We classify all the surfaces of general type whose canonical map is composed with a pencil if they are the quotient of the diagonal action by an Abelian group acting over the product of two curves. As far as we know all the previous examples with isotrivial canonical map are very particular cases of our construction; moreover we find an unexpected action by a group of order 16. This construction can be easily generalizable to all dimensions.
We construct, for every prime p, a function field K of characteristic p and an ordinary abelian variety A over K, with no isotrivial factors, that admits an etale self-isogeny of p-power degree. As a consequence, we deduce that there exist ordinary abelian varieties over function fields whose groups of points over the maximal purely inseparable extension is not finitely generated, answering in the negative a question of Thomas Scanlon.
The cohomological support locus of pluricaonical sheaves and the Iitaka fibration
设$alb_X: X \rightarrow A$是光滑射影簇的Albanese映射，$f: X \rightarrow Y$是$alb_X$的Stein分解的纤维化.对于正整数$m$，如果$f$和$m$满足假设AS（1，2），则平移通过上同调轨迹$V^0(ω_X^m, alb_X)$的所有分量的原点生成$I^*Pic^0(S)$，其中$I: X \rightarrow S$表示Iitaka纤维化。这一结果适用于研究复正则映射。我们还考虑了纤维化是否是等平凡的和同源的产品的问题。
Families of Group Actions, Generic Isotriviality, and Linearization
我们证明了一个“类属等价定理"，它说的是，两个仿射态射$p: S \to Y$和$q: T \to Y$的变种同构（封闭）纤维成为同构下占主导地位的etale基地的变化$ϕ: U \to Y$。一个特殊的情况是下面的结果。如果$ϕ$是平坦的，并且所有纤维都（约化并）同构于$F$，则将态射$ϕ: X \to Y$称为“具有纤维$F$的纤维化“。然后仿射纤维化纤维$F$承认一个etale显性态射$μ: U \to Y$，使得拉回是一个平凡纤维丛：$U\times_Y X \simeq U\times F$。作为应用，我们给出了以下两个（已知）结果的简短证明：（a）正规簇上的仿射$\A^1$ -纤维化在Zuraki-拓扑中是局部平凡的;（b）光滑曲线上的仿射$\A^2$ -纤维化在Zuraki-拓扑中是局部平凡的。我们还研究了由曲线参数化的$\A^2$上的约化群作用族，并证明了非有限约化群在$Å^3$上的每一个忠实作用都是可线性化的，即$G$ -同构于$G$的一个表示。
We prove that the genus $g$, the relative irregularity $q_f$ and the Clifford index $c_f$ of a non-isotrivial fibration $f$ satisfy the inequality $q_f \leq g - c_f$. This gives in particular a proof of Xiao's conjecture for fibrations whose general fibres have maximal Clifford index.
We show that there exists no smooth fibration of a smooth complex quasi-projective variety of log-general type over a quasi-abelian variety. The proof uses M. Popa and C. Schnell's construction of Higgs bundle.
Let X be an abelian scheme over a scheme B. The Fourier--Mukai transform gives an equivalence between the derived category of X and the derived category of the dual abelian scheme. We partially extend this to certain schemes X over B (which we call degenerate abelian schemes) whose generic fiber is an abelian variety, while special fibers are singular. Our main result provides a fully faithful functor from a twist of the derived category of Pic$^τ$(X/B) to the derived category of X. Here Pic$^τ$(X/B) is the algebraic space classifying fiberwise numerically trivial line bundles. Next, we show t…
We describe the minimal number of critical points and the minimal number $s$ of singular fibres for a non isotrivial fibration of a surface $S$ over a curve $B$ of genus $1$, constructing a fibration with $s=1$ and irreducible singular fibre with $4$ nodes. Then we consider the associated factorizations in the mapping class group and in the symplectic group. We describe explicitly which products of transvections on homologically independent and disjoint circles are a commutator in the Symplectic group $Sp (2g, \mathbb{Z})$.
For every fibration $f : X \to B$ with $X$ a compact Kähler manifold, $B$ a smooth projective curve, and a general fiber of $f$ an abelian variety, we prove that $f$ has an algebraic approximation.
Let $A$ be a non-isotrivial ordinary abelian surface over a global function field with good reduction everywhere. Suppose that $A$ does not have real multiplication by any real quadratic field with discriminant a multiple of $p$. We prove that there are infinitely many places modulo which $A$ is isogenous to the product of two elliptic curves.
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…
We are interested in studying the variation of the Hitchin fibration in moduli spaces of parabolic Higgs bundles, under the action of a ramified covering. Given a degree two map $π$ : Y $\rightarrow$ X between compact Riemann surfaces, we may pull back a Higgs bundle from X to Y , the lifted Higgs bundle tends to have many apparent singularities, then we perform a suitable birational transformation in order to eliminate them. This correspondence preserves the Hitchin fibrations and then its restriction to a general fiber gives a map between Abelian varieties. The aim of this paper is to descri…
The geometry of elliptic fibrations translates to the physics of gauge theories in F-theory. We systematically develop the dictionary between arithmetic structures on elliptic curves as well as desingularized elliptic fibrations and symmetries of gauge theories on a circle. We show that the Mordell-Weil group law matches integral large gauge transformations around the circle in Abelian gauge theories and explain the significance of Mordell-Weil torsion in this context. We also use Higgs transitions and circle large gauge transformations to introduce a group law for genus-one fibrations with mu…
In this paper we describe the fundamental group-scheme of a proper variety fibered over an abelian variety with rationally connected fibers over an algebraically closed field. We use old and recent results for the Nori fundamental group-scheme, and of finite group-schemes in general to prove that the kernel of such a fibration is finite, and that the homotopy exact sequence holds in this case. As an application, we describe the fundamental group-schemes of certain proper varieties that are connected by curves or are related to them.
A vector bundle on a smooth projective variety, if it is generically generated by global sections, yields a rational map to a Grassmannian, called Kodaira map. We investigate the asymptotic behaviour of the Kodaira maps for the symmetric powers of a vector bundle, and we show that these maps stabilize to a map dominating all of them, as it happens for a line bundle via the Iitka fibration. Through this Iitaka-type construction, applied to the cotangent bundle, we give a new characterization of Abelian varieties.
Mumford constructed a family of abelian fourfolds with special stucture not characterized by endomorphism ring. Galluzzi showed that the weight 2 Hodge structure of such a variety decomposes into Hodge substructures via the action of Mumford-Tate group, one of which is of K3 type with Hodge number (1,7,1). We will compute the intersection form of such a Hodge structure and provide a canonical integral Hodge structure on this subspace. Furthermore, we shall show two applications of this invariant. Firstly, using the above formula we shall show that every K3 surface obtained in this way will adm…
We construct infinitely many non-isotrivial families of abelian varieties over given four punctured projective lines. These families lead to algebraic solutions of Painleve VI equation. Finally, based on a recent paper by Lin-Sheng-Wang, we prove a complete characterization for the locus of motivic Higgs bundles in the moduli space as fixed points of an ``additive'' self-map. This is a note based on the lecture given by the second named author on 04 Nov. 2022 at Tsinghua University.
We prove some new results on the arithmetic of abelian varieties over function fields of one variable over finitely generated (infinite) fields. Among other things, we introduce certain new natural objects `discrete Selmer groups' and `discrete Shafarevich-Tate groups', and prove that they are finitely generated $\Bbb Z$-modules. Further, we prove that in the isotrivial case, the discrete Shafarevich-Tate group vanishes and the discrete Selmer group coincides with the Mordell-Weil group. One of the key ingredients to prove these results is a new specialisation theorem à la Néron for first Galo…
