Let $C$ be a closed Riemann surface of genus $g\geq 1$. Fix a holomorphic 1-form on $C$; it endows $C$ with a flat structure (i.e. a metric of trivial holonomy which has conical singularities at a finite set of points $C_{\mathrm{sing}}$).

Let's fix a geodesic segment of unit length $X \subset C/C_{\mathrm{sing}}$ and a point $x \in X$. Generically, the geodesic starting from $x$ and orthogonal to $X$ will be dense in $C$. It will return infintely many times to $X$: let's denote the consecutive points of intersecttion with $X$ as $x=x_0, \: x_1, \:x_2, \dots$. If we join $x_0$ and $x_n$ along $X$, we get a closed loop defining a homology class $c_n \in H_1(C, \mathbb{Z})$.

It's known that for generic choice of parameters, the limit $l=\lim_{n\rightarrow \infty}\frac{c_n}{n}\in H_1(C, \mathbb{R})$ exists (this is the so-called asymptotic cycle construction). Moreover, it turns out that there exists a sequence of numbers $v_1=1>\dots>v_g>0$ and homology classes $l_1=l,\dots, l_g \in H_1(C, \mathbb{R})$ such that following asymptotic formula holds $$ c_n=nl_1+\dots n^{v_g}l_g+O(1). $$ It also can be shown that the linear span of $l_1, \dots l_g$ is a Lagrangian subspace of $H_1(C, \mathbb{R})$ (with the symplectic structure determined by intersection form).

This picture can be globalized in the following sense. The vector spaces $H_1(C, \mathbb{R})$ form a symplectic vector bundle $\mathrm{H}_{1}$ over the moduli space of flat surfaces with fixed profile of zeroes of the holomorphic 1-form $\mathcal{H}$ ("stratum of abelian differentials"). The Lagrangian subspaces constructed above give us a Lagrangian subbundle $L \subset \mathrm{H}_1$.

My question is: has there been any recent progress on the problem of describing $L$ in terms of the geometry of $\mathcal{H}$? Apparently, this is called 'dynamical Hodge decomposition' (see Zorich's notes).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy