Algebraic intersection for translation surfaces in the stratum \(\mathcal{H}(2)\) (Q2656161)

Let \((X,g)\) be a closed, orientable, Riemannian surface with area \(\text{Vol}(X,g)\). Then the algebraic intersection of closed curves in \(X\) endows the first homology \(H_1(X,\mathbb{R})\) with an antisymmetric, non degenerate, bilinear form, denoted \(\text{Int}(\cdot,\cdot)\). For a curve \(\alpha\), let furthermore \(l_g(\alpha)\) be its length. This article studies the interesting quantity \[ \text{KVol}(X,g) = \text{Vol}(X,g) \cdot \sup \left\{ \frac{\text{Int}(\alpha,\beta)}{ l_g(\alpha) \cdot l_g(\beta)} : \alpha,\beta \text{ piecewise smooth closed curves } \right\}. \] The factor \(\text{Vol}(X,g)\) assures that \(\text{KVol}\) is invariant under scaling. This invariant was used in [the third author and \textit{B. Muetzel}, Manuscr. Math. 143, No. 1--2, 19--49 (2014; Zbl 1296.53071)] to compare the stable norm and the \(L^2\)-norm of elements in \(H_1(X,\mathbb{R})\). Let \(\text{sys}(X,g)\) and \(\text{sys}_h(X,g)\) be the length of a systole and a homological systole of \((X,g)\), respectively. Let furthermore \(\operatorname{diam}(X,g)\) be its diameter. In [loc. cit.] it was further shown that \[ \max\left\{ 1, \frac{ \text{Vol}(X,g)}{2\text{sys}_h(X,g) \cdot \operatorname{diam}(X,g)} \right\} \leq \text{KVol}(X,g) \leq \frac{9 \text{Vol}(X,g)}{\text{sys}_h(X,g)^2} \leq \frac{9 \text{Vol}(X,g)}{\text{sys}(X,g)^2}. \quad \quad (1) \] To make \(\text{KVol}\) go to infinity, it is sufficient to pinch a non-separating closed curve \(\alpha\) to make its length go to zero, so the interesting surfaces are those for which \(\text{KVol}\) is small. In genus one the value one is already obtained in the flat torus \((T,g_E) = (\mathbb{R}^2 / \mathbb{Z}^2,g_E)\), where \(g_E\) denotes the Euclidean metric. In all other genera, however, the value of the infimum \[ \text{KVol}(m) = \inf \{ \text{KVol}(X,g), (X,g) \text{ closed, orientable, Riemannian surface of genus } m \} \] is unknown. The corresponding infimum for \( \displaystyle \frac{\text{Vol}(X,g)}{\text{sys}(X,g)^2}\) is called the systolic area in genus \(m\). It follows from the upper bound in (1) and the known upper bounds for the systolic area, that \[ 1 \leq \text{KVol}(m) \leq c \cdot \frac{ m}{\log(m)^2}. \] It follows from [\textit{M. G. Katz} and \textit{S. Sabourau}, Proc. Am. Math. Soc. 134, No. 4, 1189--1195 (2006; Zbl 1090.53045)], that \(\text{KVol}(2) \leq \frac{2}{\sqrt{3}}\). In this article the authors explore this invariant in the important case of translation surfaces of genus \(m=2\). They study the quantity \(\text{KVol}\) on the stratum \(\mathscr{H}(2)\) of translation surfaces of genus two with a single conical point. There they provide an explicit sequence \(L(n,n)\) of surfaces such that \(\text{KVol}(L(n, n))\) tends to two when \(n\) goes to infinity and conjecture that the infimum in \(\mathscr{H}(2)\) is equal to two. Interestingly, as shown in the companion article [the authors, ``Algebraic intersection for translation surfaces in a family of Teichmüller disks'', Preprint, \url{arXiv:2007.10847}] in this stratum the surface \(M\) that realizes the infimum of the systolic area has \(\text{KVol}(M) = 3\), so the two infima are not realized by the same surface.