Birational Geometry is a classical mathematical discipline whose roots go back to ancient Greece. Nevertheless, it still offers many hard unsolved problems. The core part of this project is tackling these problems with cutting edge modern methods coming from the Homological Mirror Symmetry (HMS) program.
HMS is a deep geometric duality which originates in Quantum Field Theory. Traditionally, HMS is used in studying novel phenomena and proving unexpected results in Symplectic Geometry suggested by algebraic geometry.
This project uses HMS to produce new unexpected applications of symplectic topology to algebraic geometry, to answer classical open problems in birational geometry, and to ultimately bring a quite new prospective on the way geometry is done today.
Technically our approach is based on Categorical Kaehler geometry – a direction which is being developed by M. Kontsevich and the principal investigator. The most notable application of this approach is the proof of the non rationality of generic four dimensional cubic – arguably the central problem in rationality.
The detailed study of the singularities of the quantum D module produces a completely new type of birational invariant. This new invariant is a canonical decomposition of the cohomologies of the four dimensional cubic based on simultaneous use of both (algebraic and symplectic) sides of HMS.
The example of 4 dimensional cubic is only the tip of the iceberg. There are many other examples and applications of the above approach – e.g. applications to uniformization problems.
The progress in all these directions will be disseminated during several events in the International Center for Mathematical Sciences at the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences (ICMS – Sofia).
Bulgarian National Scientific Program “VIHREN”, Project № KP-06-DV-7
Principal Investigator (PI):
Ludmil Katzarkov
Host institution:
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences
Project Period:
January 2020 – December 2024
Team

Stefan Ivanov
Established Researcher

Alexander Petkov
Recognized Researcher

Enrique Ruby Becerra
Postdoctoral fellow

Nicola Stanchev
PhD Student

Erik Paemurru
Postdoctoral fellow

Tokio Sasaki
Postdoctoral fellow

Jiachang Xu
Postdoctoral fellow

Marin Genov
PhD Student
Major Results
The unifying theme of our research is categorical Kaehler Geometry initiated by M. Kontsevich and L. Katzarkov. During the first year we have obtained results in the following directions:
- Developing foundations of new Birational invariants – spectra: Noncommutative, quantum, Givental Steenbrink. These invaraints are part of a new categorical singularity theory.
- Study relations between these spectra and their relations with other parts of Mathematics and theoretical Physics.
- Developing the foundations of Quantum toric geometry. This is a part of a bigger program – creating Chimeric Geometry.
- Developing theory of spectral networks and initiating theory of invariants they produce. This will lead to a new approach to classical uniformation questions.
- Developing the theory of moduli spaces of objects. We also investigate the stratification inflicted on this moduli spaces by Noncommutative spectra.
- New results on special metrics – classical and categorical.
In the second and third year we obtained the following results:
- The non-rationality of many 3- and 4-dimensional Fanos unknown before is proved.
- We extended the methods and results to the case of Fano complete intersections in weighted projective spaces and more general Fano stacks.
- We extended these results to the case of algebraically non-closed fields making a connection between Physics and Logics from one side and Arithmetics on the other.
- Proof of Blow up Formulae.
- Proof of Semi-continuity of Spectra.
- The existence of Moduli spaces of objects for 3- and 4-dimensional CY manifolds is proved.
- The proof is based on a recent work by T. Pantev, B. Toen, G. Vessozi, M. Vaquette and makes a connection between shifted symplectic structures and derived categories.
- We expect that the stratification defined on these moduli spaces and the spectra for the corresponding categories are closely related. In that respect, we outline the connection between non-commutative spectrum and Vafa-Witten invariants.
- We defined the Quantum Toric Geometry in the new cite of o-minimal structures.
- We defined the theory of deformations for quantum stacks.
- The existence of Fukaya-Seidel category with coefficients is established.
- The proof of the Homological Mirror Symmetry (HMS) for Fukaya-Seidel category with coefficients is obtained in several important cases.
- We developed the study of the discriminanats of toric varieties.
- New examples of complete Einstein Riemannian manifolds of negative scalar curvature are constructed as a hyperbolic extension of a complete paraholomorphic paracomplex Riemannian Einstein space of negative scalar curvature. For example, we show that the hyperbolic extension of the product of two n-dimensional discs with the Poincaré metric is a complete odd dimensional Riemannian Einstein space with negative scalar curvature.
- An Obata type theorem on the 3-Sasakian sphere is proved. All solutions to the quaternionic contact Yamabe equation on the qc sphere of dimension 4n + 3 as well as on the quaternionic Heisenberg group are determined.
- A uniqueness theorem for the qc Yamabe problem in a compact locally 3-Sasakian manifold is shown. Consequently, all extremals of the Folland-Stein inequality on the quaternionic Heisenberg groups are explicitly determined.
- A new version of the CR almost Schur Lemma is established. It gives an estimation of the pseudohermitian scalar curvature on a compact strictly pseudoconvex pseudohermitian manifold, which in the torsion-free case is a better estimate compared with the known ones.
- Quaternionic contact (qc) versions of the so called Almost Schur Lemma are also established.
- Curvature properties in the geometries with skew-symmetric torsion and special holonomy SU(3) are established and the reflection on the exterior derivative and covariant derivative of the torsion are obtained.
In the fourth and fifth year, we obtained the following results:
- We introduced a new type of “Hodge theory” – Theory of atoms. This is a new type of quasitanakian category by which we defined new birational and symplectic invariants.
- Based on the Theory of atoms, we obtained obstruction to rationality of orbifolds and stacks.
- Combined with classical invariants we obtained even stronger invariants.
- We have given a new proof of a theorem of Kapranov. Using that we described new stratification on moduli space of objects.
- We extend classical Hodge theory to a new definable structures. Many of the classical results of Hodge theory were extended to this new definable structure.
- We developed a new Fukaya-Seidel category using perverse sheaf of categories. We proved Homological Mirror Symmetry (HMS) for Fukaya-Seidel category in these new terms.
- Curvature properties in the geometries with skew-symmetric torsion and special holonomy SU(3), G2 and Spin(7) are established and the reflection on the exterior derivative and covariant derivative of the torsion are obtained. If the torsion is closed, it is shown that these geometries are generalized Ricci solitons.
- The quaternionic contact heat equation is investigated and an entropy-type formula is established towards finding geometric inequalities, in particular, a quaternionic contact version of the famous Li-Yau inequality, which is one of the main ingredients in the famous Hamilton Ricci flow and Perelman entropy.
- Paraquaternionic conformal deformations of a paraquaternionic contact manifold as well as a pqc curvature tensor are defined. It is proved that the pqc curvature is invariant under paraquaternionic contact deformations. It is shown that the pqc curvature vanishes if and only if the pqc manifold is locally paraquaternionic contact equivalent to the paraquaternionicHeisenberg group as well as to the paraquaternionic psedosphere.
Events
Public communication
CKGA project at the 11th Sofia Science Festival
The CKGA project was presented at the 11th Sofia Science Festival which took place on 15th and 16th May 2021 at Sofia Tech Park. The Festival was organized by the British Council - Bulgaria under the patronage of the Ministry of Education and Science.