Project Details
Projekt
Log Hodge Structures in Mirror Symmetry
Applicant
Simon Felten
Subject Area
Mathematics
Term
since 2021
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 451094600
In geometry, we study spaces: geometric structures like lines, planes or the surface of a ball. In Mirror Symmetry - an area of geometry - we are interested in pairs of spaces that are complementary in an abstract sense. Our vision is to understand exactly why these pairs exist. As a method to reach this goal, we can associate a so-called Hodge structure to each space; this Hodge structure encodes information about the space. In doing so, we have two advantages: Firstly, many spaces are already determined by their Hodge structures. Secondly, it is often easier to study the Hodge structure than the space itself.When we modify a space somewhat, then its - with a view towards mirror symmetry - relevant properties remain the same. This remains true even when we modify the space so much that it is degenerate. For example, when we change the parameter of a hyperbola in such a way that the hyperbola gets closer to the coordinate axes until they coincide, then we have a degeneration. Gross and Siebert have established a new approach to Mirror Symmetry. According to this approach, we don't study the hyperbola itself, but its degeneration, namely the two coordinate axes. This degeneration is not only a space, but indeed a logarithmic space. Roughly speaking, this means that the space "knows" that it originates from a hyperbola, and that it has not always been degenerate.Since we can associate a Hodge structure to every space, we expect that we can associate a logarithmic Hodge structure (LHS) to every logarithmic space. We already know what a LHS should be, but we don't know yet how to associate one to a log space. Currently, we only know this in the simplest cases. The main goal of this project is to find out how we can associate a LHS to a log space.Gross and Siebert have shown how we can get mirror pairs of log spaces. However, these log spaces have a very special form. In contrast, LHS describe - conjecturally - much more general log spaces. Thus, if we can translate the mirror construction of Gross and Siebert to the realm of LHS, then we might see more clearly why mirror pairs exist.
DFG Programme
WBP Fellowship
International Connection
USA
