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Ludmil Katzarkov

Publications and source records attributed to Ludmil Katzarkov.

At least 19 recordsLinked to original sources
arXiv2026

On the atomic decomposition of complete intersection in flag varieties

We develop a root-theoretic localization formalism for genus-zero Gromov--Witten invariants of flag varieties and of smooth zero loci of globally generated homogeneous vector bundles. The resulting invariants are expressed as finite sums over decorated trees whose contributions are determined by the root system of the ambient flag variety and by the torus weights of the defining bundle. We connect these computations with the Theory of Hodge Atoms of Katzarkov--Kontsevich--Pantev--Yu via the matrix of small quantum multiplication by first Chern class. Let $X$ be a Fano fourfold with $h^{3,1}(X)=1$ arising as a hyperplane section of a suitable Fano fivefold. Its cohomology decomposes into a monodromy-fixed part and the middle vanishing cohomology. This decomposition is preserved by quantum multiplication by the first Chern class. Moreover, it acts by a scalar on the vanishing summand. We show, for the monodromy-fixed block, that (a) if every eigenvalue has algebraic multiplicity at most two and $X$ is Hodge general, then $X$ is irrational; (b) if every eigenvalue has Jordan defect at most one and $X$ is rational, then every weak factorization contains a smooth surface center whose minimal model is a projective K3 surface. Many applications are presented.

math.AG↗
arXiv2026

Towards Categorical Kähler Geometry

We outline the contours of an emerging theory of Kähler metrics in derived noncommutative geometry. This is a refinement of the theory of Bridgeland stability conditions encoding underlying differential-geometric structures. We propose elements of such a structure in both Archimedean and non-Archimedean settings, including metrized objects, mass measures satisfying a BPS inequality, harmonic metrics, minimizing flows, and complexified Kähler potentials. We develop the framework through examples and constructions involving Fukaya categories, quiver representations and associated C$^*$-algebras, spectral networks, and comonadic adjunctions of stable $\infty$-categories.

math.AG↗
arXiv2026

A Geometric Realization of Spherical T-Duality via $\star$-Diagrams

We relate spherical T-duality for oriented linear $\mathrm{S}^3$-bundles over $\mathrm{S}^4$ (the Milnor bundles $M_{m,n}$, which are $\mathrm{S}^3$-principal exactly when $m=0$ or $n=0$, and whose total spaces are homotopy $7$-spheres exactly when $m+n=\pm1$) to $\star$-diagrams and to a higher-dimensional generalization of the logarithmic transformations of $4$-manifold topology. For an $\mathrm{S}^3$-principal pair $(P,H)$, $(\widehat P,\widehat H)$ over $\mathrm{S}^4$, we show that the T-duality correspondence space $P\times_{\mathrm{S}^4}\widehat P$ is itself a $\star$-diagram of a distinguished type, which we call \emph{bifree}, and that bifree $\star$-diagrams are precisely the fiber products of principal bundles; spherical T-duality of the decorated pair is then a condition on the fluxes carried by that diagram. For bundles of equal Euler class $ku$, principal or not, we show that the two bundles are spherical T-dual with the diagonal fluxes $[k]$, and that they occur as the two base manifolds of an explicit $\star$-diagram, obtained by pulling back a principal Milnor bundle; this diagram is never bifree. We then introduce product-preserving generalized logarithmic transformations on products $Σ\times\mathrm{S}^1$ of homotopy spheres with the circle, and prove that, after stabilization by $\mathrm{S}^1$, the spherical T-dualities between homotopy $7$-spheres are realized by such transformations. In particular, $Σ^7_{GM}\times\mathrm{S}^1$ is obtained from $\mathrm{S}^7\times\mathrm{S}^1$ by one of them, where $Σ^7_{GM}$ denotes the Gromoll--Meyer exotic sphere: spherical T-duality relates distinct smooth structures on the topological $7$-sphere, and the relation is implemented by an explicit cut-and-paste operation.

math.DG↗
arXiv2026

The Cohomology of Solvmanifold SYZ Mirrors

This paper investigates the geometric and cohomological properties of non-Kähler SYZ mirror symmetry for dual torus fibrations over solvmanifolds in the sense of Lau, Tseng and Yau. We are mainly concerned with three questions: \textbf{(a)} How the Lau-Tseng-Yau notion of non-Kähler SYZ is related to the mapping of supersymmetric branes between symplectic and complex sides; \textbf{(b)} Finding explicit non-Kähler SYZ mirror pairs determined purely by Lie-theoretic data; \textbf{(c)} better understand the cohomological correspondence in the Lau-Tseng-Yau framework (given by a Fourier-Mukai transform), especially concerning the role of Tseng-Yau cohomology. We prove that the Fourier-Mukai transform introduced by Lau-Tseng-Yau exchanges type-A supersymmetric cycles, which are given by special Lagrangian sections equipped with flat $\mathrm{U}(1)$ connections, with type-B cycles, corresponding to line bundles whose connections satisfy the deformed Hermitian-Yang-Mills (dHYM) equation. We provide pure Lie-theoretic criteria for the existence of non-Kähler SYZ mirror pairs whose base manifolds are solvmanifolds. Applying these criteria, we construct new explicit families of mirror pairs from almost abelian and generalized Heisenberg Lie groups, and provide a complete classification of such pairs arising from nilpotent Lie groups. To contextualize the role of the Tseng-Yau cohomology, we link it to noncommutative geometry. We introduce the Tseng-Yau and Bott-Chern mirror bicomplexes. We show that (some of) their enclosed cohomologies reduce to the primitive Tseng-Yau and Bott-Chern cohomologies and that for basic forms they are isomorphic under the Fourier-Mukai transform. As a last contribution, we discuss how to explicitly compute the Tseng-Yau and the Bott-Chern cohomology for the non-Kähler SYZ mirror pairs constructed here.

hep-th↗
arXiv2026

Atoms meet symbols

This paper introduces a novel framework for constructing invariants in $G$-equivariant birational geometry by unifying two recent approaches: the theory of atoms recently developed by Katzarkov, Kontsevich, Pantev, and Yu, and the theory of modular symbols due to Kontsevich, Tschinkel, and Pestun. We initiate the theory of Chen-Ruan atoms. Assuming the blowup formula for the quantum Chen-Ruan cohomology, we outline how to extend the theory of atoms to global quotient orbifolds and present some striking applications. In addition, we develop a separate class of purely geometric invariants for $\mathbb{Z}/2$- and $\mathbb{Z}/3$-actions on surfaces and threefolds. We provide many examples of non-$G$-linearizable $G$-actions on projective varieties treated with these new techniques.

math.AG↗
arXiv2026

Birational Invariants from Hodge Structures and Quantum Multiplication

We introduce new invariants of smooth complex projective varieties, called Hodge atoms. Their construction combines rational Gromov-Witten invariants with classical Hodge theory and relies on the notion of an F-bundle, which is a non-archimedean version of a non-commutative Hodge structure. The Hodge atoms arise from the spectral decomposition of the F-bundle under the Euler vector field action, and behave additively under blowups, in accordance with Iritani's blowup theorem. We compute several examples and demonstrate applications to birational geometry. In particular, we prove that a very general cubic fourfold is not rational. We also obtain a new proof of the equality of Hodge numbers of birational Calabi-Yau manifolds in any dimension. Furthermore, we show that the framework naturally extends to representations of other motivic Galois groups. This enables the theory of atoms to produce new obstructions to rationality over non-algebraically closed fields of characteristic zero as well.

math.AG↗
arXiv2025

A Gromov-Witten approach to $G$-equivariant birational invariants

In arXiv:2404.19088, we initiated a program linking birational invariants with smooth ones and offering new interpretations of classical invariants, such as the Kervaire-Milnor invariants. Here, we rely on the profound geometric reasoning provided by Lupercio and Uribe in the early 00s to establish a connection between Chen-Ruan cohomology and several $G$-birational invariants introduced in the pioneering works Kontsevich, Kresch, Pestun, Tschinkel, along with presenting applications. Combined with the theory of atoms by Katzarkov, Kontsevich, Pantev, and Yu, the proposal in this paper program will lead to a theory of equivariant atoms.

math.AG↗
arXiv2025

$\widehat{Z}$ and Splice Diagrams

We study quantum $q$-series invariants of 3-manifolds $\widehat{Z}_σ$ of Gukov-Pei-Putrov-Vafa, using techniques from the theory of normal surface singularities such as splice diagrams. We show that the (suitably normalized) sum of all $\widehat{Z}_σ$ depends only on the splice diagram, and in particular, it agrees for manifolds with the same universal abelian cover. We use these ideas to find simple formulas for $\widehat{Z}_σ$ invariants of Seifert manifolds. Applications include a better understanding of the vanishing of the $q$-series $\widehat{Z}_σ$. Additionally, we study moduli spaces of flat $\operatorname{SL}_2(\mathbb{C})$ connections on Seifert manifolds and their relation to spectra of surface singularities, extending a result of Boden and Curtis for Brieskorn spheres to Seifert rational homology spheres with 3 singular fibers and to Seifert homology spheres with any number of fibers.

math.GT↗
arXiv2025

Modularity of Landau-Ginzburg models

For each Fano threefold, we construct a family of Landau-Ginzburg models which satisfy many expectations coming from different aspects of mirror symmetry; they are log Calabi-Yau varieties with proper potential maps; they admit open algebraic torus charts on which the potential function $w$ restricts to a Laurent polynomial satisfying a deformation of the Minkowski ansatz; the general fibres of $w$ are Dolgachev-Nikulin dual to the anticanonical hypersurfaces in $X$. To do this, we study the deformation theory of Landau-Ginzburg models in arbitrary dimension, following the third-named author, Kontsevich, and Pantev, specializing to the case of Landau-Ginzburg models obtained from Laurent polynomials. Our proof of Dolgachev-Nikulin mirror symmetry is by detailed case-by-case analysis, refining work of Cheltsov and the fifth-named author.

math.AG↗
arXiv2024

Revisiting the Classical McKay Correspondence, Derived Equivalences and the Spectrum of Kleinian Surface Singularities: A Look Through the Mirror

In this article, we revisit the classical McKay correspondence via homological mirror symmetry. Specifically, we demonstrate how this correspondence can be articulated as a derived equivalence between the category of vanishing cycles associated with a Kleinian surface singularity and the category of perfect complexes on the corresponding quotient orbifold. We further illustrate how this equivalence allows for the interpretation of the spectrum of a Kleinian surface singularity solely in terms of the representation-theoretic data of the associated binary polyhedral group.

math.AG↗
arXiv2024

Lagrangian Floer theory for trivalent graphs and homological mirror symmetry for curves

Mirror symmetry for higher genus curves is usually formulated and studied in terms of Landau-Ginzburg models; however the critical locus of the superpotential is arguably of greater intrinsic relevance to mirror symmetry than the whole Landau-Ginzburg model. Accordingly, we propose a new approach to the A-model of the mirror, viewed as a trivalent configuration of rational curves together with some extra data at the nodal points. In this context, we introduce a version of Lagrangian Floer theory and the Fukaya category for trivalent graphs, and show that homological mirror symmetry holds, namely, that the Fukaya category of a trivalent configuration of rational curves is equivalent to the derived category of a non-Archimedean generalized Tate curve. To illustrate the concrete nature of this equivalence, we show how explicit formulas for theta functions and for the canonical map of the curve arise naturally under mirror symmetry.

math.SG↗
arXiv2024

Reductive Shafarevich Conjecture

In this paper, we prove the holomorphic convexity of the covering of a complex projective {normal} variety $X$, which corresponds to the intersection of kernels of reductive representations $ρ:π_1(X)\to {\rm GL}_{N}(\mathbb{C})$, therefore answering a question by Eyssidieux, Katzarkov, Pantev, and Ramachandran in 2012. It is worth noting that Eyssidieux had previously proven this result in 2004 when $X$ is smooth. While our approach follows the general strategy employed in Eyssidieux's proof, it introduces several improvements and simplifications. Notably, it avoids the necessity of using the reduction mod $p$ method in Eyssidieux's original proof. Additionally, we construct the Shafarevich morphism for complex reductive representations of fundamental groups of complex quasi-projective varieties unconditionally, and proving its algebraic nature at the function field level.

math.AG↗
arXiv2023

On pseudo-Anosov autoequivalences

Motivated by results of Thurston, we prove that any autoequivalence of a triangulated category induces a filtration by triangulated subcategories, provided the existence of Bridgeland stability conditions. The filtration is given by the exponential growth rate of masses under iterates of the autoequivalence, and only depends on the choice of a connected component of the stability manifold. We then propose a new definition of pseudo-Anosov autoequivalences, and prove that our definition is more general than the one previously proposed by Dimitrov, Haiden, Katzarkov, and Kontsevich. We construct new examples of pseudo-Anosov autoequivalences on the derived categories of quintic Calabi-Yau threefolds and quiver Calabi-Yau categories. Finally, we prove that certain pseudo-Anosov autoequivalences on quiver 3-Calabi-Yau categories act hyperbolically on the space of Bridgeland stability conditions.

math.AG↗
arXiv2023

Exceptional collections and phantoms of special Dolgachev surfaces

We provide an explicit description of exceptional collection of maximal length in the derived category $D^b(Y)$ for a particular class of elliptic surfaces $Y$. The existence of non\,-\,trivial semiorthogonal complement (a "\,phantom\,") of this collection is also established.

math.AG↗
arXiv2023

Discriminants and toric K-theory

We discuss a categorical approach to the theory of discriminants in the combinatorial language introduced by Gelfand, Kapranov and Zelevinsky. Our point of view is inspired by homological mirror symmetry and provides $K$--theoretic evidence for a conjecture presented by Paul Aspinwall in a conference talk in Banff in March 2016 and later in a joint paper with Plesser and Wang.

math.AG↗
arXiv2022

Shafarevich mappings and period mappings

We shall show that a smooth, quasi-projective variety $X$ has a holomorphically convex universal covering $\wt X$ when (i) $π_1(X)$ is residually nilpotent and (ii) there is an admissable variation of \mhs\ over $X$ whose monodromy representation has a finite kernel, and where in each case a corresponding period mapping is assumed to be proper.

math.AG↗