Moduli of real algebraic surfaces and their superanalogues. Differentials, spinors and Jacobians of real curves.

*(English. Russian original)*Zbl 1002.14012
Russ. Math. Surv. 54, No. 6, 1091-1147 (1999); translation from Usp. Mat. Nauk 54, No. 6, 3-60 (1999).

This is a survey of the important theory of real algebraic curves, which has reached maturity in great part thanks to this author’s work. The problems are classical, and currently relevant to applications to completely integrable PDEs, as well as superalgebraic curves and quantum strings [cf. e.g. Yu. I. Manin, Proc. Steklov Inst. Math. 183, 149-162 (1991); translation from Tr. Mat. Inst. Steklova 183, 126-138 (1990; Zbl 0797.14010)].

The emphasis of the treatment is on classification and moduli problems. The methods are classical function theory (complex function theory in presence of an involution that defines a real structure); theory of line bundles that provide spinor structures; and topology, which provides concrete constructions and existence proofs by orienting and glueing contours.

After thoroughly reviewing the definitions of real structures and moduli spaces, and reviewing the proof of the real analog of the uniformization theorem for Riemann surfaces, the author defines the Arf function for a real curve, and shows that there is a one-to-one correspondence between the liftings of a Fuchsian group and the Arf functions. By analyzing the analog of Fourier series he gives a bound on the number of zeros of a real tensor. Then he sets up a theory of real Jacobian and Prymian varieties, by establishing existence theorems for real holomorphic differentials. In the last three sections the author achieves a description of moduli spaces for real algebraic supercurves, giving the number of components and the (strong) diffeomorphism type (the concept of strong diffeomorphism is introduced by the author to take care of the non-commutative variables separately), in the cases \(N=1,2\) (\(N\) being the number of the non-commutative variables).

A different approach to real algebraic curves, giving constructions of families and again differential equations for their evolution, was pursued by G. Mikhalkin [Ann. Math. (2) 151, No. 1, 309-326 (2000; Zbl 1073.14555))], using the concept of amoebas introduced in the book by I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, “Discriminants, resultants, and multidimensional determinants” (1994; Zbl 0827.14036).

The emphasis of the treatment is on classification and moduli problems. The methods are classical function theory (complex function theory in presence of an involution that defines a real structure); theory of line bundles that provide spinor structures; and topology, which provides concrete constructions and existence proofs by orienting and glueing contours.

After thoroughly reviewing the definitions of real structures and moduli spaces, and reviewing the proof of the real analog of the uniformization theorem for Riemann surfaces, the author defines the Arf function for a real curve, and shows that there is a one-to-one correspondence between the liftings of a Fuchsian group and the Arf functions. By analyzing the analog of Fourier series he gives a bound on the number of zeros of a real tensor. Then he sets up a theory of real Jacobian and Prymian varieties, by establishing existence theorems for real holomorphic differentials. In the last three sections the author achieves a description of moduli spaces for real algebraic supercurves, giving the number of components and the (strong) diffeomorphism type (the concept of strong diffeomorphism is introduced by the author to take care of the non-commutative variables separately), in the cases \(N=1,2\) (\(N\) being the number of the non-commutative variables).

A different approach to real algebraic curves, giving constructions of families and again differential equations for their evolution, was pursued by G. Mikhalkin [Ann. Math. (2) 151, No. 1, 309-326 (2000; Zbl 1073.14555))], using the concept of amoebas introduced in the book by I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, “Discriminants, resultants, and multidimensional determinants” (1994; Zbl 0827.14036).

Reviewer: Emma Previato (Boston)

##### MSC:

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

14H40 | Jacobians, Prym varieties |

14M30 | Supervarieties |

14P25 | Topology of real algebraic varieties |

81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |

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

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