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**

**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.

# Events

# Papers

- Yu-Wei Fan, Simion Filip, Fabian Haiden,
**Ludmil Katzarkov**, Yijia Liu,*On pseudo-Anosov autoequivalences*, Advances in Mathematics, Vol. 384, (2021), 107732. https://doi.org/10.1016/j.aim.2021.107732,**IF: 1.675 (Q1)**

**L****udmil****Katzarkov**, Pranav Pandit, Theodore Spaide,*Calabi-Yau structures, spherical**functors, and shifted symplectic structures*, Advances in Mathematics, Vol. 392, (2021), 10803 https://doi.org/10.1016/j.aim.2021.108037,**IF: 1.675 (Q1)**

**Ludmil Katzarkov**, Ernesto Lupercio, Laurent Meersseman, Alberto Verjovsky,*Quantum (Non-Commutative) Toric Geometry: Foundations*, Advances in Mathematics, Vol. 391, (2021), https://doi.org/10.1016/j.aim.2021.107945,**IF: 1.675 (Q1)**

**Stefan Ivanov**, Ivan Minchev, Dimiter Vassilev,*Solution**o**f**t**he Qc Yamabe Equation**o**n**a**3-Sasakian Manifold**a**nd**t**he Quaternionic Heisenberg Group*, accepted for publication in Analysis & PDE, https://msp.org/soon/coming.php?jpath=apde,**IF: 2.344 (Q1)**

**Stefan Ivanov**, Hristo Manev, Mancho Manev,*Para-Sasaki-like Riemannian manifolds and new Einstein metrics*, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas (RACSAM) 115, 112 (2021). https://doi.org/10.1007/s13398-021-01053-z,**IF:****2.****276****(Q1)**

- Dennis Borisov,
**Ludmil Katzarkov**, Artan Sheshmani,*Shifted symplectic structures on derived Quot-stacks I – Differential graded manifolds*–, Advances in Mathematics, Vol. 403 (2022), https://doi.org/10.1016/j.aim.2022.108369,**IF: 1.675 (Q1)**

- Dennis Borisov,
**Ludmil Katzarkov**, Artan Sheshmani, Shing-Tung Yau,*Strictification**a**nd Gluing**o**f Lagrangian Distributions**o**n Derived Schemes**w**ith Shifted Symplectic Forms*, (accepted in Advances in Math, 2022),**IF: 1.675 (Q1)** **Katzarkov**, K.S. Lee, J. Svoboda,**A. Petkov**,*Interpretations of Spectra*, to appear in Birational Geometry, Kaehler Einstein Metrics and Degenerations, accepted 2022, the book will be available on April 9, 2023, Springer, Series Title Springer Proceedings in Mathematics & Statistics,**SJR (2021): 0.204**, https://www.barnesandnoble.com/w/birational-geometry-k-hler-einstein-metrics-and-degenerations-ivan-cheltsov/1142863567- Haiden.
**L. Katzarkov**, C. Simpson,*Spectral networks and stability conditions for Fukaya categories**with coefficients,*preprint https://doi.org/10.48550/arXiv.2112.13623 - Auroux, A. Efimov,
**L. Katzarkov,***Lagrangian Floer theory for trivalent graphs and homological mirror symmetry for curves,*preprint https://doi.org/10.48550/arXiv.2107.01981 **S****.****Ivanov**,**A****.****Petkov**,*The CR Almost Schur Lemma and the positivity condition*s, preprint https://doi.org/10.48550/arXiv.2204.03461**S****.****Ivanov**,**A****.****Petkov**,*The Almost Schur Lemma in Quaternionic Contact Geometry*, preprint https://doi.org/10.48550/arXiv.2204.04482**Katzarkov**, K.-S. Lee,*Vector bundles on elliptic surfaces and logarithmic transformations*, preprint https://arxiv.org/pdf/2204.08927.pdf- Paul Horja,
**L. Katzarkov**,*Discriminants and toric K-theory*, preprint arXiv:2205.00903 - Dennis Borisov,
**Ludmil Katzarkov**, Artan Sheshmani, Shing-Tung Yau,*Global Shifted Potentials for Moduli Stacks of Sheaves on Calabi-Yau Four-Folds*, preprint https://arxiv.org/abs/2007.13194v2 - Green, P. Griffiths,
**L. Katzarkov**,*Shafarevich mappings and period mappings*, preprint https://arxiv.org/abs/2209.14088 - Gukov,
**L. Katzarkov**, J. Svoboda,*for Plumbed Manifolds and Splicing*, in preparation - Kontsevich,
**L. Katzarkov**, T. Pantev, T. Yu,*F − bundles, descendants, and**blow – up**formulas*, in preparation - Matheus Silva Costa, Lino Grama,
**Ludmil Katzarkov**,*Generalized toric varieties, LVMB manifolds and Lie groupoids*, in preparation - Ilya Karzhemanov,
**Ludmil Katzarkov**,*Exceptional collections and phantoms of special Dolgachev surfaces*, 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.

## Functions holomorphic over finite-dimensional commutative associative algebras

Marin Genov gave this talk on the Second Annual Meeting of Young Bulgarian Mathematicians conference, June 13-14 2022.

*Abstract:* We shall discuss the basic function theory arising from replacing the field of complex numbers by a more general finite-dimensional commutative associative algebra both in the domain and codomain in the definition of single-variable complex differentiability. What we get is a function theory that lives and thrives on the border between the kingdom of the classical single-variable complex analysis and the realm of Several Complex Variables.

# 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.