Periodic Reporting for period 1 - SCGA (Derived categories, stability conditions and geometric applications)
Berichtszeitraum: 2021-09-01 bis 2023-08-31
Geometry studies higher-dimensional curved spaces. We can describe these spaces by equations, but the only case where we have any hope to use them for calculation is when the equations are polynomials. The resulting spaces are the objects of algebraic geometry, which are called varieties. The geometric information of varieties can be encoded in algebraic objects, known as derived categories. Inspired by ideas in string theory in physics, Bridgeland introduced the notion of stability conditions on derived categories.
The main goal of this intradisciplinary research was to employ ideas and tools in algebra, geometry and mathematical physics to develop new techniques for solving long-standing problems in algebraic geometry. The central theme is the deformation of stability conditions on derived categories and wall-crossing.
The main impact of this research programme will be to enhance the existing expertise in pure mathematics in Europe. My projects employed ideas and tools in different fields of mathematics to develop modern techniques for answering many fundamental questions in algebraic geometry. In particular, it strengthens existing links between algebraic geometry, homological algebra, birational geometry, symplectic geometry, enumerative geometry and mathematical physics, which are all very active areas of research in Europe. Hence my projects will increase the possibility of collaborations between researchers in these fields. A great occasion to transfer ideas and techniques between different research areas is intradisciplinary workshops which I have attended and delivered talks several times during my fellowship. These workshops will have a significant impact on Europe and international mathematical communities.
Impact on the younger generation of scientists: I delivered several talks in Junior Geometry seminars and schools around the world. I have also introduced new projects to several Master/Phd students and I plan to supervise these projects.
Although my research programme is focussing on topics in mathematics, it has the potential of influencing other sciences. Lots of objectives in the proposal are closely related to string theory, hence, the results and techniques would be helpful for physicists. Moreover, the second project involves a detailed study of stable vector bundles holding a specific number of global sections on algebraic curves, which will be useful for algebraic-geometric codes in coding theory.
The main goal of this intradisciplinary research was to employ ideas and tools in algebra, geometry and mathematical physics to develop new techniques for solving long-standing problems in algebraic geometry. The central theme is the deformation of stability conditions on derived categories and wall-crossing.
The main impact of this research programme will be to enhance the existing expertise in pure mathematics in Europe. My projects employed ideas and tools in different fields of mathematics to develop modern techniques for answering many fundamental questions in algebraic geometry. In particular, it strengthens existing links between algebraic geometry, homological algebra, birational geometry, symplectic geometry, enumerative geometry and mathematical physics, which are all very active areas of research in Europe. Hence my projects will increase the possibility of collaborations between researchers in these fields. A great occasion to transfer ideas and techniques between different research areas is intradisciplinary workshops which I have attended and delivered talks several times during my fellowship. These workshops will have a significant impact on Europe and international mathematical communities.
Impact on the younger generation of scientists: I delivered several talks in Junior Geometry seminars and schools around the world. I have also introduced new projects to several Master/Phd students and I plan to supervise these projects.
Although my research programme is focussing on topics in mathematics, it has the potential of influencing other sciences. Lots of objectives in the proposal are closely related to string theory, hence, the results and techniques would be helpful for physicists. Moreover, the second project involves a detailed study of stable vector bundles holding a specific number of global sections on algebraic curves, which will be useful for algebraic-geometric codes in coding theory.
Due to some visa issues, I could stay in Paris and use my fellowship only for four months, so the following is a summary of the two projects that I have finished in this short period.
1) Hyperkahler manifolds as Brill-Noether loci on K3 curves.
Hyperkahler manifolds play a crucial role in complex algebraic geometry. In this project, I have shown for the first time that we can obtain Hyperkahler manifolds of arbitrary dimensions as moduli spaces of special vector bundles on curves. The latter are spaces that parametrise these special bundles.
Fix a K3 surface X and curve C on surface X. Consider the moduli space M of stable vector bundles on C with canonical determinant which have the maximum number of linearly independent global sections. I have shown that if the genus of C is high enough, then the moduli space M is a smooth projective Hyperkahler manifold which is isomorphic to a moduli space of stable vector bundles on X.
2) Explicit formulae for rank zero and rank two Donaldson-Thomas invariants; and the OSV conjecture.
One of the very active modern topics in mathematics is Donaldson–Thomas theory which is the theory of Donaldson–Thomas invariants. Given a compact moduli space of sheaves on a Calabi–Yau threefold, its Donaldson–Thomas invariant is the virtual number of its points.
In the first part of this project, I have described an explicit formula for rank zero Donaldson–Thomas invariant in terms of rank one Donaldson–Thomas invariant. This is significant in light of the prediction by physicists, which gives hope of expressing rank zero Donaldson-Thomas invariants in terms of modular forms. This would be a major advance in the area.
In the second part of the project, I have proved a mathematical formulation of a famous conjecture by physicists (Ooguri, Strominger and Vafa) for rank zero Donaldson-Thomas invariants.
In the final part, I have described an explicit formula for rank two Donaldson-Thomas invariants in terms of rank one Donaldson-Thomas invariants, i.e. curve counting. This is the first time that such a general formula is proved for an arbitrary rank 2 class on an arbitrary Calabi–Yau threefold.
1) Hyperkahler manifolds as Brill-Noether loci on K3 curves.
Hyperkahler manifolds play a crucial role in complex algebraic geometry. In this project, I have shown for the first time that we can obtain Hyperkahler manifolds of arbitrary dimensions as moduli spaces of special vector bundles on curves. The latter are spaces that parametrise these special bundles.
Fix a K3 surface X and curve C on surface X. Consider the moduli space M of stable vector bundles on C with canonical determinant which have the maximum number of linearly independent global sections. I have shown that if the genus of C is high enough, then the moduli space M is a smooth projective Hyperkahler manifold which is isomorphic to a moduli space of stable vector bundles on X.
2) Explicit formulae for rank zero and rank two Donaldson-Thomas invariants; and the OSV conjecture.
One of the very active modern topics in mathematics is Donaldson–Thomas theory which is the theory of Donaldson–Thomas invariants. Given a compact moduli space of sheaves on a Calabi–Yau threefold, its Donaldson–Thomas invariant is the virtual number of its points.
In the first part of this project, I have described an explicit formula for rank zero Donaldson–Thomas invariant in terms of rank one Donaldson–Thomas invariant. This is significant in light of the prediction by physicists, which gives hope of expressing rank zero Donaldson-Thomas invariants in terms of modular forms. This would be a major advance in the area.
In the second part of the project, I have proved a mathematical formulation of a famous conjecture by physicists (Ooguri, Strominger and Vafa) for rank zero Donaldson-Thomas invariants.
In the final part, I have described an explicit formula for rank two Donaldson-Thomas invariants in terms of rank one Donaldson-Thomas invariants, i.e. curve counting. This is the first time that such a general formula is proved for an arbitrary rank 2 class on an arbitrary Calabi–Yau threefold.
The two projects that I have finished during the four-month fellowship in Paris are both far beyond what was originally expressed in the state of the art. The key technique of deforming stability conditions and wall-crossing was much more useful than what I originally thought.
My first project made a new bridge between differential geometers and algebraic geometers using stability conditions.
The second project turn out to be very interesting for physicists and made new clear connections between the predictions in string theory and the modern area of research in algebraic geometry.
My first project made a new bridge between differential geometers and algebraic geometers using stability conditions.
The second project turn out to be very interesting for physicists and made new clear connections between the predictions in string theory and the modern area of research in algebraic geometry.