jatil vibhedak swaroop

ganitat, eka jatil vibhedak swaroop heey anekpatanvar (samanyatah eka jatil bahugunak) eka vibhedak swaroop aahe, jyala jatil gunank asanyachi parvanagi aahe.
vibhedak bhoomitimadhye jatil swarupanche vyapak upayog aahet. jatil anek patanvar, teey moolbhut aahet aanee bahutek bijganitiy bhoomiti, kohler bhoomiti aanee hodge siddhantasathi aadhar mhanoon kaam karatat. jatil nasalelya anek patanvar, teey jawaljawal jatil sanrachana, spinnerscha siddhant aanee see. aar. sanrachananchya abhyasat dekhil bhoomika bajavtat.
samanyatah, form swikartat asha kahee vanchhaniy vighatanamule jatil swarupancha vichar keela jaato. udaharanarth, eka jatil bahugunit prakaravar, konatyahi jatil 'keey'-swarupache tathakathit ('pii', 'kyu')-swarupanchya bereejmadhye adviteey vighatana kelee jaau shakateh andaje, holomorphic nirdeshankanchya 'pii' bhinnatanchya wages aanee tyanchya jatil sanyugmanchya 'kyu' bhinnatansah. ("pii", "kyu")-swarupancha samooh abhyasacha adim uddesh banato aanee "keey"-swarupanpeksha anek patanvar eka sookshm bhaumitik rachana nirdharit karato. agadi sookshm rachana astitvat aahet, udaharanarth, hodge siddhant lagoo hoto asha prakarananmadhye.
jatil anekpatanvar bhinn rupe.
samaja qee "m" heey "n" yaa jatil parimananche eka jatil anekpat aahe. mug eka sthanik samanvay pranali aahe jyamadhye "n" jatil-moolya functions "z" 1, asatat..., zed "aine" asey qee eka patchpasoon aahet patchmadhye honare samanvay sankraman hii yaa chalanchi samaroop karye aahet. hii sankraman karye keval gulgulit nasoon, samaroop aahet yaa vastusthitivar moolbhootpane avalambun, jatil swarupanchi jaaga eka samruddha rachana dharan karate.
eka swaroop.
aapan eka-swarupachya prakaranapasun suruvat karato. pratham, pratyek "jeey" sathee, guntaguntichya nirdeshankanna tyanchya vastavik aanee kalpanik bhaganmadhye vighatit qaraa. parvanagi dya
ekhadyala asey disate qee jatil gunank asalele konatehi vibhedak swaroop eka berries mhanoon advitiyapane lihile jaau shakate
ω1,0 hii keval sootra _ 3 asaleli jatil vibhedak swarupanchi jaaga asoo dya aanee ω0,1 hii keval sootra _ 4 asaleli swarupanchi jaaga asoo dya. cotchy-reeman samikaranandware, ekhadi vyakti heey darshavu shakate qee ω1,0 aanee ω0,1 hya jaaga holomorphic samanvay badalankhali sthir aahet. aahet shabdant, jarr konee holomorphic samanvay pranalichi vegali nivad kelee tarr ω1,0 chey ghatak tensorial paddhatine rupantarit hotat, jase qee ω0,1 chey ghatak karatat. asha prakare ω0,1 aanee ω1,0 rikt jaaga complexes manifoldver complexes vector bundle tharavatat.
uchch-stariy form.
jatil vibhedak swarupanche ways utpadan vastavik swarupanpramanech paribhashit kele jaate. p" aanee "q" hii nakaratmak nasalelya purnankanchi jodee ≤ "n" asoo dya. ωp, q chee jaaga, ("p", "q")-rupe ω1,0 pasoon "p" ghatakanchya ways utpadananchi resheey jodani aanee ω0,1 pasoon "q" ghatakanchi resheey jodani gheuun paribhashit kelee jaatat.
jethe ω1,0 chey "p" ghatak aanee ω0,1 chey "q" ghatak aahet. 1-swarupanchya done jaganpramanech, heey nirdeshankanchya holomorphic badalankhali sthir aahet aanee mhanoon vector bundle nishchit karatat.
jarr "e" k "hii ekoon degree" k "chya serve jatil vibhedak swarupanchi jaaga asel, tarr" e "k" chaa pratyek ghatak ωp, q saha asalelya jaganmadhil ghatakanche resheey sanyojan mhanoon adviteey paddhatine vyakt keela jaau shakato. adhik sankshiptapane, thet berries vighatan aahe
kaaran hii thet berries vighatan heey holomorphic samanvay badalankhali sthir asate, tyamule teey vector bundle vighatan dekhil nirdharit karate.
visheshtah, pratyek 'keey' aanee pratyek 'pii' aanee 'kyu' sathee, vector bundleche eka pramanit prakshepan aahe.
dolbalt operator.
nehamiche bahya vyutpann, sootra _ 8 dware vibhaganche mapping paribhashit karate.
bahya vyutpann swatach anekpatanchi adhik kathor jatil rachana pratibimbit karat nahee.
dii" aanee mageel upavibhagat paribhashit kelele andaj vaparoon, dolbuilt operator paribhashit karane shakya aaheh
yaa sanchalakanche sthanik nirdeshankat varnan karnyasathi, chala
khaleel gunadharm dharan kartana disatatah
heey chalak aanee tyanche gunadharm heey dolbiault samarupata aanee hodge siddhantachya anek pailunsathi aadhar tayar karatat.
jatil manifoldchya taara-aakarachya kshetravar, dolbalt operatorkade duheri homotopy operator asatat jeey formula _ 14 sathee homotopy operatorchya vibhajanamule hotat. hii complexes manifoldvaril poincare lemmachi samagri aahe.
sootra _ 15 aanee sootra _ 16 sathee poinker lemma sthanik sootra _ 17-lemma madhye aanakhi sudharle jaau shakate, jeey darshavite qee pratyek sootra _ 14-achook jatil vibhedak swaroop pratyakshat sootra _ 17-achook aahe. compact kehlerver sthanik sootra 17-lemma dharan karnyache jaagtik swaroop anek patinni tayar hote, jyala sootra 17-lemma mhanoon olakhale jaate. haah hodge siddhantacha parinam aahe aanee asey namood karate qee eka jatil vibhedak swaroop jeey jaagtik staravar sootra 14-barobar aahe aahe shabdant, jyacha varg dii raam cohomologymadhye shoonya aahe) heey jaagtik staravar sootra 17-barobar aahe.
holomorphic rupe.
pratyek "pii" sathee, eka holomorphic "pii"-form haah bandalcha eka holomorphic vibhag aahe, ω "pii", 0. sthanik nirdeshankat, nantar, eka holomorphic "pii"-form yaa swarupat lihila jaau shakato.
jethe sootra _ 25 hii holomorphic functions aahet. samatulya, aanee jatil sanyugmachya swatantryamule, ("pii", 0)-swaroop "α" heey jarr aanee fakt jarr asel tarr samaroop aahe.
jaree yamule kadhikadhi gondhal nirman houu shakato, tyamule anek lekhak paryayi sanketan swikaranyacha cull dakhavtat, taree samarupi 'pii'-swarupanche aavaran anekada 'pii' asey lihile jaate.

