Oscillating integrals and a generalization

• Applicant: Professor Dr. Claus Hertling
• Subject Area: Mathematics
• Term: from 2005 to 2014
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 16442363

Givental worked out cases of mirror symmetry, in which the quantum cohomology of Fano manifolds is on one side, and oscillating integrals of holomorphic tame functions on affine manifolds are on the other side. These oscillating integrals generalize the period integrals leading to variations of Hodge structures. In fact, by work of Sabbah, they give rise to mixed Hodge structures. The first goal of the project is to study arithmetic properties of these oscillating integrals and of Sabbah´s mixed Hodge structures. This will enrich Givental´s mirror symmetry by incorporating the real structures and the mixed Hodge structures which live on both sides. It will also complete a conjecture of Dubrovin on the quantum cohomology of Fano manifolds. The second goal is to develop the structure theory of a generalization of oscillating integrals. It consists in the direct images of elementary D-modules and provides a very natural geometric class of D-modules with irregular singularities.

• DFG Programme: Research Grants