The “Hodge conjecture in toric varieties” sits at the crossroads of algebraic geometry, combinatorics, and topology — with emphasis on intersection cohomology, Hodge modules, and combinatorial structures on fans.
Below is a consolidated roster of leading figures and lines of research, drawn from all three sources.
🧩 Foundational Figures in Intersection Cohomology
Robert MacPherson (IAS) & Mark Goresky
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Co-inventors of Intersection Homology (IH) — the replacement for ordinary cohomology on singular spaces.
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Foundational for all later work on toric and singular varieties.
Top papers:
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Intersection Homology II — Invent. Math. (1980) https://doi.org/10.1007/BF01389773
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Intersection Homology Theory — Topology (1980) https://doi.org/10.1016/0040-9383(80)90003-9
G. Barthel, J.-P. Brasselet, K.-H. Fieseler, L. Kaup (BBFK)
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Created the combinatorial model of intersection cohomology for fans (a discrete version of IH).
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Their formalism is the backbone for computational and toric approaches.
Top papers:
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Combinatorial Intersection Cohomology for Fans — Compositio Math. (2002) https://arxiv.org/abs/math/0203142
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Toric Varieties and Intersection Cohomology — J. Reine Angew. Math. (1991) https://doi.org/10.1515/crll.1991.418.91
🔶 Combinatorial and Geometric IH Researchers
Tom C. Braden (UMass Amherst)
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Developed combinatorial and representation-theoretic interpretations of IH.
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Key figure in hypertoric and toric IH theory.
Top papers:
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Combinatorial Intersection Cohomology of Fans — https://arxiv.org/abs/math/9907088
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Hypertoric Varieties and Hodge Theory — https://arxiv.org/abs/math/0207154
Nicholas Proudfoot (University of Oregon)
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Works on hypertoric and conical symplectic varieties, generalizing toric geometry.
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Frequent collaborator with Braden.
Top papers:
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A Survey of Hypertoric Geometry — https://arxiv.org/abs/1909.08412
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Hypertoric Intersection Cohomology — https://arxiv.org/abs/math/0207155
Kalle Karu (UBC)
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Proved the Hard Lefschetz Theorem for combinatorial IH of polytopes.
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Bridges polyhedral combinatorics and algebraic geometry.
Top papers:
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Hard Lefschetz Theorem for Nonrational Polytopes — Invent. Math. (2004) https://arxiv.org/abs/math/0311027
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The Kähler Package for Intersection Cohomology of Nonrational Polytopes — https://arxiv.org/abs/math/0405345
🧮 Hodge–Theoretic & Toric Directions
Hyunsuk Kim & Sridhar Venkatesh (2024–2025, arXiv preprints)
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Study Hodge filtrations on intersection cohomology Hodge modules for toric varieties.
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Give explicit, algorithmic Hodge decompositions from fan data.
Top papers:
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Hodge Modules on Toric Varieties and Intersection Cohomology Filtrations — https://arxiv.org/abs/2404.08122
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Algorithmic Descriptions of Hodge Structures for Fans — https://arxiv.org/abs/2502.03115
Laurentiu Maxim (U. Wisconsin)
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Works on mixed Hodge modules, giving theoretical foundations that support toric Hodge theory.
Top papers:
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Intersection Homology and Perverse Sheaves (Lecture notes) — https://web.math.wisc.edu/~maxim/ih.pdf
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Intersection Homology, Perverse Sheaves, and Applications — Notices AMS (2019) https://doi.org/10.1090/noti1829
🌐 Hodge Conjecture in Toric & Quasi-Smooth Settings
Ugo Bruzzo (SISSA, Italy) & Antonella Grassi (University of Pennsylvania)
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Explore Hodge conjecture for hypersurfaces and intersections in toric varieties.
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Connect algebraic cycles and combinatorial Hodge theory.
Top papers:
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The Hodge Conjecture for Hypersurfaces in Simplicial Projective Toric Varieties — https://arxiv.org/abs/math/9803147
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Hodge Conjecture for Quasi-smooth Intersections in Toric Varieties — Springer (2019) https://doi.org/10.1007/s00029-019-0515-5
András Szenes & Olga Trapeznikova (University of Geneva / SwissMAP)
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Recently (2025) produced explicit combinatorial models of IH for type-A toric varieties.
Top papers:
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Intersection Cohomology of Type-A Toric Varieties — Alco (2025) https://alco.centre-mersenne.org/item/10.5802/alco.456/
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Combinatorial Models for Toric Intersection Cohomology — https://arxiv.org/abs/2503.02118
Shihoko Ishii (University of Tokyo)
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Expert on arc spaces and singularities of toric varieties, contributing to understanding their Hodge-theoretic and motivic structure.
Top papers:
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Arc Spaces and the Nash Problem for Toric Varieties — https://doi.org/10.1007/s002220050230
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Jet Schemes and Singularities of Toric Varieties — https://arxiv.org/abs/math/0403151
📘 Standard References and Supporting Works
David A. Cox, John B. Little, Hal Schenck
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Authors of the canonical textbook Toric Varieties, the go-to reference in the field.
Top references:
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Toric Varieties — AMS Graduate Studies in Mathematics (2011) https://pi.math.cornell.edu/~david/CoxLittleSchenck-ToricVarieties.pdf
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Ideals, Varieties, and Algorithms (context for computations) — https://doi.org/10.1007/978-3-319-16721-3
Claire Voisin (Collège de France)
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Broke ground on counterexamples to the generalized Hodge conjecture for compact Kähler varieties.
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Foundational authority in variations of Hodge structures.
Top papers:
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Hodge Theory and Complex Algebraic Geometry I & II — Cambridge University Press (2007) https://doi.org/10.1017/CBO9780511615344
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Counterexamples to the Generalized Hodge Conjecture for Compact Kähler Varieties — https://arxiv.org/abs/math/0209265
🧭 Summary Table: Key Lines of Research
| Theme | Leading Researchers | Core Focus | Representative URLs |
|---|---|---|---|
| Intersection Homology Foundations | MacPherson, Goresky, BBFK | Topological and combinatorial IH | Invent. Math. 1980, arXiv:math/0203142 |
| Combinatorial IH & Hard Lefschetz | Braden, Proudfoot, Karu | Fans, polytopes, Lefschetz | arXiv:9907088, arXiv:0311027 |
| Hodge Modules & Filtrations | Kim, Venkatesh, Maxim | IH Hodge structures | arXiv:2404.08122, web.math.wisc.edu/~maxim |
| Hodge Conjecture (Toric) | Bruzzo, Grassi, Ishii | Hypersurfaces, quasi-smooth intersections, singularities | arXiv:9803147, Springer 2019 |
| Modern Combinatorial IH | Szenes, Trapeznikova | Type-A toric IH | Alco 2025, arXiv:2503.02118 |
| General Hodge Theory | Voisin | Hodge structures & Kähler geometry | arXiv:0209265, CUP 2007 |