The Theta Divisor and Three-Manifold Invariants

The theta divisor and the Casson-Walker invariant

We use Heegaard decompositions and the theta divisor on a Riemannian surface to define a three-manifold invariant for rational homology three-spheres. This invariant is defined on the set of $Spin^c$ structures $$ {\hat θ}\colon Spin^c(Y)\longrightarrow Q. $$ In the first part of the paper, we give the definition of the invariant (which builds on the theory developed in the article, "The theta divisor and three-manifold invariants"). In the second part, we establish a relationship between this invariant and the Casson-Walker invariant.

Cutting out arithmetic Teichmueller curves in genus two via Theta functions

We compute the class of arithmetic genus two Teichmueller curves in the Picard group of pseudo-Hilbert modular surfaces, distinguished according to their torsion order and spin invariant. As an application, we compute the number of genus two square-tiled surfaces with these invariants. The main technical tool is the computation of divisor classes of Hilbert Jacobi forms on the universal abelian surface over the pseudo-Hilbert modular surface.

A semigroup of theta-curves in 3-manifolds

We establish an existence and uniqueness theorem for prime decompositions of theta-curves in $3$-manifolds.

Theta series and generalized special cycles on Hermitian locally symmetric manifolds

We study generalized special cycles on Hermitian locally symmetric spaces $Γ\backslash D$ associated to the groups $G=\mathrm{U}(p,q)$, $\mathrm{Sp}(2n,\mathbb{R}) $ and $\mathrm{O}^*(2n)$. These cycles are (covered by) locally symmetric spaces associated to subgroups of $G$ which are of the same type. Using oscillator representation and a construction which essentially comes from the thesis of Greg Anderson, we show that Poincaré duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermtia…

Theta-graph and diffeomorphisms of some 4-manifolds

In this article, we construct countably many mutually non-isotopic diffeomorphisms of some closed non simply-connected 4-manifolds that are homotopic to but not isotopic to the identity, by surgery along $Θ$-graphs. As corollaries of this, we obtain some new results on codimension 1 embeddings and pseudo-isotopies of 4-manifolds. In the proof of the non-triviality of the diffeomorphisms, we utilize a twisted analogue of Kontsevich's characteristic class for smooth bundles, which is obtained by extending a higher dimensional analogue of Marché--Lescop's "equivariant triple intersection" in conf…

高维切触流形中除子的稳定化

Stabilization of divisors in high-dimensional contact manifolds

该研究针对维数不小于5的切触流形中的余维2切触子流形定义稳定化操作，推导其两条核心性质，并证明大量横截链环具有非简单性，推进了高维切触拓扑领域的相关研究。