Topological rigidity for certain families of differentiable plane curves

Rigidity of the mod 2 families Seiberg-Witten invariants and topology of families of spin 4-manifolds

We show a rigidity theorem for the Seiberg-Witten invariants mod 2 for families of spin 4-manifolds. A mechanism of this rigidity theorem also gives a family version of 10/8-type inequality. As an application, we prove the existence of non-smoothable topological families of 4-manifolds whose fiber, base space, and total space are smoothable as manifolds. These non-smoothable topological families provide new examples of 4-manifolds $M$ for which the inclusion maps $\operatorname{Diff}(M) \hookrightarrow \operatorname{Homeo}(M)$ are not weak homotopy equivalences. We shall also give a new series…

Rigidity for Families of Polarized Calabi-Yau Varieties

In this paper, we study the analogue of the Shafarevich conjecture for polarized Calabi-Yau varieties. We use variations of Hodge structures and Higgs bundles to establish a criterion for the {\it rigidity} of families. We then apply the criterion to obtain that some important and typical families of Calabi-Yau varieties are rigid, for examples., Lefschetz pencils of Calabi-Yau varieties, {\it strongly degenerated} families (not only for families of Calabi-Yau varieties), families of Calabi-Yau varieties admitting a degeneration with {\it maximal unipotent monodromy}.

曼哈顿曲线、拓扑流的遍历理论与刚性

The Manhattan curve, ergodic theory of topological flows and rigidity

本文为一般非初等双曲群引入曼哈顿曲线，利用拓扑流遍历理论和Patterson-Sullivan测度证明其可微性与刚性，并给出三角形群等显式计算。

Cohomological rigidity of manifolds defined by right-angled 3-dimensional polytopes

A family of closed manifolds is called cohomologically rigid if a cohomology ring isomorphism implies a diffeomorphism for any two manifolds in the family. We establish cohomological rigidity for large families of 3-dimensional and 6-dimensional manifolds defined by 3-dimensional polytopes. We consider the class P of 3-dimensional combinatorial simple polytopes, different from a tetrahedron, whose facets do not form 3- and 4-belts. This class includes mathematical fullerenes, i.e. simple 3-polytopes with only 5-gonal and 6-gonal facets. By a theorem of Pogorelov, any polytope from P admits a r…

On the sharp lower bounds of modular invariants and fractional Dehn twist coefficients

Modular invariants of families of curves are Arakelov invariants in arithmetic algebraic geometry. All the known uniform lower bounds of these invariants are not sharp. In this paper, we aim to give explicit lower bounds of modular invariants of families of curves, which is sharp for genus 2. According to the relation between fractional Dehn twists and modular invariants, we give the sharp lower bounds of fractional Dehn twist coefficients and classify pseudo-periodic maps with minimal coefficients for genus 2 and 3 firstly. We also obtain a rigidity property for families with minimal modular …

Duality and vanishing theorems for topologically trivial families of smooth h-cobordisms

Using the work of Dwyer, Weiss, and Williams we associate an invariant to any topologically trivial family of smooth h-cobordisms. This invariant is called the smooth structure class, and is closely related to the higher Franz--Reidemeister torsion of Igusa and Klein. We compute the smooth structure class in terms of a fiberwise generalized Morse function using fiberwise Poincaré--Hopf theory. This computation gives rise to a duality theorem for the smooth structure class that generalizes Milnor's duality theorem for the Whitehead torsion. From this result we deduce a vanishing theorem that im…