Smoothings from log resolutions and applications

• Applicant: Dr. Tim Gräfnitz
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 567054097

Fano varieties are fundamental building blocks in algebraic geometry. A conjecture, partially motivated by mirror symmetry in string theory, suggests a correspondence between Fano varieties with mild singularities and Laurent polynomials exhibiting certain mutation properties. For dimensions up to 3, a classification of Fano varieties is known and the conjecture has been verified by explicit computations. In higher dimensions, however, no classification is known, and the mentioned correspondence is an important progress in this regard.In my program, I pursue a new approach towards a proof of this conjecture. I construct Fano varieties via deformations of toric varieties with Gorenstein singularities. First, the toric variety is degenerated into a union of several toric varieties that intersect in toric divisors. This union is endowed with a log structure, as defined in logarithmic geometry. The log singular locus of this structure is contained within the toric divisors, and its precise form is determined by a Laurent polynomial with the aforementioned mutation properties. Log resolutions of the log singularities are constructed via divisorial extractions, a class of birational maps, of the log singular locus. The result is a generally toroidal crossing space with a log structure which is, except at isolated points, log smooth over the standard log point. Generalizing existing techniques, a smoothing of this object into a Fano variety with terminal singularities is constructed.The approach outlined above is a natural extension of the Gross-Siebert program, in which similar toric degenerations are constructed, but in which the log singular locus must be intersecting transversally. My approach allows for a generalization of tropical correspondence theorems to higher-dimensional non-toric varieties. On this basis, Gromov-Witten invariants can be computed, which naturally appear in mirror symmetry and intuitively count curves on the variety. Moreover, my approach allows for the construction of degenerations of Fano varieties, from which statements about the (stable) rationality of the variety can be made. Stable rationality is a central concept in birational geometry, a branch of algebraic geometry aiming at a classification of varieties up to isomorphism outside of lower-dimensional subvarieties.

• DFG Programme: Emmy Noether Independent Junior Research Groups