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Spectral curves, simple Lie algebras, and Prym-Tjurin varieties. (English) Zbl 0707.14041
Theta functions, Proc. 35th Summer Res. Inst. Bowdoin Coll., Brunswick/ME 1987, Proc. Symp. Pure Math. 49, part 1, 627-645 (1989).
[For the entire collection see Zbl 0672.00003.]
The Lax equations $\frac{d}{dt}A(\xi,t)=[B(\xi,t),A(\xi,t)]$ where A and B are polynomials in $$\xi$$, $$\xi^{-1}$$ and take values in an irreducible representation of simple Lie algebra $${\mathfrak g}\subset gl_ n$$ are considered. An accurate definition of the spectral curve C: $$\det(\eta I-A(\xi,t))=0$$ is suggested. The symmetries of spectral curves for which A($$\xi$$,t) take values in simple Lie algebras of types $$A_ n$$, $$D_ n$$, $$E_ 6$$, $$E_ 7$$, $$E_ 8$$ are investigated. It turns out that in case $${\mathfrak g}\subsetneqq sl_ n$$ an endomorphism $$i: I(C)\to I(C)$$ exists, which satisfies the equation $$(i-1)(i+q-1)=0$$ for some $$q\geq 2$$. Here I(C) is a Jacobian of C. The number q depends on $${\mathfrak g}$$ and on the representation $${\mathfrak g}\hookrightarrow gl_ n$$. The abelian variety $$P(C,i)=(1-i)I(C)$$ is called Prym-Tjurin variety. It is proved that if certain conditions on A are satisfied then, provided A and B take values in $${\mathfrak g}\subsetneqq sl_ n$$, the flow generated by the dynamical system (*) can be linearized on the Prym-Tjurin variety $$P(C,i)\subset I(C)$$. It is shown also that the isogeny class of Prym- Tjurin varieties obtained from spectral curves does not depend on the representation of $${\mathfrak g}$$. In the construction above theta functions on Jacobi varieties correspond to $$A_ n$$, theta functions on Prym varieties $$(q=2)$$ correspond to $$D_ n$$, theta functions on Prym-Tjurin varieties with $$q=6$$ (resp. $$q=12)$$ correspond to $$E_ 6$$ (resp. $$E_ 7)$$.
A.Bobenko
Let the following data be given: (i) a complex, reductive Lie-algebra $${\mathfrak g}$$ and a rational map $$f: {\mathbb{C}}\to {\mathfrak g}$$ such that for almost all $$\xi\in {\mathbb{C}}$$ the element $$f(\xi$$) is regular and semi- simple: (ii) a non-trivial, semi-simple, irreducible representation $$\rho: {\mathfrak g}\to gl(V)$$, where $$\dim (V)=n$$. Then, (i) and (ii) determine uniquely a non-singular, projective, possibly non-connected curve C, the so-called spectral curve, with a d-sheeted covering $$p: C\to {\mathbb{P}}^ 1.$$ C can be considered as the desingularization of certain irreducible components of the closure (in $${\mathbb{P}}^ 2)$$ of the plane curve $$X\subset {\mathbb{C}}^ 2$$ determined by a completely integrable Hamiltonian system with Lax representation: $(1)\quad \frac{d}{dt}A(\xi,t)= [B(\xi,t),A(\xi,t)],$ where A and B are polynomials in $$\xi,\xi^{-1}$$ and take values in $$gl_ n$$. An interesting example of a spectral curve arises when $${\mathfrak g}$$ is of type $$E_ 6$$ and one imposes a suitable condition on the weights of $$\rho$$. In this case C is a 27-sheeted covering of $${\mathbb{P}}^ 1$$, and the general fibre can be identified with the 27 lines on a cubic surface.
An important role is played by the Prym-Tjurin varieties. Assume $${\mathfrak g}\subsetneqq sl_ n$$, then one can construct a non-zero correspondence D on C that induces an endomorphism $$i: Jac(C)\to Jac(C)$$, such that $$(i- 1)(i+q-1)=0$$ for some $$q\geq 2$$. The integer q depends on $${\mathfrak g}$$ and the representation $$\rho$$. In the aforementioned example $$q=6$$. For other root systems one can work with Del Pezzo surfaces and one finds e.g. for $$A_ 4, D_ 4, E_ 7, E_ 8$$ the q-values 3, 4, 12 and 60, respectively. The abelian variety $$P(C,i)=(1-i)Jac(C)$$ is called the Prym-Tjurin variety.
One has the following theorem: Let $${\mathfrak g}$$ be a Lie-algebra of type $$A_ n, D_ n, E_ 6, E_ 7$$ or $$E_ 8$$, and let f be as in (i). Let $$\rho$$ and $$\rho'$$ be two representations as in (ii). Then the corresponding Prym-Tjurin varieties $$P(C,i)$$ and $$P(C',i')$$ are isogenous. If, furthermore, f is in ‘general position’, and $$\rho$$, $$\rho'$$ correspond to so-called minuscule weights, then there is a polarized isomorphism $$(P(C,i),\Xi)\overset \sim \rightarrow (P(C',i'),\Xi')$$. - Another result of the paper says that under suitable conditions on A and B the flow generated by the dynamical system (1) can be linearized on $$P(C,i)$$. This result may be considered with respect to the well-known fact that for $${\mathfrak g}=so(2n)$$ the flow determined by (1) can be linearized on a complex subtorus, the Prym variety, $$P(C,\sigma)=(1-\sigma)Jac(C)$$, where $$\sigma$$ is an involution on C. In this case the Prym-Tjurin variety coincides with the Prym variety.
W.Hulsbergen

##### MSC:
 14K30 Picard schemes, higher Jacobians 14H40 Jacobians, Prym varieties 14K25 Theta functions and abelian varieties 14K10 Algebraic moduli of abelian varieties, classification 17B20 Simple, semisimple, reductive (super)algebras 14K02 Isogeny