Finite, fiber-preserving group actions on elliptic 3-manifolds

Topological uniqueness results for Lefschetz fibrations over the disc

We prove that a Lefschetz fibration over the disc that, after compactification, has the same singular fibers as an extremal rational elliptic surface can be obtained by deleting a singular fiber and a section from the rational extremal elliptic surface, i.e. such a Lefschetz fibration is determined up to topological equivalence by its set of singular fibers. We get a complete clasification of Lefschetz fibrations with 2 $I_1$ fibers as a byproduct of our results. The proof is inspired by homological mirror symmetry and Karpov--Nogin's theorem on constructivity of helices on del Pezzo surfaces.…

Toledo invariants of Higgs bundles on elliptic surfaces associated to base orbifolds of Seifert fibered homology 3-spheres

To each connected component in the space of semisimple representations from the orbifold fundamental group of the base orbifold of a Seifert fibered homology 3-sphere into the Lie group U(2,1), we associate a real number called the "orbifold Toledo invariant." For each such orbifold, there exists an elliptic surface over it, called a Dolgachev surface. Using the theory of Higgs bundles on these Dolgachev surfaces, we explicitly compute all values taken on by the orbifold Toledo invariant.

椭圆纤维化4流形的光滑窄Mordell-Weil群

The Smooth Narrow Mordell-Weil Group of Elliptically Fibered 4-Manifolds

本文推广Farb与Looijenga的工作，定义椭圆纤维化4流形的光滑窄Mordell-Weil群，构造其与H²(M)平凡格正交补的同构，给出其秩的显式公式并证明有理椭圆曲面是两类Mordell-Weil群秩相同的唯一椭圆曲面。

Diffeomorphisms of Elliptic 3-Manifolds

The elliptic 3-manifolds are the closed 3-manifolds that admit a Riemannian metric of constant positive curvature, that is, those that have finite fundamental group. The (Generalized) Smale Conjecture asserts that for any elliptic 3-manifold M, the inclusion from the isometry group of M to the diffeomorphism group of M is a homotopy equivalence. The original Smale Conjecture, for the 3-sphere, was proven by J. Cerf and A. Hatcher, and N. Ivanov proved the generalized conjecture for many of the elliptic 3-manifolds that contain a geometrically incompressible Klein bottle. Our main results are 1…

Nonisotopic Symplectic Tori in the Fiber Class of Elliptic Surfaces

The purpose of this note is to present a construction of an infinite family of symplectic tori T_{p} representing an arbitrary multiple of the homology class of the fiber of an elliptic surface E(n), for n > 2, such that, for i \neq j, there is no orientation-preserving diffeomorphism between (E(n),T_{i}) and (E(n),T_{j}). In particular, these tori are mutually nonisotopic. This complements previous results of Fintushel and Stern, showing in particular the existence of such phenomenon for a primitive class.

On negative spheres in elliptic surfaces

Using elliptic fibrations with specific singular fibers, we find spheres with very negative self-intersections in elliptic surfaces and in their blow-ups.

复复利椭圆保留拉格朗日平面

Complex Hyperbolic Elliptics Preserving Lagrangian Planes

本文证明复复利椭圆保留拉格朗日平面的充要条件是其为实椭圆，且此类保留连续一参数家族拉格朗日平面，通过固定点形成固定环面。

Notes on branched coverings of Seifert manifolds

In a paper published in 2002, the author gave a criterion to determine whether there is a fiber-preserving branched covering between two given orientable Seifert manifolds with orientable bases. Here we supply some details of the proof of two claims in that paper. We give an explicit construction of fiber-preserving branched covering between two Seifert fibered solid tori when their Seifert invariants satisfy certain relation, and we show the factorability of fiber-preserving branched coverings between two closed Seifert manifolds.

Symplectic tori in rational elliptic surfaces

Let E(1)_p denote the rational elliptic surface with a single multiple fiber f_p of multiplicity p. We construct an infinite family of homologous non-isotopic symplectic tori representing the primitive class [f_p] in E(1)_p when p>1. As a consequence, we get infinitely many non-isotopic symplectic tori in the fiber class of the rational elliptic surface E(1) (complex projective plane blown-up at nine branch points of a generic pencil of cubic curves). We also show how these tori can be non-isotopically and symplectically embedded in many other symplectic 4-manifolds.

关于尖点和简单椭圆奇点的Milnor纤维的Lefschetz纤维结构

Lefschetz fibrations on the Milnor fibers of cusp and simple elliptic singularities

研究显示，通过构造S^1参数的genus-一Lefschetz纤维结构，揭示了尖点和简单椭圆奇点的Milnor纤维与Lawson叶状结构的关系，并证明其可粘合成K3表面。

光滑Mordell-Weil群与椭圆曲面的映射类群

The smooth Mordell-Weil group and mapping class groups of elliptic surfaces

该研究受数论Néron-Lang定理启发，证明椭圆纤维化的光滑Mordell-Weil群有限生成，显式计算一类单连通复曲面的对应群，证明其纤维化结构拓扑同痕唯一，结合Donaldson结果得到高亏格曲面微分同胚的同痕性质。

Isometries of elliptic 3-manifolds

The closed 3-manifolds of constant positive curvature were classified long ago by Seifert and Threlfall. Using well-known information about the orthogonal group O(4), we calculate their full isometry groups Isom(M), determine which elliptic 3-manifolds admit Seifert fiberings that are invariant under all isometries, and verify that the inclusion of Isom(M) to Diff(M) is a bijection on path components.

Spaces of elliptic differentials

We study modular fibers of elliptic differentials, which are roughly spaces of torus-coverings over a fixed base torus. For genus 2 torus covers with fixed degree we show, that the modular fibers F_d(1,1) are itself connected torus covers with Veech group SL_2(Z). Using results of Eskin, Masur and Schmoll we calculate the Euler Characteristic and the parity of the spin structure of the quadratic differential (F_d(1,1)/(-id),q_d). We state and apply formulas for the asymptotic quadratic growth rates of various types of geodesic segments on a surface S "contained" in F_d(1,1). The quadratic grow…