Bott–Samelson resolution

In algebraic geometry, the Bott–Samelson resolution of a Schubert variety is a resolution of singularities. It was introduced by Bott & Samelson (1958) in the context of compact Lie groups.^[1] The algebraic formulation is independently due to Hansen (1973) and Demazure (1974).

Definition

Let G be a connected reductive complex algebraic group, B a Borel subgroup and T a maximal torus contained in B.

Let $w\in W=N_{G}(T)/T.$ Any such w can be written as a product of reflections by simple roots. Fix minimal such an expression:

{\underline {w}}=(s_{i_{1}},s_{i_{2}},\ldots ,s_{i_{\ell }})

so that $w=s_{i_{1}}s_{i_{2}}\cdots s_{i_{\ell }}$ . (ℓ is the length of w.) Let $P_{i_{j}}\subset G$ be the subgroup generated by B and a representative of $s_{i_{j}}$ . Let $Z_{\underline {w}}$ be the quotient:

Z_{\underline {w}}=P_{i_{1}}\times \cdots \times P_{i_{\ell }}/B^{\ell }

with respect to the action of $B^{\ell }$ by

(b_{1},\ldots ,b_{\ell })\cdot (p_{1},\ldots ,p_{\ell })=(p_{1}b_{1}^{-1},b_{1}p_{2}b_{2}^{-1},\ldots ,b_{\ell -1}p_{\ell }b_{\ell }^{-1}).

It is a smooth projective variety. Writing $X_{w}={\overline {BwB}}/B=(P_{i_{1}}\cdots P_{i_{\ell }})/B$ for the Schubert variety for w, the multiplication map

\pi :Z_{\underline {w}}\to X_{w}

is a resolution of singularities called the Bott–Samelson resolution. $\pi$ has the property: $\pi _{*}{\mathcal {O}}_{Z_{\underline {w}}}={\mathcal {O}}_{X_{w}}$ and $R^{i}\pi _{*}{\mathcal {O}}_{Z_{\underline {w}}}=0,\,i\geq 1.$ In other words, $X_{w}$ has rational singularities.^[2]

There are also some other constructions; see, for example, Vakil (2006).

Notes

^ Gorodski & Thorbergsson (2002).
^ Brion (2005, Theorem 2.2.3.)

References

Bott, Raoul; Samelson, Hans (1958), "Applications of the theory of Morse to symmetric spaces", American Journal of Mathematics, 80: 964–1029, doi:10.2307/2372843, MR 0105694.
Brion, Michel (2005), "Lectures on the geometry of flag varieties", Topics in cohomological studies of algebraic varieties, Trends Math., Birkhäuser, Basel, pp. 33–85, arXiv:math/0410240, doi:10.1007/3-7643-7342-3_2, MR 2143072.
Demazure, Michel (1974), "Désingularisation des variétés de Schubert généralisées", Annales Scientifiques de l'École Normale Supérieure (in French), 7: 53–88, MR 0354697.
Gorodski, Claudio; Thorbergsson, Gudlaugur (2002), "Cycles of Bott-Samelson type for taut representations", Annals of Global Analysis and Geometry, 21 (3): 287–302, arXiv:math/0101209, doi:10.1023/A:1014911422026, MR 1896478.
Hansen, H. C. (1973), "On cycles in flag manifolds", Mathematica Scandinavica, 33: 269–274 (1974), doi:10.7146/math.scand.a-11489, MR 0376703.
Vakil, Ravi (2006), "A geometric Littlewood-Richardson rule", Annals of Mathematics, Second Series, 164 (2): 371–421, arXiv:math.AG/0302294, doi:10.4007/annals.2006.164.371, MR 2247964.

Read other articles:

Chojnice

ChojniceKota Pasar BenderaLambang kebesaranNegara PolandiaProvinsiPomorskiePowiatChojniceGminaChojniceHak kota1360Pemerintahan • Wali kotaArseniusz FinsterLuas • Kota Pasar21,05 km2 (8,13 sq mi)Ketinggian tertinggi152 m (499 ft)Ketinggian terendah150 m (490 ft)Populasi (2016) • Kota Pasar39.961 • Kepadatan1,9/km2 (4,9/sq mi) • Metropolitan45,000Zona waktuUTC+1 (CET) •...

Альтернативные версии Человека-паука

В связи с тем, что комиксы о Человеке-пауке в рамках вселенной Marvel продаются вполне успешно, издатели приняли решение ввести несколько параллельных серий, в которых частично изменяются привычный вид персонажа и окружающая обстановка в рамках так называемой Мультивселе�...

CNP Assurances

CNP AssurancesJenisSociété AnonymeKode emitenEuronext: CNPIndustriJasa keuanganDidirikan1959KantorpusatPrancisWilayah operasiEropa, Amerika SelatanTokohkunciFrédéric Lavenir(CEO), Jean-Paul Faugère (Chairman)ProdukAsuransiPendapatan$36,2 miliarTotal aset$455,1 miliarKaryawan4,800Situs webhttp://www.cnp.fr/en CNP Assurances adalah sebuah perusahaan asal Prancis yang bergerak di sektor finansial.[1] Fokus utama CNP Assurances adalah industri asuransi.[1] Pada tahun 201...

Jihad: A Story of the Others

TV series or program Jihad: A Story of the OthersDirected byDeeyah KhanStarring Abu Muntasir Alyas Karmani Munir Zamir Usama Hasan Shahid Butt Yasmin Mulbocus Country of originUnited Kingdom / NorwayOriginal languageEnglishProductionProducers Darin Prindle Andrew Smith CinematographyNeil HarveyEditorKevin ThomasRunning time50 minutesOriginal releaseRelease15 June 2015 (2015-06-15) Jihad: A Story of the Others is a 2015 documentary film by Norwegian director Deeyah Khan. The fi...

Kaki

Untuk kegunaan lain, lihat Kaki (disambiguasi). kaki Kaki merupakan salah satu anggota tubuh hewan atau manusia yang digunakan untuk berjalan, berlari atau melompat. Kaki terdiri dari beberapa bagian, termasuk telapak kaki, sendi yang bekerja dalam suatu sistem terpadu sehingga memungkinkan bagi inang untuk berjalan. Referensi France, Diane L. (2008). Human and Nonhuman Bone Identification: A Color Atlas. CRC Press. ISBN 1-4200-6286-7. Marieb, Elaine Nicpon; Hoehn, Katja (2007). Hu...

The Image of You

2004 song by Anjeza Shahini The Image of YouSingle by Anjeza ShahiniReleased2004 (2004)GenrePopLength3:00LabelRTSHComposer(s)Edmond ZhulaliLyricist(s)Agim DoçiAnjeza Shahini singles chronology The Image of You (2004) Mes nesh (2005) Eurovision Song Contest 2004 entryCountryAlbaniaArtist(s)Anjeza ShahiniLanguageEnglishComposer(s)Edmond ZhulaliLyricist(s)Agim DoçiFinals performanceSemi-final result4thSemi-final points167Final result7thFinal points106Entry chronologyTomorrow I Go (2005) ...

Diah Pitaloka

Artikel ini bukan mengenai Dyah Pitaloka Citraresmi atau Rieke Diah Pitaloka. Diah PitalokaS.Sos., M.Si. Anggota Dewan Perwakilan RakyatRepublik IndonesiaPetahanaMulai menjabat 1 Oktober 2014Daerah pemilihanJawa Barat III Informasi pribadiLahir30 November 1977 (umur 46)Cilacap, Jawa Tengah, IndonesiaKebangsaanIndonesiaPartai politikPartai Demokrasi Indonesia PerjuanganSuami/istriCipto PranotoAlma materUniversitas PadjadjaranUniversitas Katolik ParahyanganPekerjaanPolitikusSunting kot...

72nd Mechanized Brigade (Ukraine)

Ukrainian Ground Forces formation This article needs to be updated. Please help update this article to reflect recent events or newly available information. (September 2022) 72nd Mechanized Brigade(2016–present) 72nd Guards Mechanized Brigade(2002–2016) 72nd Guards Mechanized Division(1992–2002) 72nd Guards Motor Rifle Division(1957–1992) 72nd Guards Rifle Division(1943–1957) 29th Rifle Division (2nd Formation)(1941–1943)72nd Mechanized Brigade Sleeve PatchActiveDecember 5, 1941&#...

هنود

هنودمعلومات عامةنسبة التسمية الهند التعداد الكليالتعداد قرابة 1.21 مليار[1][2]تعداد الهند عام 2011ق. 1.32 مليار[3]تقديرات عام 2017ق. 30.8 مليون[4]مناطق الوجود المميزةبلد الأصل الهند البلد الهند الهند نيبال 4,000,000[5] الولايات المتحدة 3,982,398[6] الإمار...

2016年美國總統選舉

2016年美國總統選舉 ← 2012 2016年11月8日 2020 → 538個選舉人團席位獲勝需270票民意調查投票率55.7%[1][2] ▲ 0.8 % 获提名人唐納·川普希拉莉·克林頓政党共和黨民主党家鄉州紐約州紐約州竞选搭档迈克·彭斯蒂姆·凱恩选举人票 304[3][4][註 1] 227[5] 胜出州/省 30 + 緬-2 20 + DC 民選得票 62,984,828[6] 65,853,514[6]...

土库曼斯坦总统

土库曼斯坦总统土库曼斯坦国徽土库曼斯坦总统旗現任谢尔达尔·别尔德穆哈梅多夫自2022年3月19日官邸阿什哈巴德总统府（Oguzkhan Presidential Palace）機關所在地阿什哈巴德任命者直接选举任期7年，可连选连任首任萨帕尔穆拉特·尼亚佐夫设立1991年10月27日土库曼斯坦土库曼斯坦政府与政治国家政府土库曼斯坦宪法国旗国徽国歌立法機關（英语：National Council of Turkmenistan） ...

內閣總理大臣

关于与「內閣總理大臣」標題相近或相同的条目页，請見「內閣總理大臣 (消歧義)」。日本國內閣總理大臣內閣總理大臣紋章現任岸田文雄自2021年10月4日在任尊称總理、總理大臣、首相、阁下官邸總理大臣官邸提名者國會全體議員選出任命者天皇任期四年，無連任限制[註 1]設立法源日本國憲法先前职位太政大臣（太政官）首任伊藤博文设立1885年12月22日，...

2023 Rally Mexico

19th edition of Rally Mexico 2023 Rally MexicoRally Guanajuato México 2023Round 3 of 13 in the 2023 World Rally Championship← Previous eventNext event →The Rally Mexico returned to the championship after a two-year absence.Host country MexicoRally baseLeón, GuanajuatoDates run16 – 19 March 2023Start locationCity of Guanajuato, GuanajuatoFinish locationLeón, GuanajuatoStages23 (315.69 km; 196.16 miles)[1]Stage surfaceGravelTransport distance65...

SMA Immanuel Kalasan

Artikel ini tidak memiliki referensi atau sumber tepercaya sehingga isinya tidak bisa dipastikan. Tolong bantu perbaiki artikel ini dengan menambahkan referensi yang layak. Tulisan tanpa sumber dapat dipertanyakan dan dihapus sewaktu-waktu.Cari sumber: SMA Immanuel Kalasan – berita · surat kabar · buku · cendekiawan · JSTOR SMA Immanuel Kalasan adalah sebuah sekolah menengah atas swasta Kristen di Kalasan, Kabupaten Sleman, Daerah Istimewa Yogyakarta, ...

Archimedes Palimpsest

Greek parchment codex manuscript A typical page from the Archimedes Palimpsest. The text of the prayer book is seen from top to bottom, the original Archimedes manuscript is seen as fainter text below it running from left to right Discovery reported in the New York Times on July 16, 1907 The Archimedes Palimpsest is a parchment codex palimpsest, originally a Byzantine Greek copy of a compilation of Archimedes and other authors. It contains two works of Archimedes that were thought to have bee...

Our Times (film)

Our TimesPoster di bioskopNama lainTradisional我的少女時代Sederhana我的少女时代MandarinWǒ De Shàonǚ Shídài SutradaraFrankie ChenProduserYeh Ju-fenPemeranVivian SungDarren WangDino LeeDewi ChienPenata musikHou Chih-chienPerusahaanproduksiFocus Films Hualien Media International Huace Pictures (Tianjin)[1]DistributorHualien Media InternationalTanggal rilis 13 Agustus 2015 (2015-08-13) (Taiwan) Durasi134 minutesNegaraTaiwanBahasaBahasa TionghoaAnggara...

The Washington Papers

Official logo This article is part of a series aboutGeorge Washington Early life Family Military career Electoral history American Revolution Virginia Association Commander in Chiefof the Continental Army Valley Forge Battle of Trenton Mount Vernon Conference 1787 Constitutional Convention 1st President of the United States Presidency (Timeline) First term 1788–89 election 1st inauguration Judiciary Act Whiskey Rebellion Thanksgiving Presidential title Coinage Act Residence Act District of ...

Alessandro Tartagni

Alessandro Tartagni Alessandro Tartagni, latinizzato come Alexander de Tartagnis e conosciuto anche come Alessandro da Imola (Imola, 1424 – 1477), è stato un giurista italiano, soprannominato nella sua epoca doctor aureus e doctor veritatis. Codicis partem commentaria, 1570 (Fondazione Mansutti, Milano). Indice 1 Biografia 2 Opere 2.1 Manoscritti 3 Note 4 Bibliografia 5 Altri progetti 6 Collegamenti esterni Biografia Studiò filosofia e diritto all'Università di Bologna, dove fu forse all...

Varian tekstual dalam Perjanjian Baru

Varian tekstual adalah variasi bacaan yang ditemukan dalam berbagai naskah kuno yang memuat bagian Perjanjian Baru dalam Alkitab Kristen. Varian tekstual muncul ketika seorang penyalin membuat sebuah perubahan yang disengaja atau tidak disengaja pada teks yang sedang direproduksi olehnya. Beberapa perubahan yang umum termasuk penghapusan, penataan ulang, pengulangan, atau penggantian satu atau beberapa kata ketika mata si penyalin beralih kembali dari salinannya ke naskah asli tetapi pada kat...

Associazione Calcio Siena 1937-1938

Associazione Calcio SienaStagione 1937-1938 Sport calcio Squadra Siena Allenatore Vittorio Faroppa Presidente Raniero Ricci Serie C1º nel girone D (promosso in Serie B) Coppa ItaliaPrimo turno eliminatorio Maggiori presenzeCampionato: Vasco Lenzi (30) Miglior marcatoreCampionato: Vasco Lenzi (23) StadioCampo di Piazza Vittorio Emanuele II 1936-1937 1938-1939 Si invita a seguire il modello di voce Questa voce raccoglie le informazioni riguardanti l'Associazione Calcio Siena nelle competi...