Quadratic differentials in low genus: exceptional and non-varying strata.
(Différentielles quadratiques en petit genre : strates exceptionnelles et strates sans variance.)

*(English. French summary)*Zbl 1395.14008Let \(\mathcal M_g\) be the moduli space parametrising compact Riemann surfaces of genus \(g\). The moduli space \(\Omega \mathcal M_g\) parametrising abelian differentials \(\omega\) on the Riemann surfaces can be decomposed into strata \(\Omega \mathcal M_g(m_1, \ldots, m_k)\) according to the number of multiplicities of the zeros of \(\omega\). Analogously, consider now the moduli space parametrising pairs \((X, q)\) where \(X\) is a compact Riemann surface and \(q\) is a quadratic differential on \(X\) having at most simple poles. Then we can define strata \(\mathcal Q(d_1, \ldots, d_n)\) according to the multiplicity of the zeros or poles of these differentials. E. Lanneau [Ann. Sci. Éc. Norm. Supér. (4) 41, No. 1, 1–56 (2008; Zbl 1161.30033)] determined the connected components for these strata of quadratic differentials: some have hyperelliptic components and apart from a short list of exceptional cases the other strata are connected. The first result of this paper gives an algebraic description of the connected components of these exceptional strata: given a quadratic differential \(q\) having a zero or pole of order \(d_i\) at \(p_i\) denote by \(\mathrm{ div}(q)_0 = \sum_{d_i>0} d_i p_i\) the zero divisor of \(q\). In genus three the exceptional strata have exactly two connected components and they are distinguished by the dimension of \(H^0(X, \operatorname{div}(q)_0/3)\). In genus four the exceptional strata have exactly two non-hyperelliptic connected components and they are distinguished by the dimension of \(H^0(X, \operatorname{div}(q)/3)\) where \(\operatorname{div}(q)\) is the divisor associated to \(q\).
The second result deals with the Lyapunov exponents of Teichmüller curves in the strata of quadratic differentials. The authors call a stratum non-varying if the sum of the Lyapunov exponents is the same for all Teichmüller curves in the stratum. In their earlier work [Geom. Topol. 16, No. 4, 2427–2479 (2012; Zbl 1266.14018)] they showed that many components of abelian differentials are non-varying for curves of low genus. In this paper they give a list of non-varying strata (both exceptional and non-exceptional) for quadratic differentials over curves of genus at most four.

Reviewer: Andreas Höring (Nice) (MR3215925)

##### MSC:

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

30F30 | Differentials on Riemann surfaces |

14H15 | Families, moduli of curves (analytic) |

30F60 | Teichmüller theory for Riemann surfaces |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

14H51 | Special divisors on curves (gonality, Brill-Noether theory) |

37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |