A note on surfaces with $p_g=q=2$ and an irrational fibration

A note on Lefschetz fibrations on algebraic surfaces

Let $S$ be a smooth projective complex algebraic surface and $f\, :\, S\, \longrightarrow\, {\mathbb C}{\mathbb P}^2$ a finite map. Consider a pencil of hyperplane sections on ${\mathbb C}{\mathbb P}^2$ and pull it back to $S$. We address the question whether such a pencil is a Lefschetz pencil on $S$.

关于曲面上Campana一般型纤维化的注记

A note on fibrations of Campana general type on surfaces

本文构造了射影直线上亏格2的Campana一般型纤维化的单连通曲面例子，其特殊纤维分量重数最小≥2且gcd=1，并推广到任意偶数亏格。

Notes on hyperelliptic fibrations of genus 3, I

We shall study the structure of hyperelliptic fibrations of genus 3, from the view point given by Catanese and Pignatelli in arXiv:math/0503294. In this part I, we shall give a structure theorem for such fibrations for the case of f : S \to B with all fibers 2-connected. We shall also give, for the case of B projective line, sufficient conditions for the existence of such fibrations from the view point of our structure theorem, prove the uniqueness of the deformation type and the simply connectedness of S for some cases, and give some examples including those with minimal regular S with geomet…

Notes on hyperelliptic fibrations of genus 3, II

This is the part II of the series under the same title. In part I, using the approach developed by Catanese--Pignatelli arXiv:math/0503294, we gave a structure theorem for hyperelliptic genus 3 fibrations all of whose fibers are 2-connected (arXiv:1209.6278 [math.AG]). In this part II, we shall give a structure theorem for smooth deformation families of these to non-hyperelliptic genus 3 fibrations. As an application, we shall give a set of sufficient conditions for our genus 3 hyperelliptic fibration above to allow deformation to non-hyperelliptic fibrations, and use this to study certain min…

Birational models of del Pezzo fibrations

This is a preliminary version of the paper for the Lecture Notes Series.

Hyper-Kaehler compactification of the intermediate Jacobian fibration of a cubic fourfold : the twisted case

The starting point of this note is our recent paper with Laza and Saccà on the construction of deformations of O'Grady's $10$-dimensional manifolds as compactifications of intermediate Jacobian fibrations associated to cubic fourfolds. The note provides a complement to that paper consisting in the analogous construction in the twisted case, leading to isogenous but presumably not isomorphic or birational hyper-Kähler manifolds.

关于 lc-trivial 纤维化的一个注记

A note on lc-trivial fibrations

本文研究 lc pair 上的 lc-trivial 纤维化，证明经基变换后可选取仅依赖维数、Cartier 指数与一般纤维的正整数 $n$，使 $n(K_X+Δ)$ 等价于 Cartier 除子的拉回。

A remark on Ehresmann's Fibration Theorem

This note records that in the setting of complex varieties, the cohomological consequence of Ehresmann's fibration theorem holds without the smooth assumption on the base or the total space.

A remark on a question of Beauville about lagrangian fibrations

This note is a proof of the fact that a lagrangian torus on a hyperkaehler fourfold is always a fiber of an almost holomorphic lagrangian fibration.

A note on local trigonal fibrations

We show that some of centeral fibers of degenerations of hyperelliptic curves are realized as those trigonal curves. In particular, any hyperelliptic curve can be the central fiber of a degeneration of trigonal curves.

A Note on Lefschetz Fibrations on Compact Stein 4-Manifolds

Loi-Piergallini and Akbulut-Ozbagci showed that every compact Stein surface admits a Lefschetz fibration over the 2-disk with bounded fibers. In this note we give a more intrinsic alternative proof of this result.

One remark on the b-semiampleness of the moduli part

In this short note we reduce the b-semiampleness conjecture for lc-trivial fibrations to the b-semiampleness conjecture for klt-trivial fibrations.

关于Fano纤维化纤维乘积的乘积

On multiplicities of fibers of Fano fibrations

本文将birational几何中的多个猜想，包括Shokurov关于log Calabi-Yau纤维化基的奇点猜想和有理连通Calabi-Yau品种的有界猜想，归结为Fano纤维化过曲线的纤维乘积猜想。

A note on a question of Shioda about integral sections

We consider a rational elliptic surface with a relatively minimal fibration. We compute the number of integral sections in the above rational elliptic surface. As an application, we obtain an estimate of polynomial solutions of some equations.

A note on the fundamental group of Kodaira fibrations

The fundamental group $π$ of a Kodaira fibration is, by definition, the extension of a surface group $Π_b$ by another surface group $Π_g$, i.e. \[ 1 \rightarrow Π_g \rightarrow π\rightarrow Π_b \rightarrow 1. \] Conversely, we can inquire about what conditions need to be satisfied by a group of that sort in order to be the fundamental group of a Kodaira fibration. In this short note we collect some restriction on the image of the classifying map $m \colon Π_b \to Γ_g$ in terms of the coinvariant homology of $Π_g$. In particular, we observe that if $π$ is the fundamental group of a Kodaira fibr…

Maximal rationally connected fibrations and movable curves

A well known result of Miyaoka asserts that a complex projective manifold is uniruled if its cotangent bundle restricted to a general complete intersection curve is not nef. Using the Harder-Narasimhan filtration of the tangent bundle, it can moreover be shown that the choice of such a curve gives rise to a rationally connected foliation of the manifold. In this note we show that, conversely, a movable curve can be found so that the maximal rationally connected fibration of the manifold may be recovered as a term of the associated Harder-Narasimhan filtration of the tangent bundle.

拉格朗日纤维化到具有孤立商奇点的变分上

Lagrangian Fibrations onto Varieties with Isolated Quotient Singularities

本文证明了具有孤立商奇点的变分在拉格朗日纤维化下必为光滑，特别地，超Kähler四维叠的拉格朗日纤维化目标表面必为$\mathbb{P}^2$。

Linear Toric Fibrations

These notes are based on three lectures given at the 2013 CIME/CIRM summer school. The purpose of this series of lectures is to introduce the notion of a toric fibration and to give its geometrical and combinatorial characterizations. Polarized toric varieties which are birationally equivalent to projective toric bundles are associated to a class of polytopes called Cayley polytopes. Their geometry and combinatorics have a fruitful interplay leading to fundamental insight in both directions. These notes will illustrate geometrical phenomena, in algebraic geometry and neighboring fields, which …

关于Nguyen关于曲面纤维化猜想的一个新反例

A new counterexample to Nguyen's conjecture on surface fibration

本文针对Nguyen关于具有3条奇异纤维的曲面纤维化的猜想，构造了亏格$g=2$时的一个新反例。

A note on affine quotients and equivariant double fibrations

We consider two linear reductive algebraic groups $ G $ and $ G' $ over $ C $. Take a finite dimensional rational representation $ W $ of $ G \times G' $. Let $ Y = W // G := Spec C[W]^G $ and $ X = W // G' := \Spec C[W]^{G'} $ be the affine quotients. The quotient space $ X $ (respectively $ Y $) naturally inherits the action of $ G $ (respectively $ G' $). In this note, we study the interrelation between the orbit structures of $ X / G $ and $ Y / G' $. In a good situation, we can embed $ Y / G' $ into $ X / G $, and the embedding map preserves important properties such as the closure relati…

Kimura-finiteness of quadric fibrations over smooth curves

In this short note, making use of the recent theory of noncommutative mixed motives, we prove that the Voevodsky's mixed motive of a quadric fibration over a smooth curve is Kimura-finite.

A short proof of a conjecture of Matsushita

In this note we build on the arguments of van Geemen and Voisin to prove a conjecture of Matsushita that a Lagrangian fibration of an irreducible hyperkähler manifold is either isotrivial or of maximal variation. We also complete a partial result of Voisin regarding the density of torsion points of sections of Lagrangian fibrations.

直线上的雅可比纤维化截面

Sections of Jacobian fibrations over lines

本文研究光滑曲面线性系中直线对应的通用雅可比截面的上同调类，针对K3曲面的通用紧化雅可比，修正了排印错误与猜想7，探讨结果与文献步骤的协调问题。

A Note on a Conjecture of Xiao

Let f:S ->B be a relatively minimal fibred surface. In this note we give a partial affirmative answer to a conjecture of Xiao, proving that the direct image of the relative dualizing sheaf of $f$ is ample when the slope of the fibration is less than 4, if the general fibre of $f$ is non-hyperelliptic or the genus of the fibre or of the base curve is low.

A note on Iitaka's conjecture $C_{3,1}$ in positive characteristics

Let $f:X\to Y$ be a fibration from a smooth projective 3-fold to a smooth projective curve, over an algebraically closed field $k$ of characteristic $p >5$. We prove that if the generic fiber $X_η$ has big canonical divisor $K_{X_η}$, then $$κ(X)\geκ(Y) + κ(X_η).$$

The Kodaira Dimension of Lefschetz Fibrations

In this note, we verify that the complex Kodaira dimension $κ^h$ equals the symplectic Kodaira dimension $κ^s$ for smooth 4-manifolds with complex and symplectic structures. We also calculate the Kodaira dimension for many Lefschetz fibrations.

The Kobayashi pseudometric on algebraic manifold and a canonical fibration

Given a compact complex manifold X of dimension n, we define a bimeromorphic invariant $κ_+(X)$ as the maximum p for which there is a saturated line subsheaf L of the sheaf of holomorphic p forms whose Kodaira dimension $κ(L)$ equals p. We call X special if $κ_+(X)=0$. We give some evidence that this condition characterizes X to have identically vanishing Kobayashi pseudometric. We use the well-known construction of F. Campana to give, for each projective X a canonical fibration $f: X\to Y$ holomorphic outside a proper subvariety of Y and whose general fibers are special. We show that the inhe…

A note on homological mirror symmetry for singularities of type D

We prove homological mirror symmetry for Lefschetz fibrations obtained as disconnected sums of polynomials of types A or D. The proof is based on the behavior of the Fukaya category under the addition of a polynomial of type D.

A Note On Lagrangian Vanishing Sphere Bundles

A classical way to construct a Lagrangian in a symplectic manifold $Σ$ is to let $Σ$ appear as a smooth fiber in a Lefschetz fibration. If this is possible the singularities of the fibration induce Lagrangian spheres in $Σ$ and these spheres, in turn, are representatives of the corresponding vanishing cycles in the homology of $Σ$. In this paper our aim is twofold: The first is to describe a generalization of the above mentioned construction to the "Morse-Bott" case. This leads, whenever such a degeneration exists, to the existence of Lagrangian sphere bundles rather than just spheres. In the …

Triviality of fibered Calabi-Yau manifolds without singular fibers

In this note we show that if a compact Kahler manifold with trivial canonical bundle is the total space of a holomorphic fibration without singular fibers, then the fibration is a holomorphic fiber bundle. In the algebraic case, the fibration becomes trivial after a finite base change.

Hermitian Metrics of Positive Holomorphic Sectional Curvature on Fibrations

The main result of this note essentially is that if the base and fibers of a compact fibration carry Hermitian metrics of positive holomorphic sectional curvature, then so does the total space of the fibration. The proof is based on the use of a warped product metric as in the work by Cheung in case of negative holomorphic sectional curvature, but differs in certain key aspects, e.g., in that it does not use the subadditivity property for holomorphic sectional curvature due to Grauert-Reckziegel and Wu.

Kähler态射的射影性准则

Projectivity criteria for Kähler morphisms

本文证明了轻微奇异紧Kähler空间间纤维化的两个射影性准则，即Kodaira和Moishezon准则的相对版本，并得出MRC纤维化总存在射影态射模型。

A note on rank two stable bundles over surfaces

Let $π: X \longrightarrow C$ be a fibration with reduced fibers over a curve $C$ and consider a polarization $H$ on the surface $X$. Let $E$ be a stable vector bundle of rank $2$ on $C$. We prove that the pullback $π^*E$ is a $H-$stable bundle over $X$. This result allows us to relate the corresponding moduli spaces of stable bundles $\mathcal{M}_C(2,d)$ and $\mathcal{M}_{X,H}(2,df,0)$ through an injective morphism. We study the induced morphism at the level of Brill-Noether loci to construct examples of Brill-Noether loci on fibered surfaces. Results concerning the emptiness of Brill-Noether …

自反多边形与有理椭圆曲面

Reflexive polygons and rational elliptic surfaces

本文研究了与十六个自反多边形相关的八个有理椭圆曲面，其椭圆纤维化在镜像对称下对应八个光滑 del Pezzo 曲面族。

关于零循环的 Albanese 映射的注记

Notes sur l'application d'Albanese pour les zéro-cycles

本文研究任意域上光滑射影簇零循环 Chow 群到 Albanese 簇有理点群的映射满射性，并借助 Severi–Brauer 纤维化与函数域扩张联系普遍零循环及三维簇有理性。

A note on Harris Morrison sweeping families of maximal gonality

Harris and Morrison constructed semistable families f:F \to Y of k-gonal curves of genus g such that for every k the corresponding modular curves give a sweeping family in the k-gonal locus in the moduli space. Their construction depends on the choice of a smooth curve X. We show that if the genus g(X) is sufficiently high with respect to g, then the ratio K_F^2 / χ(O_F) is 8 asymptotically with respect to g(X). We show also that if the gonality is maximal and some conjectured estimates of Harris and Morrison hold, the slope of the fibration f: F\to Y is 12 asymptotically with respect to g and…

A refined Kodaira dimension and its canonical fibration

Given a (meromorphic) fibration $f:X\to Y$ where $X$ and $Y$ are compact complex manifolds of dimensions $n$ and $m$, we define $L_f$ to be the invertible subsheaf of the sheaf of holomorphic $m$-forms of $X$ given by the saturation of $f^*K_Y$, where $K_Y$ is the canonical sheaf of $Y$. We define the Kodaira dimension of the orbifold base $(Y,f)$ by that of $L_f$ and call the fibration to be of general type if this dimension is the maximal $\dime Y$. We call $X$ special if it does not have a general type fibration with positive dimensional base and call $f$ special if its general fibers are. …

On Degenerations of Fibre Spaces of Curves of Genus >= 2

In this note, we show that for surfaces admitting suitable fibrations, any given degeneration X / Delta is bimeromorphic to a fiber space over Delta and we apply this result to the study of the degenerate fiber.