Mirror symmetry and toric degenerations of partial flag manifolds

In this paper we propose and discuss a mirror construction for complete intersections in partial flag manifolds F (n1, . . . , nl, n). This construction includes our previous mirror construction for complete intersection in Grassmannians and the mirror construction of Givental for complete flag manifolds. The key idea of our construction is a degeneration of F (n1, . . . , nl, n) to a certain Gorenstein toric Fano variety P (n1, . . . , nl, n) which has been investigated by Gonciulea and Lakshmibai. We describe a natural small crepant desingularization of P (n1, . . . , nl, n) and prove a generalized version of a conjecture of Gonciulea and Lakshmibai on the singular locus of P (n1, . . . , nl, n). Mathematisches Institut, Eberhard-Karls-Universitat Tubingen, D-72076 Tubingen, Germany, email address: batyrev@bastau.mathematik.uni-tuebingen.de Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA email address: ciocan@math.okstate.edu Department of Mathematics, University of California Davis, Davis, CA 95616, USA email address: bumsig@math.ucdavis.edu FB 17, Mathematik, Johannes Gutenberg-Universitat Mainz, D-55099 Mainz, Germany, email address: straten@mathematik.uni-mainz.de