Vector bundles on the cubic threefold.

*(English)*Zbl 1056.14059
Bertram, Aaron (ed.) et al., Symposium in honor of C. H. Clemens. A weekend of algebraic geometry in celebration of Herb Clemens’s 60th birthday, University of Utah, Salt Lake City, UT, USA, March 10–12, 2000. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-2152-0/pbk). Contemp. Math. 312, 71-86 (2002).

From the introduction: Let \(X\) be a smooth cubic hypersurface in \(\mathbb{P}^4\). In their seminal paper [Ann. Math. (2) 95, 281–356 (1972; Zbl 0214.48302)], H. Clemens and P. Griffiths showed that the intermediate Jacobian \(J(X)\), an abelian variety defined analytically through Hodge theory, is a fundamental tool to understand the geometry of \(X\). They studied the Fano surface \(F\) of lines contained in \(X\), proving that the Abel-Jacobi map embeds \(F\) into \(J(X)\) and induces an isomorphism \(\text{Alb}(F)@>\sim>> J(X)\). They were able to deduce from this the Torelli theorem and the non-rationality of \(X\)

Mumford noticed that one can expess \(J(X)\) as a Prym variety and thus gave an alternate proof for the non-rationality of \(X\) [H. Clemens and P. Griffiths, loc. cit., Appendix C]. Later Clemens observed that one could use the twisted cubics as well, giving an elegant parametrization of the theta divisor. At this point the cubic threefold could be considered as well understood, and the emphasis shifted to other Fano threefolds.

On the other hand vector bundles on low-dimensional varieties attracted much attention in the last two decades, notably because of their relation with mathematical physics. Since rank 2 vector bundles are well known to be connected to codimension 2 cycles, it seems obvious to try to relate them to the intermediate Jacobian. Surprisingly this was done only recently [cf. D. Markushevich and A. Tikhomirov, J. Algebr. Geom. 10, 37–62 (2001; Zbl 0987.14028)], followed by [A. Iliev and D. Markushevich, Doc. Math., J. DMV 5, 23–47 (2000; Zbl 0938.14021)] and [S. Druel, Int. Math. Res. Not. 2000, No. 19, 985–1004 (2000; Zbl 1024.14004)]. The author explains these results in this paper and gives some applications.

He looks at the simplest possible case, namely stable rank 2 vector bundles on \(X\) with trivial determinant. Using the isomorphism \(\deg: H^4(X,\mathbb{Z})@>\sim>>\mathbb{Z}\) it is easily seen that the second Chern class \(c_2\) is a number \(\geq 2\). This leads the author to consider the moduli space \({\mathcal M}\) of stable rank 2 vector bundles on \(X\) with \(c_1= 0\), \(c_2= 2\). It is a quasi-projective variety that admits a natural compatification \(\overline{\mathcal M}\) consisting of classes of semi-stable sheaves with \(c_1= 0\), \(c_2= 2\), \(c_3= 0\). The main result is:

Theorem 1.1. The moduli space \(\overline{\mathcal M}\) is isomorphic to the intermediate Jacobian of \(X\) blown up along the Fano surface.

The theorem as stated is due to S. Druel [loc. cit.]; the proof relies heavily on the results in the papers by Markushevich and Tikhomirov, respectively Iliev and Markushevich cited above. These papers are rather technical, making it worthwhile to have a simplified version, that focuses on the geometric ideas, which is the main goal of this paper. The last sections contain some applications, in particular the construction of a completely integrable Hamiltonian system related to the situation.

For the entire collection see [Zbl 1002.00010].

Mumford noticed that one can expess \(J(X)\) as a Prym variety and thus gave an alternate proof for the non-rationality of \(X\) [H. Clemens and P. Griffiths, loc. cit., Appendix C]. Later Clemens observed that one could use the twisted cubics as well, giving an elegant parametrization of the theta divisor. At this point the cubic threefold could be considered as well understood, and the emphasis shifted to other Fano threefolds.

On the other hand vector bundles on low-dimensional varieties attracted much attention in the last two decades, notably because of their relation with mathematical physics. Since rank 2 vector bundles are well known to be connected to codimension 2 cycles, it seems obvious to try to relate them to the intermediate Jacobian. Surprisingly this was done only recently [cf. D. Markushevich and A. Tikhomirov, J. Algebr. Geom. 10, 37–62 (2001; Zbl 0987.14028)], followed by [A. Iliev and D. Markushevich, Doc. Math., J. DMV 5, 23–47 (2000; Zbl 0938.14021)] and [S. Druel, Int. Math. Res. Not. 2000, No. 19, 985–1004 (2000; Zbl 1024.14004)]. The author explains these results in this paper and gives some applications.

He looks at the simplest possible case, namely stable rank 2 vector bundles on \(X\) with trivial determinant. Using the isomorphism \(\deg: H^4(X,\mathbb{Z})@>\sim>>\mathbb{Z}\) it is easily seen that the second Chern class \(c_2\) is a number \(\geq 2\). This leads the author to consider the moduli space \({\mathcal M}\) of stable rank 2 vector bundles on \(X\) with \(c_1= 0\), \(c_2= 2\). It is a quasi-projective variety that admits a natural compatification \(\overline{\mathcal M}\) consisting of classes of semi-stable sheaves with \(c_1= 0\), \(c_2= 2\), \(c_3= 0\). The main result is:

Theorem 1.1. The moduli space \(\overline{\mathcal M}\) is isomorphic to the intermediate Jacobian of \(X\) blown up along the Fano surface.

The theorem as stated is due to S. Druel [loc. cit.]; the proof relies heavily on the results in the papers by Markushevich and Tikhomirov, respectively Iliev and Markushevich cited above. These papers are rather technical, making it worthwhile to have a simplified version, that focuses on the geometric ideas, which is the main goal of this paper. The last sections contain some applications, in particular the construction of a completely integrable Hamiltonian system related to the situation.

For the entire collection see [Zbl 1002.00010].