type with $p_g=0$, and gives an updated list of the existing surfaces, in the case where $K^2= 1,...,7$. It also focuses on certain important aspects of this classification.
In this note, we report some progress we made recently on the automorphisms groups of K3 surfaces. A short and straightforward proof of the impossibility of Z/(60) acting purely non-symplectically on a K3 surface, is also given, by using Lefschetz fixed point formula for vector bundles.
K3 surfaces with maximal complex multiplication
本文继Valloni之后，研究了具有整数环复乘的复射影K3曲面。
This paper surveys recent progress towards the Manin conjecture for (singular and non-singular) del Pezzo surfaces. To illustrate some of the techniques available, an upper bound of the expected order of magnitude is established for a singular del Pezzo surface of degree four.
We determine the minimum positive entropy of complex Enriques surface automorphisms. This together with McMullen's work completes the determination of the minimum positive entropy of complex surface automorphisms in each class of Enriques-Kodaira classification of complex surfaces.
We propose the new construction of complex surfaces with $h^{1,0} = h^{2,0} = 0$ from smoothings of normal crossing surfaces with non-collapsible dual complexes and carry it out for the simplest case of the duncehat complex, obtaining the surface with $h^{1,1} = 9$ (presumably Barlow surface).
In this paper we show that the moduli space of nodal cubic surfaces is isomorphic to a quotient of a 4-dimensional complex ball by an arithmetic subgroup of the unitary group. This complex ball uniformization uses the periods of certain K3 surfaces which are naturally associated to cubic surfaces. A similar uniformization is given for the covers of the moduli space corresponding to geometric markings of the Picard group or to the choice of a line on the surface. We also give a detailed description of the boundary components corresponding to singular surfaces.
In this paper we discuss recent progress on the modularity of Calabi-Yau varieties. We focus mostly on the case of surfaces and threefolds. We will also discuss some progress on the structure of the L-function in connection with mirror symmetry. Finally, we address some questions and open problems.
We establish Manin's conjecture for a cubic surface split over Q and whose singularity type is 2A_2+A_1. For this, we make use of a deep result about the equidistribution of the values of a certain restricted divisor function in three variables in arithmetic progressions. This result is due to Friedlander and Iwaniec (and was later improved by Heath-Brown) and draws on the work of Deligne.
In this article we explain the theory of rigid residue complexes in commutative algebra and algebraic geometry, summarizing the background, recent results and anticipated future results. Unlike all previous approaches to Grothendiec Duality, the rigid approach concentrates on the construction of rigid residue complexes over rings, and their intricate yet robust properties. The geometrization, i.e. the passage to rigid residue complexes on schemes and Deligne-Mumford (DM) stacks, by gluing, is fairly easy. In the geometric part of the theory, the main results are the Rigid Residue Theorem and t…
We prove Manin's conjecture for two del Pezzo surfaces of degree four which are split over Q and whose singularity types are respectively 3A_1 and A_1+A_2. For this, we study a certain restricted divisor function and use a result about the equidistribution of its values in arithmetic progressions. In this task, Weil's bound for Kloosterman sums plays a key role.
Let X be a compact complex surface with a real foliation. If all leaves are compact complex curves, the foliation must be holomorphic.
Gersten-Witt cohomologies of Hirzebruch surfaces are computed.
The subject of this paper is the problem of arrangement of real algebraic curves on real algebraic surfaces. In this paper we extend Rokhlin, Kharlamov-Gudkov-Krakhnov and Kharlamov-Marin congruences for curves on surfaces and give some applications of this extension. For some pairs consisting of a surface and a curve on this surface (in particular for M-pairs) we introduce a new structure --- the complex separation that is separation of the complement of curve into two surfaces. In accordance with Rokhlin terminology the complex separation is a complex topological characteristic of real algeb…
In this paper we prove that no complex surface of general type is diffeomorphic to a rational surface, thereby completing the smooth classification of rational surfaces and the proof of the Van de Ven conjecture on the smooth invariance of Kodaira dimension.
We survey recent work, published since 2015, on equivariant Oka theory. The main results described in the survey are as follows. Homotopy principles for equivariant isomorphisms of Stein manifolds on which a reductive complex Lie group $G$ acts. Applications to the linearisation problem. A parametric Oka principle for sections of a bundle $E$ of homogeneous spaces for a group bundle $\mathscr G$, all over a reduced Stein space $X$ with compatible actions of a reductive complex group on $E$, $\mathscr G$, and $X$. Application to the classification of generalised principal bundles with a group a…
Based on Teichmüller theory, we construct a degenerating family $\overline{Y}_g^{orb} \rightarrow \overline{M}_g^{orb}$ over the Deligne-Mumford compactification of the moduli space with the natural orbifold structure such that any fibered complex surface admitting unstable fibers can be pulled back from this family.
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