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**Real normalized differentials and Arbarello’s conjecture.**
*(English.
Russian original)*
Zbl 1278.30041

Funct. Anal. Appl. 46, No. 2, 110-120 (2012); translation from Funkts. Anal. Prilozh. 46, No. 2, 37-51 (2012).

Let \(M_g\) denote the moduli space of compact Riemann surfaces of genus \(g\geq 2\). About forty years ago, E. Arbarello considered in \(M_g\) the natural flag \(W_2\subset W_3\subset\cdots\subset W_{g-1}\subset W_g= M_g\) consisting of the subvarieties \(W_n=\{[C]\in M_g\mid\exists p\in C: h^0(C,{\mathcal O}_C(np)> 1\}\) for \(2\leq n\leq g\), the so-called Weierstrass flag in \(M_g\). In the course of his study of the Weierstrass flag, he conjectured that any compact complex cycle in \(M_g\) of dimension at least \(g-n\) must intersect the \((2g+ n-3)\)-dimensional, irreducible subvariety \(W_n\), cf. [E. Arbarello, Compos. Math. 29, 325–342 (1974; Zbl 0355.14013)]. As \(W_2\) is the affine subvariety of hyperelliptic curves in \(M_g\), Arbarello’s conjecture implies that there are no compact complex cycles of dimension greater than \(g-2\) in \(M_g\). This statement was indeed proved by S. Diaz [Mem. Am. Math. Soc. 327, 69 p. (1985; Zbl 0581.14018)], while Arbarello’s general conjecture remained open until 2012. Actually, in the paper under review, the author finally presents a complete proof of Arbarello’s conjecture, the highly non-trivial nature of which is due to the fact that the higher Weierstrass loci \(W_n\) are almost never affine, cf. [E. Arbarello and G. Mondello, Contemp. Math. 564, 137–144 (2012; Zbl 1255.14021)].

The author’s proof of Arbarello’s conjecture is based on applications of the so-called Whitham perturbation theory for integrable systems, which he himself had developed in the 1990s with a view toward topological quantum field theories and superymmetric gauge theories. Later on, the application of this framework to the study of moduli spaces of curves was initiated by I. M. Krichever and S. Grushevsky in their paper [in: Ji, Lizhen (ed.) et al., Geometry of Riemann surfaces and their moduli spaces. Somerville, MA: International Press. Surveys in Differential Geometry 14, 111–129 (2010; Zbl 1213.14055)]. Building upon this previous work in Whitham theory, especially on the central concept of real normalized meromorphic-differentials on Riemann surfaces, the author constructs certain foliations on moduli spaces of Riemann surfaces equipped with appropriate additional data, describes the corresponding leaves of these foliations defined by real normalized differentials as smooth complex varieties, and introduces then yet another new ingredient for the proof of Arbarello’s conjecture, namely the concept of cycles dual to critical points of real normalized meromorphic differentials on a Riemann surface. In this context, it is shown that the homology classes of these dual cycles generate the first homology group \(H_1(C,\mathbb{Z})\) of the given surface \(C\).

In the last section, the author gives the proof of Arbarello’s conjecture in its full generality, thereby making ingenious use of the new methods as developed in the previous sections. As an additional illustration of the power of these methods, the author has included a new proof of S. Diaz’s theorem (mentioned above) that is based on the properties of real normalized differentials with one pole of second-order. No doubt, this paper provides a decisive breakthrough in the study of the geometry of the Weierstrass flag in the moduli spaces of Riemann surfaces.

The author’s proof of Arbarello’s conjecture is based on applications of the so-called Whitham perturbation theory for integrable systems, which he himself had developed in the 1990s with a view toward topological quantum field theories and superymmetric gauge theories. Later on, the application of this framework to the study of moduli spaces of curves was initiated by I. M. Krichever and S. Grushevsky in their paper [in: Ji, Lizhen (ed.) et al., Geometry of Riemann surfaces and their moduli spaces. Somerville, MA: International Press. Surveys in Differential Geometry 14, 111–129 (2010; Zbl 1213.14055)]. Building upon this previous work in Whitham theory, especially on the central concept of real normalized meromorphic-differentials on Riemann surfaces, the author constructs certain foliations on moduli spaces of Riemann surfaces equipped with appropriate additional data, describes the corresponding leaves of these foliations defined by real normalized differentials as smooth complex varieties, and introduces then yet another new ingredient for the proof of Arbarello’s conjecture, namely the concept of cycles dual to critical points of real normalized meromorphic differentials on a Riemann surface. In this context, it is shown that the homology classes of these dual cycles generate the first homology group \(H_1(C,\mathbb{Z})\) of the given surface \(C\).

In the last section, the author gives the proof of Arbarello’s conjecture in its full generality, thereby making ingenious use of the new methods as developed in the previous sections. As an additional illustration of the power of these methods, the author has included a new proof of S. Diaz’s theorem (mentioned above) that is based on the properties of real normalized differentials with one pole of second-order. No doubt, this paper provides a decisive breakthrough in the study of the geometry of the Weierstrass flag in the moduli spaces of Riemann surfaces.

Reviewer: Werner Kleinert (Berlin)

### MSC:

30F10 | Compact Riemann surfaces and uniformization |

30F30 | Differentials on Riemann surfaces |

14H10 | Families, moduli of curves (algebraic) |

14H55 | Riemann surfaces; Weierstrass points; gap sequences |

14H70 | Relationships between algebraic curves and integrable systems |

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

### Keywords:

Riemann surfaces; moduli of Riemann surfaces; moduli of algebraic curves; foliations; meromorphic differentials; Weierstrass points; Arbarello’s conjecture
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\textit{I. M. Krichever}, Funct. Anal. Appl. 46, No. 2, 110--120 (2012; Zbl 1278.30041); translation from Funkts. Anal. Prilozh. 46, No. 2, 37--51 (2012)

### References:

[1] | E. Arbarello, ”Weierstrass points and moduli of curves,” Compositio Math., 29 (1974), 325–342. · Zbl 0355.14013 |

[2] | E. Arbarello and G. Mondello, ”Two remarks on the Weierstrass flag,” in: Compact Moduli spaces and Vector Bundles, Contemp. Math. (Proceedings) series, vol. 564, Amer. Math. Soc., Providence, RI, 2012, 137–144. · Zbl 1255.14021 |

[3] | Integrability: The Seiberg-Witten and Whitham Equations (eds. H. W. Braden and I. M. Krichever), Gordon and Breach Science Publishers, Amsterdam, 2000. |

[4] | S. Diaz, ”Exceptional Weierstrass points and the divisor on moduli space that they define,” Mem. Amer. Math. Soc., 56:327 (1985). · Zbl 0581.14018 |

[5] | A. Gorsky, I. Krichever, A. Marshakov, A. Mironov, and A. Morozov, ”Integrability and Seiberg-Witten exact solution,” Phys. Lett. B, 355 (1995), 466–474. · Zbl 0997.81567 |

[6] | J. Harris and I. Morrison, Moduli of Curves, Graduate Texts in Math., vol. 187, Springer-Verlag, New York, 1998. · Zbl 0913.14005 |

[7] | S. Grushevsky and I. Krichever, ”The universalWhitham hierarchy and geometry of the moduli space of pointed Riemann surfaces,” in: Surveys in Differ. Geom., vol. 14, Int. Press, Somerville, MA, 2010, 111–129. · Zbl 1213.14055 |

[8] | S. Grushevsky and I. Krichever, Foliations on the Moduli Space of Curves, Vanishing in Cohomology, and Calogero-Moser Curves, http://arxiv.org/abs/1108.4211 . · Zbl 1213.14055 |

[9] | I. M. Krichever, ”Spectral theory of finite-zone nonstationary Schrödinger operators. A nonstationary Peierls model,” Funkts. Anal. Prilozhen., 20:3 (1986), 42–54; English transl.: Funct. Anal. Appl., 20:3 (1986), 203–214. · Zbl 0637.35060 |

[10] | I. M. Krichever, ”Method of an averaging for two-dimensional ”integrable” equations,” Funkts. Anal. Prilozhen., 22:3 (1988), 37–52; English transl.: Funct. Anal. Appl., 22:3 (1988), 200–213. · Zbl 0688.35088 |

[11] | I. Krichever, ”The {\(\tau\)} -function of the universal Whitham hierarchy, matrix models, and topological field theories,” Comm. Pure Appl. Math., 47:4 (1994), 437–475. · Zbl 0811.58064 |

[12] | I. Krichever and D. H. Phong, ”On the integrable geometry of soliton equations and N = 2 supersymmetric gauge theories,” J. Differential Geom., 45:2 (1997), 349–389. · Zbl 0889.58044 |

[13] | I. Krichever and D. H. Phong, ”Symplectic forms in the theory of solitons,” in: Surveys in Differ. Geom., vol. 4, Int. Press, Boston, MA, 1998, 239–313. · Zbl 0931.35148 |

[14] | E. Looijenga, ”On the tautological ring of Mg,” Invent. Math., 121:2 (1995), 411–419. · Zbl 0851.14017 |

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