{"publisher":"Deutsche Mathematiker Vereinigung","user_id":"ea97e931-d5af-11eb-85d4-e6957dddbf17","type":"journal_article","main_file_link":[{"url":"https://ems.press/journals/dm/articles/8965058","open_access":"1"}],"day":"01","abstract":[{"text":"Extending work of Bielawski-Dancer 3 and Konno 14, we develop a theory of toric hyperkähler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study of semi-projective toric varieties, meaning GIT quotients of affine spaces by torus actions, and specifically, of Lawrence toric varieties, meaning GIT quotients of even-dimensional affine spaces by symplectic torus actions. A toric hyperkähler variety is a complete intersection in a Lawrence toric variety. Both varieties are non-compact, and they share the same cohomology ring, namely, the Stanley-Reisner ring of a matroid modulo a linear system of parameters. Familiar applications of toric geometry to combinatorics, including the Hard Lefschetz Theorem and the volume polynomials of Khovanskii-Pukhlikov 11, are extended to the hyperkähler setting. When the matroid is graphic, our construction gives the toric quiver varieties, in the sense of Nakajima 17.","lang":"eng"}],"date_published":"2002-01-01T00:00:00Z","external_id":{"arxiv":["math/0203096"]},"doi":"10.4171/DM/130","citation":{"short":"T. Hausel, B. Sturmfels, Documenta Mathematica 7 (2002) 495–534.","mla":"Hausel, Tamás, and Bernd Sturmfels. “Toric Hyperkähler Varieties.” <i>Documenta Mathematica</i>, vol. 7, no. 1, Deutsche Mathematiker Vereinigung, 2002, pp. 495–534, doi:<a href=\"https://doi.org/10.4171/DM/130\">10.4171/DM/130</a>.","ista":"Hausel T, Sturmfels B. 2002. Toric hyperkähler varieties. Documenta Mathematica. 7(1), 495–534.","ieee":"T. Hausel and B. Sturmfels, “Toric hyperkähler varieties,” <i>Documenta Mathematica</i>, vol. 7, no. 1. Deutsche Mathematiker Vereinigung, pp. 495–534, 2002.","ama":"Hausel T, Sturmfels B. Toric hyperkähler varieties. <i>Documenta Mathematica</i>. 2002;7(1):495-534. doi:<a href=\"https://doi.org/10.4171/DM/130\">10.4171/DM/130</a>","chicago":"Hausel, Tamás, and Bernd Sturmfels. “Toric Hyperkähler Varieties.” <i>Documenta Mathematica</i>. Deutsche Mathematiker Vereinigung, 2002. <a href=\"https://doi.org/10.4171/DM/130\">https://doi.org/10.4171/DM/130</a>.","apa":"Hausel, T., &#38; Sturmfels, B. (2002). Toric hyperkähler varieties. <i>Documenta Mathematica</i>. Deutsche Mathematiker Vereinigung. <a href=\"https://doi.org/10.4171/DM/130\">https://doi.org/10.4171/DM/130</a>"},"article_processing_charge":"No","article_type":"original","publist_id":"5741","quality_controlled":"1","title":"Toric hyperkähler varieties","oa":1,"intvolume":"         7","publication":"Documenta Mathematica","author":[{"first_name":"Tamas","full_name":"Hausel, Tamas","last_name":"Hausel","id":"4A0666D8-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Bernd","full_name":"Sturmfels, Bernd","last_name":"Sturmfels"}],"extern":"1","page":"495 - 534","acknowledgement":"Both authors were supported by the Miller Institute for Basic Research in Science, in the form of a Miller Research Fellowship (1999-2002) for the first author and a Miller Professorship (2000-2001) for the second author. The second author was also supported by the National Science\r\nFoundation (DMS-9970254).","_id":"1451","status":"public","publication_status":"published","month":"01","scopus_import":"1","oa_version":"Published Version","year":"2002","volume":7,"publication_identifier":{"issn":["1431-0635"]},"issue":"1","language":[{"iso":"eng"}],"date_updated":"2023-07-26T09:16:33Z","date_created":"2018-12-11T11:52:06Z"}