Surfaces with K^2=8, p_g=4 and canonical involution

Involutions on algebraic surfaces and the Generalised Bloch's conjecture

In this note we are going to consider a smooth projective surface equipped with an involution and study the action of the involution at the level of Chow group of zero cycles.

Symplectic involutions of K3 surfaces act trivially on CH_0

Symplectic involutions of a K3 surface are those involutions which leave the holomorphic 2-form invariant. We show, as predicted by Bloch's conjecture, that they act trivially on the CH_0 group of the K3 surface. This was recently proved by Huybrechts and Kemeny for one of the three types of symplectic involutions.

Elliptic canonical bases for toric hyper-Kahler manifolds

Lusztig defined certain involutions on the equivariant K-theory of Slodowy varieties and gave a characterization of certain bases called canonical bases. In this paper, we give a conjectural generalization of these involutions and K-theoretic canonical bases to conical symplectic resolutions which have good Hamiltonian torus actions and state several conjectures related to them which we check for toric hyper-Kahler manifolds. We also propose an elliptic analogue of these bar involutions. As a verification of our proposal, we explicitly construct elliptic lifts of K-theoretic canonical bases an…

Bloch's conjecture on surfaces of general type with an involution

In this short note we prove that the Bloch's conjecture holds for a surface of general type of numerical Godeaux type or some class of numerical Campedelli type, with geometric genus zero equipped with an involution, when the quotient of the surface by the involution is a rational surface.

K3曲面上的对合与导出等价

Involutions on K3 surfaces and derived equivalence

本文研究K3曲面上对合在导出等价共轭下的行为，并将其应用于等变双有理几何。

Surfaces with involution and Prym constructions

An involution on a surface induces involutions on the cohomology, the Chow group and the Brauer group of the surface. We give a detailed study of those actions. We show that the odd part of these groups can be used to describe the geometry of cubic fourfolds and conic bundles over $\mathbb{P}^3$.

Involutions of second kind on Shimura surfaces and surfaces of general type with $q=p_g=0$

Quaternionic Shimura surfaces are quotient of the bidisc by an irreducible cocompact arithmetic group. In the present paper we are interested in (smooth) quaternionic Shimura surfaces admitting an automorphism with one dimensional fixed locus; such automorphisms are involutions. We propose a new construction of surfaces of general type with $q=p_{g}=0$ as quotients of quaternionic Shimura surfaces by such involutions. These quotients have finite fundamental group.

Classification of involutions on Enriques surfaces

We present the classification of involutions on Enriques surfaces. We classify those into 18 types with the help of the lattice theory due to Nikulin. We also give all examples of the classification.

Beilinson关于带反演的K3表面的猜想

Beilinson's conjecture on K3 surfaces with an involution

本文证明了特定K3表面在带有反演时，Beilinson猜想成立，其商结构为射影平面分支于六次曲线。

Involutions on numerical Campedelli surfaces

Numerical Campedelli surfaces are minimal surfaces of general type with p_g=0 (and so q=0) and K^2=2. Although they have been studied by several authors, their complete classification is not known. In this paper we classify numerical Campedelli surfaces with an involution, i.e. an automorphism of order 2. First we show that an involution on a numerical Campedelli surface S has either four or six isolated fixed points, and the bicanonical map of S is composed with the involution if and only if the involution has six isolated fixed points. Then we study in detail each of the possible cases, desc…

Van Geemen--Sarti involutions and elliptic fibrations on K3 surfaces double cover of $\mathbb{P}^2$

In this paper we classify the elliptic fibrations on K3 surfaces which are the double cover of a blow up of $\mathbb{P}^2$ branched along rational curves and we give equations for many of these elliptic fibrations. Thus we obtain a classification of the van Geemen--Sarti involutions (which are symplectic involutions induced by a translation by a 2-torsion section on an elliptic fibration) on such a surface. Each van Geemen--Sarti involution induces a 2-isogeny between two K3 surfaces, which is described in this paper.

Fourier-Mukai transformations on K3 surfaces with $ρ=1$ and Atkin-Lehner involutions

We show that there is a surjection from the Fourier-Mukai transformations on projective K3 surfaces with the Picard number $ρ(X)=1$ to so called to the group of Atkin-Lehner involutions. This was expected in Hosono-Lian-Oguiso-Yau's paper.

Enriques involutions on singular K3 surfaces of small discriminants

We classify Enriques involutions on a K3 surface, up to conjugation in the automorphism group, in terms of lattice theory. We enumerate such involutions on singular K3 surfaces with transcendental lattice of discriminant smaller than or equal to 36. For 11 of these K3 surfaces, we apply Borcherds method to compute the automorphism group of the Enriques surfaces covered by them. In particular, we investigate the structure of the two most algebraic Enriques surfaces.

Bloch's conjecture on certain surfaces of general type with $p_g=0$ and with an involution: the Enriques case

In this short note we prove that an involution on certain examples of surfaces of general type with $p_g=0$, acts as identity on the Chow group of zero cycles of the relevant surface. In particular we consider examples of such surfaces when the quotient is an Enriques surface and show that the Bloch's conjecture holds for such surfaces.

Involutions on a surface of general type with $p_g=q=0$, $K^2=7$

In this paper we study on the involution on minimal surfaces of general type with $p_g=q=0$ and $K^2=7$. We focus on the classification of the birational models of the quotient surfaces and their branch divisors induced by an involution.

Godeaux surfaces with an Enriques involution and some stable degenerations

We give an explicit description of the Godeaux surfaces that admit an involution such that the quotient surface is birational to an Enriques surface; these surfaces give a 6-dimensional unirational irreducible subset of the moduli space of surfaces of general type. In addition, we describe the Enriques surfaces that are birational to the quotient of a Godeaux surface by an involution and we show that they give a 5-dimensional unirational irreducible subset of the moduli space of Enriques surfaces. Finally, by degenerating our description we obtain some examples of non-normal stable Godeaux sur…

Bloch's conjecture for surfaces with involutions and of geometric genus zero

Let $S$ be a smooth projective surface with $p_g=0$, let $ι$ be a regular involution acting on $S$, and let $W$ be the resolution of singularities of the quotient surface $S/ι$. In the paper we prove that Bloch's conjecture holds for the surface $S$ if and only if it holds for the surface $W$. This yields Bloch's conjecture for all surfaces $S$ whenever the same conjecture is true for the desingularized quotient $W$. In particular, Bloch's conjecture holds true for all numerical Godeaux surfaces with involutions, a "half" of Campedelli surfaces with involutions, the surface of Craighero and Ga…

Involutions on surfaces with $p_g=q=1$

In this paper some numerical restrictions for surfaces with an involution are obtained. These formulas are used to study surfaces of general type $S$ with $p_g=q=1$ having an involution $i$ such that $S/i$ is a non-ruled surface and such that the bicanonical map of $S$ is not composed with $i.$ A complete list of possibilities is given and several new examples are constructed, as bidouble covers of surfaces. In particular the first example of a minimal surface of general type with $p_g=q=1$ and $K^2=7$ having birational bicanonical map is obtained.

Classification of K3-surfaces with involution and maximal symplectic symmetry

K3-surfaces with antisymplectic involution and compatible symplectic actions of finite groups are considered. In this situation actions of large finite groups of symplectic transformations are shown to arise via double covers of Del Pezzo surfaces. A complete classification of K3-surfaces with maximal symplectic symmetry is obtained.

On the nonsymplectic involutions of the Hilbert square of a K3 surface

We investigate the interplay between the moduli spaces of ample <2>-polarized IHS manifolds of type K3^[2] and of IHS manifolds of type K3^[2] with a nonsymplectic involution with invariant lattice of rank one. In particular we geometrically describe some new involutions of the Hilbert square of a K3 surface, whose existence was proven in a previous work of Boissiere, Cattaneo, Nieper-Wisskirchen and Sarti.

Special divisors on curves on K3 surfaces carrying an Enriques involution

We study the pencils of minimal degree on the smooth curves lying on a K3 surface X which carries a fixed-point free involution. Generically, the gonality of these curves is totally governed by the genus 1 fibrations of X

Involutions of polynomially parametrized surfaces

We provide an algorithm for detecting the involutions leaving a surface defined by a polynomial parametrization invariant. As a consequence, the symmetry axes, symmetry planes and symmetry center of the surface, if any, can be determined directly from the parametrization, without computing or making use of the implicit representation. The algorithm is based on the fact, proven in the paper, that any involution of the surface comes from an involution of the parameter space (the real plane, in our case); therefore, by determining the latter, the former can be found. The algorithm has been implem…

阿贝尔曲面中的曲线对合及其雅可比簇

Involutions of curves in abelian surfaces and their Jacobians

该研究针对复阿贝尔曲面上型为$(1,d)$的极化线丛线性系统中的亏格二曲线平展覆盖，给出了曲线对合不动点计数及雅可比簇同源分解的相关结果。

Enriques involutions on pencils of K3 surfaces

The three pencils of K3 surfaces of minimal discriminant whose general element covers at least one Enriques surface are Kondō's pencils I and II, and the Apéry--Fermi pencil. We enumerate and investigate all Enriques surfaces covered by their general elements.

Bitangent 曲面与四次曲面的对合

Bitangent surfaces and involutions of quartic surfaces

研究任意特征下三维射影空间中不可约曲面的双切线同余，特别关注具有有理二重点的四次曲面，尤其是 Kummer 四次曲面。

通过自等价构造K3曲面上层模空间的反辛对合

Anti-symplectic involutions on moduli spaces of sheaves on K3 surfaces via auto-equivalences

本文利用K3曲面上由球面丛诱导的导出范畴自等价构造了稳定层模空间上的新反辛对合，并统一推广了Beauville对合、Markman-O'Grady反射及Beri-Manivel构造等经典例子。

Symplectic involutions on deformations of K3^[2]

Let X be a Hyperkähler variety deformation equivalent to the Hilbert square on a K3 surface and let f be an involution preserving the symplectic form. We prove that the fixed locus of f consists of 28 isolated points and 1 K3 surface, moreover the anti-invariant lattice of the induced involution on H^2(X,Z) is isomorphic to E_8(-2). Finally we prove that any couple consisting of one such variety and a symplectic involution on it can be deformed into a couple consisting of the Hilbert square of a K3 surface and the involution induced by a Nikulin involution on the K3 surface.

Elliptic fibrations on K3 surfaces with a non-symplectic involution fixing rational curves and a curve of positive genus

In this paper we complete the classification of the elliptic fibrations on K3 surfaces which admit a non-symplectic involution acting trivially on the Néron--Severi group. We use the geometric method introduced by Oguiso and moreover we provide a geometric construction of the fibrations classified. If the non-symplectic involution fixes at least one curve of genus 1, we relate all the elliptic fibrations on the K3 surface with either elliptic fibrations or generalized conic bundles on rational elliptic surfaces. This description allows us to write the Weierstrass equations of the elliptic fibr…

K3 曲面上的椭圆纤维化与对合

Elliptic fibrations and involutions on K3 surfaces

本文综述了带有非辛对合的 K3 曲面上椭圆纤维化的分类，在更一般的对合框架下重新审视了具有 2-元初 Néron--Severi 格的 K3 曲面的完整分类，并给出了有理椭圆曲面二次覆盖的 K3 曲面上极值椭圆纤维化的完整分类。

辛映射与van Geemen-Sarti映射在K3曲面中的特殊性

Specializations of symplectic and van Geemen--Sarti involutions on K3 surfaces

本文研究了K3曲面上辛映射与van Geemen-Sarti映射的特殊性质，探讨了其Néron-Severi群的等距性及上同调作用。

Quotients of K3 Surfaces Modulo Involutions

Let X be a K3 surface with an involution g which has non-empty fixed locus X^g and acts non-trivially on a non-zero holomorphic 2-form. We shall construct all such pairs (X, g) in a canonical way, from some better known double coverings of log del Pezzo surfaces of index at most 2 or rational elliptic surfaces, and construct the only family of each of the three extremal cases where X^g contains 10 (maximum possible) curves. We also classify rational log Enriques surfaces of index 2. Our approach is more geometrical rather than lattice-theoretical (see Nikulin's paper for the latter approach).

具有平凡 Brauer 映射的 Enriques 曲面与超凯勒流形上的对合

Enriques surfaces with trivial Brauer map and involutions on hyperkähler manifolds

本文利用Brauer映射的平凡性，在典范覆盖K3曲面的凝聚层范畴上定义了新对合，并将其不动点轨迹与非交换对的Picard概形相关联。

Involutions on surfaces with $p_g=q=0$ and $K^2=3$

We study surfaces of general type $S$ with $p_g=0$ and $K^2=3$ having an involution $i$ such that the bicanonical map of $S$ is not composed with $i$. It is shown that, if $S/i$ is not rational, then $S/i$ is birational to an Enriques surface or it has Kodaira dimension $1$ and the possibilities for the ramification divisor of the covering map $S\rightarrow S/i$ are described. We also show that these two cases do occur, providing an example. In this example $S$ has a hyperelliptic fibration of genus $3$ and the bicanonical map of $S$ is of degree $2$ onto a rational surface.

Real K3 Surfaces with non-symplectic Involutions and Applications

Classification of real K3 surfaces X with a non-symplectic involution τis considered. For some exactly defined and one of the weakest possible type of degeneration (giving the very reach discriminant), we show that the connected component of their moduli is defined by the isomorphism class of the action of τand the anti-holomorphic involution ϕin the homology lattice. (There are very few similar cases known.) For their classification we apply invariants of integral lattice involutions with conditions which were developed by the first author in 1983. As a particular case, we describe connected …

K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space IV: the structure of invariant

A holomorphic torsion invariant of K3 surfaces with involution was introduced by the second-named author. In this paper, we completely determine its structure as an automorphic function on the moduli space of such K3 surfaces. On every component of the moduli space, it is expressed as the product of an explicit Borcherds lift and a classical Siegel modular form. We also introduce its twisted version. We prove its modularity and a certain uniqueness of the modular form corresponding to the twisted holomorphic torsion invariant. This is used to study an equivariant analogue of Borcherds' conject…

关于超Kähler流形上反演的极大性及曲面点希尔伯特方案

On maximality of involutions of hyper-Kähler manifolds and punctual Hilbert schemes of surfaces

本文研究了超Kähler流形和曲面点希尔伯特方案上反演的极大性，推导出极大反演的条件并证明了某些流形不存在极大反演。

具有p_g=0、K^2=7和对合的光滑一般型极小曲面

Smooth minimal surfaces of general type with $p_g=0, K^2=7$ and involutions

本文改进并细化了Lee与第二作者关于p_g=0、K^2=7的光滑一般型极小曲面上对合的研究，排除k=9时κ(W)=1的情形，缩减并新给出k=9和k=11时分支除子B_0的可能性，并证明B_0有三个不可约分支时S为Inoue曲面。

Geometric lifting of the canonical basis and semitoric degenerations of Richardson varieties

In the sl\_n case, A. Berenstein and A. Zelevinsky studied the Schützenberger involution in terms of Lusztig's canonical basis, [3]. We generalize their construction and formulas for any semisimple Lie algebra. We use for this the geometric lifting of the canonical basis, on which an analogue of the Schützenberger involution can be given. As an application, we construct semitoric degenerations of Richardson varieties, following a method of P. Caldero, [6]

Symplectic involutions of holomorphic symplectic fourfolds

Let X be a holomorphic symplectic fourfold such that b_2=23 and i a symplectic involution of X . The fixed locus F of i is a smooth symplectic submanifold of X; we show that F contains at least 12 isolated points and 1 smooth surface. We conjecture that F is made of 28 isolated fixed points and 1 K3 surface and we provide evidences for the conjecture in some examples, as the Hilbert scheme of a K3 surface, the Fano variety of a cubic in P^5 and the double cover of an EPW sextic.

Numerical Godeaux surfaces with an involution

Minimal algebraic surfaces of general type with the smallest possible invariants have geometric genus zero and K^2=1 and are usually called "numerical Godeaux surfaces". Although they have been studied by several authors, their complete classification is not known. In this paper we classify numerical Godeaux surfaces with an involution, i.e. with an automorphism of order 2. We prove that they are birationally equivalent either to double covers of Enriques surfaces, or to double planes of two different types: the branch curve either has degree 10 and suitable singularities, originally suggested…

K3 Surfaces with Involution and Analytic Torsion

This is the abstruct of the revised paper. We study the equivariant analytic torsion for K3 surfaces with an anti-symplectic involution with the invariant lattice M (such a surface is called a 2-elementary K3 surface of type M in this paper), and show that it (together with the analytic torsion of the fixed curves) can be identified with the automorphic form on the moduli space characterizing the discriminant locus. Three lattices A_1, II_{1,1}(2), II_{1,9}(2) are of particular interest, because they consist of the building blocks of 2-elementary lattices. An explicit formula is given for them…

Non-symplectic involutions on manifolds of $K3^{[n]}$-type

We study irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on a $K3$ surface and admitting a non-symplectic involution. We classify the possible discriminant forms of the invariant and anti-invariant lattice for the action of the involution on cohomology, and explicitly describe the lattices in the cases where the invariant has small rank. We also give a modular description of all $d$-dimensional families of manifolds of $K3^{[n]}$-type with a non-symplectic involution for $d\geq 19$ and $n\leq 5$, and provide examples arising as moduli spaces of …

某些带对合的K3曲面上的高阶周环

Higher Chow cycles on some K3 surfaces with involution

该研究针对2<r<18的情形，在格极化K3曲面族上构造了显式(2,1)型高阶周环，证明其不可分解部分非挠，属中间皮卡秩下首批此类显式例子，构造基于奇异二重平面模型，用退化方法完成证明。

双有理对合

A birational involution

本文基于格理论预测并利用Mukai模型与例外群作用描述了18次K3曲面Hilbert立方的反辛双有理对合，证明其不定轨迹与同调射影对偶性相关，并得出该对合是Mukai flop实例的结论。

K3 surfaces with a pair of commuting non-symplectic involutions

We study K3 surfaces with a pair of commuting involutions that are non-symplectic with respect to two anti-commuting complex structures that are determined by a hyper-Kähler metric. One motivation for this paper is the role of such $\mathbb{Z}^2_2$-actions for the construction of $G_2$-manifolds. We find a large class of smooth K3 surfaces with such pairs of involutions, but we also pay special attention to the case that the K3 surface has ADE-singularities. Therefore, we introduce a special class of non-symplectic involutions that are suitable for explicit calculations and find 320 examples o…

K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space III: the case r(M)\geq18

We prove the automorphic property of the invariant of K3 surfaces with involution, which we obtained using equivariant analytic torsion, in the case where the dimension of the moduli space is less than or equal to 2.

Commuting involutions on surfaces of general type with p_g=0 and K^2=7

The aim of this article is to classify the pairs (S, G), where S is a smooth minimal surface of general type with p_g=0 and K^2=7, G is a subgroup of the automorphism group of S and G is isomorphic to the group $\mathbb{Z}_2^2$. The Inoue surfaces with K^2=7, which are finite Galois $\mathbb{Z}_2^2$-covers of the 4-nodal cubic surface, are the first examples of such pairs. More recently, the author constructed a new family of such pairs. They are finite Galois $\mathbb{Z}_2^2$-covers of certain 6-nodal Del Pezzo surfaces of degree one. We prove that the base of the Kuranishi family of deformat…

Verra fourfolds, twisted sheaves and the last involution

We study the geometry of some moduli spaces of twisted sheaves on K3 surfaces. In particular we introduce induced automorphisms from a K3 surface on moduli spaces of twisted sheaves on this K3 surface. As an application we prove the unirationality of moduli spaces of irreducible holomorphic symplectic manifolds of $K3^{[2]}$-type admitting non symplectic involutions with invariant lattices $U(2)\oplus D_4(-1)$ or $U(2)\oplus E_8(-2)$. This complements the results obtained in [Mongardi and Wandel 2015], [Bossiere et al 2016], and the results from [arXiv:1603.00403] about the geometry of IHS fou…

Non-symplectic involutions of a K3 surface

We classify K3 surfaces with a non-symplectic finite automorphism of high order. It is shown that such an automorphism cannot be of order 60, and for each of the orders 38, 44, 48, 50, 54 and 66, there exists a unique K3 surface with such an automorphism.

Enriques involutions and Brauer classes

We prove that every element of order 2 in the Brauer group of a complex Kummer surface X descends to an Enriques quotient of X. In 'generic' cases this gives a bijection between the set Enr(X) of Enriques quotients of X up to isomorphism and the set of Brauer classes of X of order 2. For some K3 surfaces of Picard rank 20 we prove that the fibres of the map from Enr(X) to Br(X)[2] above the non-zero points have the same order.

曲面上点的Hilbert概形中反辛对合不动点集的上同调

Cohomology of fixed point sets of anti-symplectic involutions in the Hilbert scheme of points on a surface

本文研究了曲面上点的Hilbert概形中反辛对合不动点集的上同调，用不动点集和商空间的混合Hodge结构计算了其上同调，分类了连通分支并确定了其结构。

更多双有理对合

More birational involutions

本文针对非常一般的极化K3曲面，用几何方法描述了其Hilbert概形上的双有理对合，并证明其关键特征对$n\ge4$仍成立，且与模空间上的$\mathbb{P}^2$-纤维化双有理等价。

K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space II: a structure theorem for r(M)>10

We study the structure of the invariant of K3 surfaces with involution, which we obtained using equivariant analytic torsion. It was known before that the invariant is expressed as the Petersson norm of an automorphic form on the moduli space. When the rank of the invariant sublattice of the K3-lattice with respect to the involution is strictly bigger than 10, we prove that this automorphic form is expressed as the tensor product of an explicit Borcherds lift and Igusa's Siegel modular form.

Spectral involutions on rational elliptic surfaces

In this paper we describe a four dimensional family of special rational elliptic surfaces admitting an involution with isolated fixed points. For each surface in this family we calculate explicitly the action of a spectral version of the involution (namely of its Fourier-Mukai conjugate) on global line bundles and on spectral data. The calculation is carried out both on the level of cohomology and in the derived category. We find that the spectral involution behaves like a fairly simple affine transformation away from the union of those fiber components which do not intersect the zero section.…