# Categorical Kaehler Geometry and Applications (CKGA)

## 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

# Abstract

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 proposal 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 PI. 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).

# Team

# Events

# Papers

## Quantum (Non-commutative) Toric Geometry: Foundations

#### Ludmil Katzarkov, Ernesto Lupercio, Laurent Meersseman, Alberto Verjovsky

In this paper, we will introduce Quantum Toric Varieties which are (non-commutative) generalizations of ordinary toric varieties where all the tori of the classical theory are replaced by quantum tori. Quantum toric geometry is the non-commutative version of the classical theory; it generalizes non-trivially most of the theorems and properties of toric geometry. By considering quantum toric varieties as (non-algebraic) stacks, we define their category and show that it is equivalent to a category of quantum fans. We develop a Quantum Geometric Invariant Theory (QGIT) type construction of Quantum Toric Varieties. Unlike classical toric varieties, quantum toric varieties admit moduli and we define their moduli spaces, prove that these spaces are orbifolds and, in favorable cases, up to homotopy, they admit a complex structure.