Monodromy Invariants and Polarization Types of Generalized Kummer Fibrations
Benjamin Wieneck
备注：
This is the final version of the paper. Some improvements, thanks to the referee. The paper was published in Mathematische Zeitschrift. Please note that my Leibniz Universität email address (@math.uni-hannover.de) does not work anymore, Math. Z. (2018)
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.
A Kodaira fibration is a non-isotrivial fibration $f\colon S\rightarrow B$ from a smooth algebraic surface $S$ to a smooth algebraic curve $B$ such that all fibers are smooth algebraic curves of genus $g$. Such fibrations arise as complete curves inside the moduli space $\mathcal{M}_g$ of genus $g$ algebraic curves. We investigate here the possible connected monodromy groups of a Kodaira fibration in the case $g=3$ and classify which such groups can arise from a Kodaira fibration obtained as a general complete intersection curve inside a subvariety of $\mathcal{M}_3$ parametrizing curves whose…
我们构造了三个新的纤维族$π: S \to B$，其中$S$是一个代数复曲面，$B$是一条曲线，它违反了肖关于一般纤维的相对不规则性和亏格的猜想. $π$的纤维是某些超椭圆曲线的三角循环覆盖，它们给出了$P^1$的具有二面角单值的覆盖。作为应用，我们还证明了Brill-Noether范围内大有效因子和nef有效因子的存在性。
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…
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.
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…
On the motive of Kummer varieties associated to $Γ_1(7)$ - Supplement to the paper: The modularity of certain non-rigid Calabi-Yau threefolds (by R. Livné and N. Yui)
We identify the perverse filtration of a Lagrangian fibration with the monodromy weight filtration of a maximally unipotent degeneration of compact hyper-Kähler manifolds.
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…
We use the mixed-twist construction of Doran and Malmendier to obtain a multi-parameter family of K3 surfaces of Picard rank $ρ\ge 16$. Upon identifying a particular Jacobian elliptic fibration on its general member, we determine the lattice polarization and the Picard-Fuchs system for the family. We construct a sequence of restrictions that lead to extensions of the polarization by two-elementary lattices. We show that the Picard-Fuchs operators for the restricted families coincide with known resonant hypergeometric systems. Second, for the one-parameter mirror families of deformed Fermat hyp…
Moduli spaces of surfaces and monodromy invariants
这篇论文是作者最近关于“abc-曲面”和他们自然的Lefschetz纤维化和投影到P^1 x P^1的单值性的结果的综述，见（arXiv：0910.2142）。调查的结果探讨各种基本问题复杂的变形与symplectomorphism与symplectomorphism从一个明确的家庭的角度来看的例子;该文件还包含一些意见，这是没有在以前的文件。
Moduli spaces and braid monodromy types of bidouble covers of the quadric
二次曲面Q的双复盖$π: S \mapsto Q$由依赖于四个正整数a，b，c，d的连通族参数化.在b=d的特殊情况下，我们称它们为abc-曲面。这样的Galois覆盖$π$允许一个小的扰动产生一个一般的4元组覆盖的Q与分支曲线$\De$，和一个自然的Lefschetz纤维化从一个小扰动的组成$ π$与第一个投影。我们证明了一个更一般的结果，这意味着对应于$\De$的辫子单值分解确定了abc-曲面的三个整数a，b，c。我们引入了一种新的方法，以区分不稳定等价的分解。这一结果与第一和第三作者以前的结果形成鲜明对比，表明对应于相应自然Lefschetz束的映射类群分解对于具有相同a+c，b值的abc-曲面是等价的。这个结果暗示了具有固定值a+c，b的abc-曲面，尽管是复纯的但不是形变等价的，可能不是正则辛同构的。
