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

Fractional Dehn twists and modular invariants

In this note, we establish a relationship between fractional Dehn twist coefficients of Riemann surface automorphisms and modular invariants of holomorphic families of algebraic curves. Specially, we give a characterization of pseudo-periodic maps with nontrivial fractional Dehn twist coefficients. We also obtain some uniform lower bounds of non-zero fractional Dehn twist coefficients.

Products of pairs of Dehn twists and maximal real Lefschetz fibrations

We address the problem of existence and uniqueness of a factorization of a given element of the modular group into a product of two Dehn twists. As a geometric application, we conclude that any maximal real elliptic Lefschetz fibration is algebraic.

论四维Dehn扭转与Milnor纤维化

On four-dimensional Dehn twists and Milnor fibrations

该研究借助磁单极Floer同调进展计算族Seiberg--Witten不变量，证明极小椭圆奇点链不定辛填充的边界Dehn扭转无限阶非平凡，揭示了四维空间中三类全新的拓扑与辛几何现象。

Dehn twists exact sequences through Lagrangian cobordism

In this paper we introduce the following new ingredients: (1) rework on part of the Lagrangian surgery theory; (2) constructions of Lagrangian cobordisms on product symplectic manifolds; (3) extending Biran-Cornea Lagrangian cobordism theory to the immersed category. As a result, we manifest Seidel's exact sequences (both the Lagrangian version and the symplectomorphism version), as well as Wehrheim-Woodward's family Dehn twist sequence (including the codimension-1 case missing in the literature) as consequences of our surgery/cobordism constructions. Moreover, we obtain an expression of the a…

A Symplectic Isotopy of a Dehn Twist on CP^n x CP^{n+1}

The complex manifold CP^n x CP^{n+1} with symplectic form σ_μ=σ_{CP^n}+μσ_{CP^{n+1}}, where σ_{CP^n} and σ_{CP^{n+1}} are normalized Fubini-Study forms, n a natural number and μ>1 a real number, contains a natural Lagrangian sphere L^μ. We prove that the Dehn twist along L^μ is symplectically isotopic to the identity for all μ>1. This isotopy can be chosen so that it pointwise fixes a complex hypersurface in CP^n x CP^{n+1} and lifts to the blow-up of CP^n x CP^{n+1} along a complex n-dimensional submanifold.

Dehn twists and invariant classes

A degeneration of compact Kaehler manifolds gives rise to a monodromy action on Betti moduli space H^1(X, G) = Hom(π_1(X),G)/G over smooth fibres with a complex algebraic structure group G being either abelian or reductive. Assume that the singularities of the central fibre is of normal crossing. When G = C, the invariant cohomology classes arise from the global classes. This is no longer true in general. In this paper, we produce large families of locally invariant classes that do not arise from global ones for reductive G. These examples exist even when G is abelian, as long as G contains mu…