We present two results about the relationship between fundamental groups of quasiprojective manifolds and linear systems on a projectivization. We prove the existence of a plane curve with non-abelian fundamental group of the complement which does not admit a mapping onto an orbifold with non-abelian fundamental group. We also find an affine manifold whose irreducible components of its characteristic varieties do not come from the pull-back of the characteristic varieties of an orbifold.
In the previous paper by Pereira and the author, it was proved that any pencil of plane curves of degree greater than one with irreducible generic fiber can have at most five completely reducible fibers although no examples with five such fibers were ever found. Recently Janis Stipins has proved that if any two fibers of a pencil intersect transversally then it cannot have five completely reducible fibers. In this paper we generalize the Stipins result to arbitrary pencils. We also include into consideration more general special fibers that are the unions of lines and non-reduced curves. These…
Let $X$ be a smooth projective surface such that linear and numerical equivalence of divisors on $X$ coincide and let $σ\subseteq |D|$ be a linear pencil on $X$ with integral general fibers. A fiber of $σ$ will be called special if either it is not integral or it has non-generic multiplicity at some of the base points (including the infinitely near ones) of the pencil. In this note we provide an algorithm to compute the integral components of the special fibers of $σ$.
We complete the study of birational geometry of Fano fiber spaces $π\colon V\to {\mathbb P}^1$, the fiber of which is a Fano double hypersurface of index 1. For each family of these varieties we either prove birational rigidity or produce explicitly non-trivial structures of Fano fiber spaces. A new linear method of studying movable systems on Fano fiber spaces $V/{\mathbb P}^1$ is developed.
We study completely reducible fibers of pencils of hypersurfaces on $\mathbb P^n$ and associated codimension one foliations of $\mathbb P^n$. Using methods from theory of foliations we obtain certain upper bounds for the number of these fibers as functions only of $n$. Equivalently this gives upper bounds for the dimensions of resonance varieties of hyperplane arrangements. We obtain similar bounds for the dimensions of the characteristic varieties of the arrangement complements.
We prove birational superrigidity of generic Fano fiber spaces $V/{\mathbb P}^1$, the fibers of which are Fano complete intersections of index 1 and dimension $M$ in ${\mathbb P}^{M+k}$, provided that $M\geq 2k+1$. The proof combines the traditional quadratic techniques of the method of maximal singularities with the linear techniques based on the connectedness principle of Shokurov and Koll\' ar. Certain related results are also considered.
We construct some infinite series of free and nearly free curves using pencils of conics with a base locus of cardinality at most two. These curves have an interesting topology, e.g. a high degree Alexander polynomial that can be explicitly determined, a Milnor fiber homotopy equivalent to a bouquet of circles, or an irreducible translated component in the characteristic variety of their complement. Monodromy eigenspaces in the first cohomology group of the corresponding Milnor fibers are also described in terms of explicit differential forms.
Consider a one-parameter family of smooth, irreducible, projective curves of genus $g\ge 2$ defined over a number field. Each fiber contains at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of the family and the Mordell--Weil rank of the fiber's Jacobian. Our proof uses Vojta's approach to the Mordell Conjecture furnished with a height inequality due to the second- and third-named authors. In addition we obtain uniform bounds for the number of torsion in the Jacobian that lie each fiber o…
We prove that a general Fano fibration $π\colon V\to {\mathbb P}^1$, the fiber of which is a double Fano hypersurface of index 1, is birationally superrigid provided it is sufficiently twisted over the base. In particular, on $V$ there are no other structures of a rationally connected fibration. The proof is based on the method of maximal singularities.
Let $(S,D)$ be a minimal log pair of general type with $S$ a smooth projective surface and $D$ a simple normal corssing reduced divisor on $S$. We assume that its log canonial linear system $|K_S+D|$ is composed of a penciel, let $f\colon S\to B$ be the fiberation induced by the linear system $|K_S+D|$ and $F$ be a general fiber of $f$. Let $b$ (resp. $g$) be the genus of the base curve $B$ (resp. general fiber $F$) and $k=D\cdot F$ the intersection number. We show that 1. If $k>0$ and $b\geq 2$ then $2\leq g+k \leq 3$, when $g+k=3$ we have $b=2$ and $h^1(S,K_S+D)=0$. 2. Suppose $p_a(D)\leq 2(…
In this work we study the connection between the existence of finite dihedral covers of the projective plane ramified along an algebraic curve C, infinite dihedral covers, and pencils of curves containing C.
Let $\mathcal{A} $ be a complexified-real arrangement of lines in $\mathbb{C}^2.$ Let $H$ be any line in $ \mathcal{A} $. Then, form a new complexified-real arrangement $ \mathcal{B}_H = \mathcal{A} \cup \mathcal{C} $ where $ \mathcal{C} \cup \{H\} $ is a pencil of lines with multiplicity $ m\geq 3 $, the intersection point in the pencil is not a multiple point in $ \mathcal{A}, $ and every line in $ \mathcal{C} $ intersects every line in $ \mathcal{A}\setminus \{H\} $ in points of multiplicity two. In this article, we show that for $ H_1, H_2 \in \mathcal{A} $ we may have that $ \mathcal{B}_{…
The paper studies a relation between fundamental group of the complement to a plane singular curve and the orbifold pencils containing it. The main tool is the use of Albanese varieties of cyclic covers ramified along such curves. Our results give sufficient conditions for a plane singular curve to belong to an orbifold pencil, i.e. a pencil of plane curves with multiple fibers inducing a map onto an orbifold curve whose orbifold fundamental group is non trivial. We construct an example of a cyclic cover of the projective plane which is an abelian surface isomorphic to the Jacobian of a curve …
It is well known that formal solutions to the Associativity Equations are the same as cyclic algebras over the homology operad $(H_*(\bar{M}_{0,n+1}))$ of the moduli spaces of $n$--pointed stable curves of genus zero. In this paper we establish a similar relationship between the pencils of formal flat connections (or solutions to the Commutativity Equations) and homology of a new series $\bar{L}_n$ of pointed stable curves of genus zero. Whereas $\bar{M}_{0,n+1}$ parametrizes trees of $\bold{P}^1$'s with pairwise distinct nonsingular marked points, $\bar{L}_n$ parametrizes strings of $\bold{P}…
Let $M$ be a complex projective manifold of dimension $n+1$ and $f$ a meromorphic function on $M$ obtained by a generic pencil of hyperplane sections of $M$. The $n$-th cohomology vector bundle of $f_0=f|_{M-\RR}$, where $\RR$ is the set of indeterminacy points of $f$, is defined on the set of regular values of $f_0$ and we have the usual Gauss-Manin connection on it. Following Brieskorn's methods in [bri], we extend the $n$-th cohomology vector bundle of $f_0$ and the associated Gauss-Manin connection to $\pl$ by means of differential forms. The new connection turns out to be meromorphic on t…
We study the deep interplay between geometry of quadrics in d-dimensional space and the dynamics of related integrable billiard systems. Various generalizations of Poncelet theorem are reviewed. The corresponding analytic conditions of Cayley's type are derived giving the full description of periodical billiard trajectories; among other cases, we consider billiards in arbitrary dimension d with the boundary consisting of arbitrary number k of confocal quadrics. Several important examples are presented in full details demonstrating the effectiveness of the obtained results. We give a thorough a…
The purpose of this paper is to exhibit a natural construction between complex geometry and symplectic geometry following the idea of mirror symmetry. Suppose we are given a family of pairs of 2-dimensional Kähler tori and stable holomorphic vector bundles on them $(\hat M_{\ep}, E_{\ep})$, \ep \in (0, 1]$, and each has structure of a Lagrangian torus fibration $π:\hat M_{\ep} \to B$ whose fibers are of diameter $O(\ep)$, and let $A_{\ep}$ be a family of hermitian Yang-Mills(HYM) connections on $E_{\ep}$. As $\ep$ goes to zero, $A_{\ep}$ will, modulo possible bubbles, converge to a connection …
In this note we determine the Bernstein-Sato polynomial $b_Q(s)$ of a generic central arrangement $Q=\prod_{i=1}^kH_i$ of hyperplanes. We establish a connection between the roots of $b_Q(s)$ and the degrees of the generators for the top cohomology of the corresponding Milnor fiber. This connection holds for all homogeneous polynomials. We also introduce certain subschemes of the arrangement determined by the roots of $b_Q(s)$.
Connections, metrics and Higgs fields on complex fiber bundles
Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心）；Tsinghua University（清华大学）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（北京雁栖湖应用数学研究院）
The characteristic varieties of a space are the jump loci for homology of rank 1 local systems. The way in which the geometry of these varieties may vary with the characteristic of the ground field is reflected in the homology of finite cyclic covers. We exploit this phenomenon to detect torsion in the homology of Milnor fibers of projective hypersurfaces. One tool we use is the interpretation of the degree 1 characteristic varieties of a hyperplane arrangement complement in terms of orbifold fibrations and multinets on the corresponding matroid. Another tool is a polarization construction, ba…
We study hyperkahler metrics and hyperholomorphic connections of Hitchin's moduli spaces after Gaiotto, Moore and Neitzke. Their construction via the twistor technique produces intricate wall crossing behaviors. For certain four dimensional Hitchin's moduli spaces local models and degeneration to local models near singular fibers of the Hitchin's fibration are understood.
The level $N$ elliptic KZB connection is a flat connection over the universal elliptic curve in level $N$ with its $N$-torsion sections removed. Its fiber over the point $(E,x)$ is the unipotent completion of $π_1(E - E[N],x)$. It was constructed by Calaque and Gonzalez. In this paper, we show that the connection underlies an admissible variation of mixed Hodge structure and that it degenerates to the cyclotomic KZ connection over the singular fibers of the compactified universal elliptic curve. These are the first steps in a larger project to compute the action of the Galois group of mixed Ta…
Let $f: W \rightarrow T$ be an elliptic threefold that is a Weierstrass model, which is locally defined by $y^2 = x^3 + fx + g$ over $T$, with a singular fiber such that $(f,g,4f^3 + 27g^2)$ vanishes of order $(4,6,12)$ over an isolated point over $T$. Such a fiber can be explicitly resolved to fibers with smaller vanishing order and the resulting model, $Y$, contains a rational elliptic surface, $S$, where some sections of $S$ are flopping curves on $Y$. As a consequence of this arithmetic and geometric connection, we are able to describe some constraints between flops of $Y$ and the properti…
