# MOTZETA Résumé de rapport

Project ID:
306610

Financé au titre de:
FP7-IDEAS-ERC

Pays:
United Kingdom

## Mid-Term Report Summary - MOTZETA (Motivic zeta functions and the monodromy conjecture)

The monodromy conjecture, formulated in the seventies by the Japanese mathematician Igusa, is one of the most important open problems in the theory of singularities. It predicts a remarkable connection between certain geometric and arithmetic invariants of a polynomial f with integer coefficients. More precisely, the monodromy conjecture states that the real parts of the poles of Igusa's zeta function associated with f correspond to local monodromy eigenvalues of f. Some special cases have been proven, but the general case remains wide open. A proof of the conjecture would unveil profound relations between singularity theory and number theory.

The aim of this ERC project is to develop a new array of tools to attack the conjecture, based on recent insights from various branches of mathematics, most prominently non-archimedean geometry, the minimal model program and mirror symmetry. Mirror symmetry is rooted in mathematical physics (string theory) but it has proven to be a very fertile heuristic for impressive breakthroughs in algebraic geometry, as well.

The biggest success of the project so far is the solution of a 1999 conjecture by Wim Veys, which predicted that the real part of every pole of maximal order of Igusa's zeta function equals minus the log canonical threshold of f, one of the most important invariants in birational geometry. In particular, such poles satisfy the monodromy conjecture. Our proof, obtained in collaboration with Chenyang Xu, is a perfect illustration of the general strategy of the ERC project. The heuristic of mirror symmetry indicated that certain objects in non-archimedean geometry should have special properties. We have proven these properties using advanced methods from the minimal model program in birational geometry, and shown that they apply not only in the traditional set-up of mirror symmetry but also to hypersurface singularities. One of these properties then implied Veys's conjecture as a special case. We expect that this strategy will lead to further important advances towards the monodromy conjecture.

The aim of this ERC project is to develop a new array of tools to attack the conjecture, based on recent insights from various branches of mathematics, most prominently non-archimedean geometry, the minimal model program and mirror symmetry. Mirror symmetry is rooted in mathematical physics (string theory) but it has proven to be a very fertile heuristic for impressive breakthroughs in algebraic geometry, as well.

The biggest success of the project so far is the solution of a 1999 conjecture by Wim Veys, which predicted that the real part of every pole of maximal order of Igusa's zeta function equals minus the log canonical threshold of f, one of the most important invariants in birational geometry. In particular, such poles satisfy the monodromy conjecture. Our proof, obtained in collaboration with Chenyang Xu, is a perfect illustration of the general strategy of the ERC project. The heuristic of mirror symmetry indicated that certain objects in non-archimedean geometry should have special properties. We have proven these properties using advanced methods from the minimal model program in birational geometry, and shown that they apply not only in the traditional set-up of mirror symmetry but also to hypersurface singularities. One of these properties then implied Veys's conjecture as a special case. We expect that this strategy will lead to further important advances towards the monodromy conjecture.