Rationality of moduli of elliptic fibrations with fixed monodromy

Bounding non-rationality of divisors on 3-fold Fano fibrations

In this paper we investigate non-rationality of divisors on 3-fold log Fano fibrations $(X,B)\to Z$ under mild conditions. We show that if $D$ is a component of $B$ with coefficient $\ge t>0$ which is contracted to a point on $Z$, then $D$ is birational to $\mathbb{P}^1\times C$ where $C$ is a smooth projective curve with gonality bounded depending only on $t$. Moreover, if $t>\frac{1}{2}$, then genus of $C$ is bounded depending only on $t$.

Birational geometry of del Pezzo fibrations with terminal quotient singularities

Del Pezzo fibrations appear as minimal models of rationally connected varieties. The rationality of smooth del Pezzo fibrations is a well studied question but smooth fibrations are not dense in moduli. Little is known about the rationality of the singular models. We prove birational rigidity, hence non-rationality, of del Pezzo fibrations with simple non-Gorenstein singularities satisfying the famous $K^2$-condition. We then apply this result to study embeddings of $\operatorname{PSL}_2(7)$ into the Cremona group.

Monodromy Invariants and Polarization Types of Generalized Kummer Fibrations

In this paper a monodromy invariant for isotropic classes on generalized Kummer type manifolds is constructed. This invariant is used to determine the polarization type of Lagrangian fibrations on such manifolds - a notion which was introduced in an earlier paper of the author. The result shows that the polarization type of a Lagrangian fibration of generalized Kummer type depends on the connected component of the moduli space.

Non-rationality of some fibrations associated to Klein surfaces

We study the polynomial fibration induced by the equation of the Klein surfaces obtained as quotient of finite linear groups of automorphisms of the plane; this surfaces are of type A, D, E, corresponding to their singularities. The generic fibre of the polynomial fibration is a surface defined over the function field of the line. We proved that it is not rational in cases D, E, although it is obviously rational in the case A. The group of automorphisms of the Klein surfaces is also described, and is linear and of finite dimension in cases D, E; this result being obviously false in case A.

Dihedral Monodromy and Xiao Fibrations

We construct three new families of fibrations $π: S \to B$ where $S$ is an algebraic complex surface and $B$ a curve that violate Xiao's conjecture relating the relative irregularity and the genus of the general fiber. The fibers of $π$ are certain étale cyclic covers of hyperelliptic curves that give coverings of $P^1$ with dihedral monodromy. As an application, we also show the existence of big and nef effective divisors in the Brill-Noether range.

On rationality of nonsingular threefolds with a pencil of Del Pezzo surfaces of degree 4

We prove a criterion of nonsingularity of a complete intersection of two fiberwise quadrics in a scroll over $P^1$. As a corollary we derive the following addition to the Alexeev theorem on rationality of standard Del Pezzo fibrations of degree 4 over $P^1$: we prove that any fibration of this kind with the topological Euler characteristic -4 is rational.

Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems

Let X -> Y be a fibration whose fibers are complete intersections of two quadrics. We develop new categorical and algebraic tools---a theory of relative homological projective duality and the Morita invariance of the even Clifford algebra under quadric reduction by hyperbolic splitting---to study semiorthogonal decompositions of the bounded derived category of X. Together with new results in the theory of quadratic forms, we apply these tools in the case where X -> Y has relative dimension 1, 2, or 3, in which case the fibers are curves of genus 1, Del Pezzo surfaces of degree 4, or Fano three…

Monodromy of elliptic surfaces

We study monodromy groups of elliptic fibrations over the projective line.

Massey Products and Fujita decompositions on fibrations of curves

Let $f:S\to B$ be a fibration of curves and let $f_\astω_{S/B}=\cU\oplus \cA$ be the second Fujita decomposition of $f.$ In this paper we study a kind of Massey products, which are defined as infinitesimal invariants by the cohomology of a curve, in relation to the monodromy of certain subbundles of $\cU$. The main result states that their vanishing on a general fibre of $f$ implies that the monodromy group acts faithfully on a finite set of morphisms and is therefore finite. In the last part we apply our result in terms of the normal function induced by the Ceresa cycle. On the one hand, we p…

Monodromy of the $SL(n)$ and $GL(n)$ Hitchin fibrations

We compute the monodromy of the Hitchin fibration for the moduli space of $L$-twisted $SL(n,\mathbb{C})$ and $GL(n,\mathbb{C})$-Higgs bundles for any $n$, on a compact Riemann surface of genus $g>1$. We require the line bundle $L$ to either be the canonical bundle or satisfy $deg(L) > 2g-2$. The monodromy group is generated by Picard-Lefschetz transformations associated to vanishing cycles of singular spectral curves. We construct such vanishing cycles explicitly and use this to show that the $SL(n,\mathbb{C})$ monodromy group is a {\em skew-symmetric vanishing lattice} in the sense of Janssen…

Arithmeticity of the monodromy of some Kodaira fibrations

A question of Griffiths-Schmid asks when the monodromy group of an algebraic family of complex varieties is arithmetic. We resolve this in the affirmative for the class of algebraic surfaces known as Atiyah-Kodaira manifolds, which have base and fibers equal to complete algebraic curves. Our methods are topological in nature and involve an analysis of the "geometric" monodromy, valued in the mapping class group of the fiber.

Singular fibers in algebraic fibrations of genus 2 and their monodromy factorizations

Kodaira's classification of singular fibers in elliptic fibrations and its translation into the language of monodromies and Lefschetz fibrations has been a boon to the study of 4-manifolds. In this article, we begin the work of translating between singular fibers of genus 2 families of algebraic curves and the positive Dehn twist factorizations of Lefschetz fibrations for a certain subset of the singularities described by Namikawa and Ueno in the 70s. We look at four families of hypersurface singularities in $\mathbb{C}^3$. Each hypersurface comes equipped with a fibration by genus 2 algebraic…

Monodromy of Rank 2 Parabolic Hitchin Systems

We study the monodromy of the Hitchin fibration for moduli spaces of parabolic G-Higgs bundles in the cases when G=SL(2,R), GL(2,R) and PGL(2,R) A calculation of the orbits of the monodromy with Z2-coefficients provides an exact count of the components of the moduli spaces for these groups.

Xiao的亏格二纤维化：分支覆盖与辫子单值性

Xiao's Genus-Two Fibration: Branched Covers and Braid Monodromy

本文通过分支覆盖构造和辫子单值性方法，直接计算Xiao亏格二Lefschetz纤维化的几何单值性分解，得到类型$(4,3)$的有序正分解及七个因子的Artin坐标。

Semistable fibrations over an elliptic curve with only one singular fibre

In this work we describe a construction of semistable fibrations over an elliptic curve with one unique singular fibre and we give effective examples using monodromy of curves.

On the rationality of algebraic monodromy groups of compatible systems

Let $E$ be a number field and $X$ a smooth geometrically connected variety defined over a characteristic $p$ finite field. Given an $n$-dimensional pure $E$-compatible system of semisimple $λ$-adic representations of the étale fundamental group of $X$ with connected algebraic monodromy groups $G_λ$, we construct a common $E$-form $G$ of all the groups $G_λ$ and in the absolutely irreducible case, a common $E$-form $G\hookrightarrow\text{GL}_{n,E}$ of all the tautological representations $G_λ\hookrightarrow\text{GL}_{n,E_λ}$ (Theorem 1.1). Analogous rationality results in characteristic $p$ ass…

Monodromy of plane curves and quasi-ordinary surfaces

We develop recursive formulas for the horizontal and vertical monodromies of a quasi-ordinary surface. These are monodromies associated to the Milnor fiber of a slice transverse to a component of the singular locus. In the course of working out these recursions, we have discovered what appears to be a new way to express the monodromy associated to the Milnor fibration of a singular plane curve.

Monodromy and birational geometry of O'Grady's sixfolds

We prove that the bimeromorphic class of a hyperkähler manifold deformation equivalent to O'Grady's six dimensional one is determined by the Hodge structure of its Beauville-Bogomolov lattice by showing that the monodromy group is maximal. As applications, we give the structure for the Kähler and the birational Kähler cones in this deformation class and we prove that the existence of a square zero divisor implies the existence a rational lagrangian fibration with fixed fibre types.

Families of lattice polarized K3 surfaces with monodromy

We extend the notion of lattice polarization for K3 surfaces to families over a (not necessarily simply connected) base, in a way that gives control over the action of monodromy on the algebraic cycles, and discuss the uses of this new theory in the study of families of K3 surfaces admitting fibrewise symplectic automorphisms. We then give an application of these ideas to the study of Calabi-Yau threefolds admitting fibrations by lattice polarized K3 surfaces.

P=W for Lagrangian fibrations and degenerations of hyper-Kähler manifolds

We identify the perverse filtration of a Lagrangian fibration with the monodromy weight filtration of a maximally unipotent degeneration of compact hyper-Kähler manifolds.

Rationality of the K-theoretical capped vertex function for Nakajima quiver varieties

In this paper we prove the rationality of the capped vertex function with descendents for arbitrary Nakajima quiver varieties with generic stability conditions. We generalise the proof given by Smirnov to the general case, which requires to use techniques of tautological classes rather than the fixed-point basis. This result confirms that the "monodromy" of the capped vertex function is trivial, which gives a strong constraint for the monodromy of the capping operators. We will also provide a GIT wall-crossing formula for the capped vertex function in terms of the quantum difference operators …

具有平凡代数单值性的超平面配置的 Milnor 纤维化

Milnor fibrations of arrangements with trivial algebraic monodromy

本文研究具有平凡代数单值性的超平面配置 Milnor 纤维化，重点分析上同调跳跃轨迹与基本群下中心序列商，并构造出 Betti 数相同但基本群不同构的一对配置。

Xiao 亏格二纤维化在次数 3、4 和 5 下的单值性

The monodromy of Xiao's genus-two fibrations in degrees 3, 4 and 5

本文通过平铺基空间和分支覆盖方法，确定了 Xiao 亏格二纤维化在次数 3、4、5 下的消失循环，并给出了新的正分解。

通过分支覆盖的扰动分裂代数奇异纤维化

Splitting Algebraic Singular Fibrations via Perturbation of Branch Covers

本文针对亏格二纤维化中另外两类奇异纤维，改用非超椭圆多项式族重建曲线，通过分支覆盖扰动将其形变为 Lefschetz 纤维化，并恢复中心纤维紧致化行为及相应单值分解。

Moduli of elliptic $K3$ surfaces: monodromy and Shimada root lattice strata

In this paper we investigate two stratifications of the moduli space of elliptically fibred K3 surfaces. The first comes from Shimada's classification of connected components of elliptically fibred K3 surfaces and is closely related to the root lattice of the fibration. The second is the monodromy stratification defined by Bogomolov, Petrov and Tschinkel. The main result of the paper is a classification of all positive-dimensional ambi-typical strata, that is strata which are both Shimada root strata and monodromy strata. We further discuss the connection with moduli spaces of lattice-polarise…

Mapping class groups, multiple Kodaira fibrations, and CAT(0) spaces

We study several geometric and group theoretical problems related to Kodaira fibrations, to more general families of Riemann surfaces, and to surface-by-surface groups. First we provide constraints on Kodaira fibrations that fiber in more than two distinct ways, addressing a question by Catanese and Salter about their existence. Then we show that if the fundamental group of a surface bundle over a surface is a ${\rm CAT}(0)$ group, the bundle must have injective monodromy (unless the monodromy has finite image). Finally, given a family of closed Riemann surfaces (of genus $\ge 2$) with injecti…

2次del Pezzo曲面丛与稳定有理性

Degree 2 del Pezzo surface bundles and stable rationality

本文研究函数域上2次del Pezzo曲面丛的算术性质，通过构造具有非平凡相对非分支上同调的例子，证明特殊化到该簇的簇不稳定有理性。

The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy

A central question in arrangement theory is to determine whether the characteristic polynomial $Δ_q$ of the algebraic monodromy acting on the homology group $H_q(F(\mathcal{A}),\mathbb{C})$ of the Milnor fiber of a complex hyperplane arrangement $\mathcal{A}$ is determined by the intersection lattice $L(\mathcal{A})$. Under simple combinatorial conditions, we show that the multiplicities of the factors of $Δ_1$ corresponding to certain eigenvalues of order a power of a prime $p$ are equal to the Aomoto--Betti numbers $β_p(\mathcal{A})$, which in turn are extracted from $L(\mathcal{A})$. When $…

The Picard-Lefschetz formula and a conjecture of Kato, the case of Lefschetz fibrations

A conjecture of Kato says that the monodromy operator on the cohomology of a semi-stable degeneration of projective varieties is represented by an algebraic cycle on the special fiber of a normal crossing model of the fiber product degeneration. We prove this conjecture in the simple case of a semi-stable degeneration arising from a Lefschetz fibration.

Symplectic Lefschetz fibrations with arbitrary fundamental groups

In this paper we give an explicit construction of a symplectic Lefschetz fibration whose total space is a smooth compact four dimensional manifold with a prescribed fundamental group. We also study the numerical properties of the sections in symplectic Lefschetz fibrations and their relation to the structure of the monodromy group.

Normal Factorization in $SL(2,Z)$ and the Confluence of Singular Fibers in Elliptic Fibrations

In this article we obtain a result about the uniqueness of factorization in terms of conjugates of the matrix $U=(\xymatrix{1 & 1 0 & 1})$, of some matrices representing the conjugacy classes of those elements of $SL(2,Z)$ arising as the monodromy around a singular fiber in an elliptic fibration (i.e. those matrices that appear in Kodaira's list).

椭圆纤维化的辫子单项群中的挠元

Torsion in the Braid Monodromy of Elliptic Fibrations

本文研究了椭圆纤维化的辫子单项群，通过将其共轭类与$\operatorname{SL}_2$-特征簇上的特殊点相联系，分类了其中的 $n$ 阶元素，并证明了该群中不存在 $n-1$ 或 $n-2$ 阶元素。

Moduli spaces of surfaces and monodromy invariants

This paper is a survey of the authors' recent results on "abc-surfaces" and the monodromy of their natural Lefschetz fibrations and projections to P^1 x P^1, see (arXiv:0910.2142). The results being surveyed explore various fundamental questions about complex deformation vs. symplectomorphism vs. diffeomorphism from the perspective of one explicit family of examples; the paper also contains some remarks which were not in earlier papers.

论四维Dehn扭转与Milnor纤维化

On four-dimensional Dehn twists and Milnor fibrations

该研究借助磁单极Floer同调进展计算族Seiberg--Witten不变量，证明极小椭圆奇点链不定辛填充的边界Dehn扭转无限阶非平凡，揭示了四维空间中三类全新的拓扑与辛几何现象。

有理纤维化三次四折体

Rational fibered cubic fourfolds

该研究在三次四折体模空间中探究多类已知有理三次除子的交，刻画其不可约分支，给出新的有理del Pezzo曲面或二次曲面纤维化三次四折体显式例子，并研究纤维化有理截面与Brauer类的对应关系。

K3曲面的退化与纤维化：格极化与镜像对称

Degenerations and Fibrations of K3 Surfaces: Lattice Polarisations and Mirror Symmetry

本文研究K3曲面Tyurin退化及椭圆纤维化的格极化理论，建立兼容现有定义的格极化概念，并探讨两种设置间的镜像对称对应。

单例和消失环对于足够 ample 线性系统在单连通表面上的 Monodromy

Monodromy and vanishing cycles for sufficiently ample linear systems on simply connected surfaces

研究了单连通表面中足够 ample 线性系统的 Monodromy 与 r-spin 映射类群的关系，并探讨了消失环在节点退化中的应用。

Veech纤维化

Veech fibrations

本文研究以Teichmüller曲线为底、一般纤维为Veech曲面的复曲面，通过纤维模m同调的单值群作用计算其拓扑与复几何不变量，得到所有已知代数本原Teichmüller曲线对应不变量的精确值。

Hyperplane arrangements and Milnor fibrations

There are several topological spaces associated to a complex hyperplane arrangement: the complement and its boundary manifold, as well as the Milnor fiber and its own boundary. All these spaces are related in various ways, primarily by a set of interlocking fibrations. We use cohomology with coefficients in rank 1 local systems on the complement of the arrangement to gain information on the homology of the other three spaces, and on the monodromy operators of the various fibrations.

Calabi--Yau 3-folds of Borcea--Voisin type and elliptic fibrations

We consider Calabi--Yau 3-folds of Borcea--Voisin type, i.e. Calabi--Yau 3-folds obtained as crepant resolutions of a quotient $(S\times E)/(α_S\times α_E)$, where $S$ is a K3 surface, $E$ is an elliptic curve, $α_S\in {\rm Aut} (S)$ and $α_E\in {\rm Aut}(E)$ act on the period of $S$ and $E$ respectively with order $n=2,3,4,6$. The case $n=2$ is very classical, the case $n=3$ was recently studied by Rohde, the other cases are less known. First we construct explicitly a crepant resolution, $X$, of $(S\times E)/(α_S\times α_E)$ and we compute its Hodge numbers; some pairs of Hodge numbers we fou…

Hitchin fibrations, abelian surfaces, and the P=W conjecture

We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus $2$ curves and arbitrary rank. In higher genus and arbitrary rank, we prove that P=W holds for the subalgebra of cohomology generated by even tautological classes. Furthermore, we show that all tautological generators lie in the correct pieces of the perverse filtration as predicted by the P=W conjecture. In combination with recent work of Mellit, this reduces the full conjecture to the multiplicativity of the perverse filtration. Our main technique is to study the Hitchin fibration as a…

The monodromy theorem for compact Kähler manifolds and smooth quasi-projective varieties

Given any connected topological space $X$, assume that there exists an epimorphism $ϕ: π_1(X) \to \mathbb{Z}$. The deck transformation group $\mathbb{Z}$ acts on the associated infinite cyclic cover $X^ϕ$ of $X$, hence on the homology group $H_i(X^ϕ, \mathbb{C})$. This action induces a linear automorphism on the torsion part of the homology group as a module over the Laurent ring $\mathbb{C}[t,t^{-1}]$, which is a finite dimensional $\mathbb{C}$-vector space. We study the sizes of the Jordan blocks of this linear automorphism. When $X$ is a compact Kähler manifold, we show that all the Jordan …

Twelve points on the projective line, branched covers, and rational elliptic fibrations

The following divisors in the space Sym^{12} P^1 of twelve points on P^1 are actually the same: (A) the possible locus of the twelve nodal fibers in a rational elliptic fibration (i.e. a pencil of plane cubic curves); (B) degree 12 binary forms that can be expressed as a cube plus a square; (C) the locus of the twelve tangents to a smooth plane quartic from a general point of the plane; (D) the branch locus of a degree 4 map from a hyperelliptic genus 3 curve to P^1; (E) the branch locus of a degree 3 map from a genus 4 curve to P^1 induced by a theta-characteristic; and several more. The corr…

$\mathbb{P}^2$ 中一般位置直线乘积的单值问题与切向中心焦点问题

Monodromy problem and Tangential center-focus problem for product of generic lines in $\mathbb{P}^2$

本文研究一般位置直线乘积商定义的有理映射相关纤维化的单值问题与切向中心焦点问题，刻画了正则纤维1-同调群中由单值作用生成的子模，以及使阿贝尔积分在中心奇点循环族上消失的亚纯1-形式。

Kodaira fibrations and beyond: methods for moduli theory

Kodaira fibred surfaces are a remarkable example of projective classifying spaces, and there are still many intriguing open questions concerning them, especially the slope question. The topological characterization of Kodaira fibrations is emblematic of the use of topological methods in the study of moduli spaces of surfaces and higher dimensional complex algebraic varieties, and their compactifications. The paper contains some new results but is mostly a survey paper, dealing with fibrations, questions on monodromy and factorizations in the mapping class group, old and new results on Variatio…

Moduli spaces and braid monodromy types of bidouble covers of the quadric

Bidouble covers $π: S \mapsto Q$ of the quadric Q are parametrized by connected families depending on four positive integers a,b,c,d. In the special case where b=d we call them abc-surfaces. Such a Galois covering $π$ admits a small perturbation yielding a general 4-tuple covering of Q with branch curve $\De$, and a natural Lefschetz fibration obtained from a small perturbation of the composition of $ π$ with the first projection. We prove a more general result implying that the braid monodromy factorization corresponding to $\De$ determines the three integers a,b,c in the case of abc-surfaces…

BMY线上的小Lefschetz纤维化

Small Lefschetz fibrations on the BMY line

本文构造了环面上的亏格19和亏格2曲面上的亏格4 Lefschetz纤维化族，均只有三个节点，位于BMY线上，并给出其拓扑、实商及单值分解等性质。

纤维积的Crepan t 消解及具有非约化纤维的有理椭圆曲面的Kummer纤维化

Crepant resolutions of fiber products and Kummer fibrations of rational elliptic surfaces with non-reduced fibers

本文研究有理椭圆曲面纤维积的 crepant 消解与 Kummer 商，给出局部模型、Hodge 数公式及特征 5、7 下的约化结果。

Trecce, Mapping class group, fibrazioni di Lefschetz ed applicazioni al diffeomorfismo di superficie algebriche

Purpose of the Conference article, intended for a wider audience, is to introduce concepts and techniques used by Bronislaw Wajnryb and the author in order to show the diffeomorphism of certain elementary algebraic surfaces, called ABC surfaces, which are not deformation equivalent. In the first part are recalled the concepts of connected sum, surgeries, Poincare' group, Braids, Riemann surfaces and Dehn's Mapping class group. Then the topics become more advanced, including Lefschetz fibrations, ABC surfaces, symplectic fibrations and braid monodromy.

高阶局部系统与范畴化的单值性等价

Higher local systems and the categorified monodromy equivalence

本研究探讨空间上$(\u221e,n)$-范畴的局部系统，证明其可由高阶单值性数据刻画，推广经典单值性等价，拓展Teleman的拓扑群作用理论，并给出可逆对象群的同构结论及在辛纤维化纤维式Fukaya范畴的应用。

柏拉图椭圆曲面

The platonic elliptic surfaces

本文构造了附属于柏拉图立体的特殊有理椭圆曲面，研究其代数几何性质等，发现需改进Dwork展开以确定其纤维化纤维的欧拉因子。