INVLOCCY · Invariants of local Calabi-Yau 3-folds
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2015-06-01 → 2017-05-31
- EU contribution
- €165,599
- Participants
- 1
- Scheme
- MSCA-IF-EF-ST
Lines connect the coordinator with its partners.
Results in brief
Invariants of local Calabi-Yau 3-folds
Enumerative geometry on complex algebraic surfaces is a classical topic dating back to the 19th century. Typical problems are: how many lines in the plane go through 2 points (answer 1), or how many rational planar cubics go through 8 general points (answer 12). Modern invariants originating from string theory have provided new tools for attacking such problems as is most strikingly illustrated by M. Kontsevich's determination of all genus zero Severi degrees of the projective plane by using Gromov-Witten invariants. Other invariants, known as Pandharipande-Thomas or stable pair invariants, are relevant for the determination of Severi degrees for sufficiently ample linear systems as was shown in joint work of the author and R.P. Thomas. Together with R.P. Thomas and V. Shende, this led to a proof of Göttsche's conjecture, which loosely states that Severi degrees for sufficiently ample linear systems on surfaces only depend on the topology. The work of Kool-Thomas and Kool-Shende-Thomas mentioned above, only concerns a special type of stable pairs on the total space of the canonical bundle K_S of a surface S, namely those stable pairs which have the property that they are scheme-theoretically supported in the zero section of K_S. Theme 1 of the project INVLOCCY aims to explore much more general stable pairs on K_S, which need not be constrained to the zero section. Inspired by the applications mentioned above, I expected this has interesting applications as well. Theme 2 of the project INVLOCCY is concerned with refinements of invariants especially with an eye towards the physics literature. These problems are purely theoretical in nature with no applications and as such their importance is part of the relevance of the development of fundamental mathematics for its own sake. The topic of this proposal has deep links with other branches of mathematics and theoretical physics, namely the theory of modular forms, enumerative geometry, combinatorics, and string theory.
Data: CORDIS, © European Union
Project objective
The study of Gromov-Witten (GW), Donaldson-Thomas, and stable pair invariants of Calabi-Yau 3-folds X forms an active area of research for geometers and physicists. These invariants play a central role in string theory and have relations with many branches of mathematics including number theory and representation theory. I am interested in questions of enumerative geometry on algebraic surfaces S. Invariants of the total space X of the canonical bundle over S can be used to answer classical enumerative questions on S. Two recent developments in stable pair theory are: (1) A better understanding of stable pairs on X not contained in the zero-section S. (2) Refinements of stable pair invariants. The first theme of my project is the study of stable pairs on X not contained in S in relation to enumerative questions. For Fano surfaces, GW invariants with sufficiently many point insertions are enumerative. By the GW/stable pairs correspondence these are equal to certain stable pair invariants of X. When the curve class is not sufficiently ample, the stable pair count may include stable pairs on X not contained in S. I propose to compute such contributions in order to obtain curve counts on S outside the ample regime. The second theme of my project is the study of refined stable pair invariants. I intend to relate the refined topological vertex appearing in the physics literature to refined invariants in the mathematics literature. Since stable pair invariants are often easiest to calculate of all the invariants of Calabi-Yau 3-folds, I expect this leads to new curve counting formulae and new calculations of refined invariants.Utrecht University, housing one of the leading schools in geometry in Europe, and Prof. Faber, one of the world's leading experts on moduli of curves, provide the perfect location and supervisor for this project. The diverse expertise of the members of the Mathematics (and Physics) Department at UU allow me to explore links with other areas.
Original text from CORDIS.
Participants
- UNIVERSITEIT UTRECHT · UtrechtCoordinatorNetherlands
Links
Data: CORDIS, © European Union