Borisov–Libgober

Borisov–Libgober is the informal name given to a set of results and constructions in algebraic geometry and topology associated with Alexander Borisov and Anatoly Libgober. The name typically refers to formulas and structures connecting elliptic genera, stringy invariants, and Hodge-theoretic data for singular varieties and orbifolds. These ideas bridge techniques from Atiyah–Singer index theorem, Hirzebruch–Riemann–Roch theorem, Mirror symmetry, and McKay correspondence to produce computable invariants used in Calabi–Yau manifold studies, Kähler manifold theory, and singularity classification.

Definition and Origins

The Borisov–Libgober constructions arose from attempts to extend classical invariants such as the Todd class, Chern class, and elliptic genus from smooth compact manifolds to singular spaces and quotient orbifolds. Drawing on work by Hirzebruch on genera, by Landweber–Stong on elliptic operators, and by Witten on elliptic genera in quantum field theory, Borisov and Libgober proposed modifications that incorporate contributions from crepant resolutions, Gorenstein singularitys, and quotient singularitys. Their approach synthesizes ideas from string theory practitioners like Vafa and Dijkgraaf and mathematical formalisms developed by Kontsevich and Batyrev for motivic and stringy invariants. The origins trace through collaborative threads connecting Yau's work on Calabi–Yau manifolds, Reid's McKay-related conjectures, and developments in equivariant cohomology.

Mathematical Properties and Theorems

Borisov–Libgober results articulate precise transformation laws and additivity properties for elliptic-type genera under birational maps such as crepant resolution and under finite group quotients. Core theorems relate the generalized elliptic genus to the orbifold elliptic genus of a quotient by a finite group action, invoking analogues of the McKay correspondence and compatibility with the Hodge decomposition for mixed Hodge structures studied by Deligne. Key properties include modularity constraints reminiscent of Modular form behavior studied by Serre and Mumford, functoriality under proper morphisms akin to the Grothendieck–Riemann–Roch theorem, and multiplicativity in fibrations similar to results by Borel and Hirzebruch. They proved equality between stringy Hodge numbers defined by Batyrev and elliptic-type invariants computed via orbifold techniques for large classes of Gorenstein singularitys and toric varietys, building on techniques of Iwanari and Fantechi in intersection theory.

Applications and Examples

Applications span enumerative predictions in mirror symmetry computations, checks of dualities conjectured by Candelas and Greene, and explicit calculations in toric and hypersurface settings studied by Varchenko and Kouchnirenko. For example, for a quotiented Calabi–Yau hypersurface in a weighted projective space with a finite abelian group action, the Borisov–Libgober elliptic genus computes invariants that match predictions from Berglund–Hübsch mirror symmetry and from orbifold conformal field theory as in works by Dixon and Lykken. In toric examples, methods of Fulton and Oda combine with Borisov–Libgober formulas to yield explicit series and q-expansions related to Jacobi forms investigated by Eichler and Zagier. Concrete uses also include comparisons of invariants across crepant resolutions appearing in examples from Reid and computational verifications using algorithms influenced by Singular and Macaulay2.

Related Concepts and Generalizations

The Borisov–Libgober framework connects to a web of generalizations: stringy invariants by Batyrev–Dais, motivic integration techniques of Kontsevich–Denef–Loeser, and orbifold cohomology developed by Chen–Ruan. Extensions include equivariant elliptic genera studied in the context of Atiyah–Bott localization, refinement to K-theoretic and derived settings inspired by Toen and Lurie, and connections to categorical approaches like Homological Mirror Symmetry of Kontsevich. The constructions interact with arithmetic aspects via Hecke operator analogues and with representation-theoretic facets via the McKay correspondence and Givental's formalism. Further generalizations treat non‑Gorenstein singularities, logarithmic pairs as in Kawamata's work, and applications to Donaldson–Thomas invariants and Gromov–Witten invariants through the interface of enumerative geometry.

Historical Development and Contributors

Principal contributors are Alexander Borisov and Anatoly Libgober; their collaborations built on foundations by Hirzebruch, Witten, Batyrev, and Kontsevich. Influential antecedents include the McKay observations popularized by Reid and the elliptic genus program advanced by Ochanine and Bott–Taubes. Subsequent developments involved researchers such as Roan, Dolgachev, Kawamata, Fantechi, Chen, Ruan, Givental, Borisov (other works), and computational efforts by groups using Macaulay2 and SageMath. Workshops and conferences at institutions like IHES, MSRI, Simons Center, and universities including Yale University, University of Illinois Urbana–Champaign, and Northeastern University fostered the exchange that shaped the theory. The interplay between mathematical physics and algebraic geometry—via figures such as Vafa, Dijkgraaf, Candelas, and Greene—remains central to ongoing extensions and applications.

Category:Algebraic geometry