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

ivanovsp@fmi.uni-sofia.bg

## Alexander Petkov

### Recognized Researcher

a.petkov@fmi.uni-sofia.bg

Enrique Ruby Becerra

### Postdoctoral fellow

ebecerra@math.cinvestav.mx

Erik Paemurru

### Postdoctoral fellow

erik.paemurru@gmail.com

Tokio Sasaki

### Postdoctoral fellow

sasakitokio@math.miami.edu

## Marin Genov

### PhD Student

m.genov@math.miami.edu

# 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.

# Events

# Papers

- Ludmil Katzarkov, Ernesto Lupercio, Laurent Meersseman, Alberto Verjovsky,
**Quantum (Non-Commutative) Toric Geometry:***Foundations, Advances in Mathematics,**Volume 391, 19 November 2021,*107945, https://www.sciencedirect.com/science/article/pii/S0001870821003844, IF: 1.688 (Q1) - Stefan Ivanov, Hristo Manev, Mancho Manev,
**Para-Sasaki-like Riemannian manifolds and new Einstein metrics**,*RACSAM 115, 112 (2021)*, https://link.springer.com/article/10.1007/s13398-021-01053-z, IF: 2.169 (Q1) - Stefan Ivanov, Ivan Minchev, Dimiter Vassilev,
**Solution of the Qc Yamabe Equation on a 3-Sasakian Manifold and the Quaternionic Heisenberg Group**, accepted for publication in*Analysis & PDE,*IF: 2.576 (Q1) - Denis Auroux, Alexander I. Efimov, Ludmil Katzarkov,
**Lagrangian Floer Theory for Trivalent Graphs and Homological Mirror Symmetry for Curves**, (submitted) - L. Katzarkov, M. Kontsevich, T. Pantev, T. Yu,
**Non Commutative Spectra and Applications**, (submitted) - Dennis Borisov, Ludmil Katzarkov, Artan Sheshmani, Shing-Tung Yau,
**Strictification and Gluing of Lagrangian Distributions on Derived Schemes with Shifted Symplectic Forms**, (submitted) - L. Katzarkov, K.S. Lee, J. Svoboda, A. Petkov,
**Interpretations of Spectra**, (submitted) - F. Haiden, L. Katzarkov, C. Simpson,
**Spectral Networks and Stability Conditions for Fukaya Categories with Coefficients**, (submitted) - L. Katzarkov, Ernesto Lupercio, Lauren Meersseman and Alberto Verjovsky,
**On the moduli space of quantum projective spaces**(in preparation) - Dennis Borisov, Ludmil Katzarkov, Artan Sheshmani, Shing-Tung Yau,
**Global Shifted Potentials for Moduli Stacks of Sheaves on Calabi-Yau Four-Folds**(in preparation)

# Presentations and Talks

## Simons Collaboration on Homological Mirror Symmetry Annual Meeting 2021

The Simons Collaboration on Homological Mirror Symmetry Annual Meeting focus on recent work which explores arithmetic, topological, and analytic refinements of the homological mirror correspondence. Alongside the foundational work needed to produce such refinements some unexpected and exciting applications of the refined mirror constructions were discussed. Talks highlight progress on novel and deeper aspects of the homological mirror correspondence.

The new discoveries and results obtained by the collaboration members and research affiliates in the past year focus on novel and deep crossover applications of the mirror correspondence.

The PI of this project, Ludmil Katzarkov, gave talk on Spectra and Applications and Maxim Kontsevich gave talk on Blow-up Equivalence which is a work in progress in a collaboration with Katzarkov.

## The Almost Schur Lemma in Quaternionic Contact Geometry -First Annual Meeting of Young Bulgarian Mathematicians

Alexander Petkov gave this talk on the First Annual Meeting of Young Bulgarian Mathematicians conference, May 19-20 2021.

Abstract: We consider the quaternionic contact (qc) version of the almost Schur Lemma. Namely, we derive an integral inequality that informally states that on a compact qc manifold of dimension bigger than seven, satisfying some positivity condition, if the traceless qc Ricci tensor and some traces of the qc torsion tensor are close to zero, then the qc scalar curvature is close to a constant.

# 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.