02664nam a22003495a 4500001001100000003001200011005001700023006001900040007001500059008004100074020001800115024002100133040001400154072001700168084001500185100002900200245010400229260008200333300003400415336002600449337002600475338003600501347002400537490004000561506006600601520143700667650002902104650005502133700002902188856003202217856006502249210-161219CH-001817-320161219234501.0a fot ||| 0|cr nn mmmmamaa161219e20170112sz fot ||| 0|eng d a978303719667070a10.4171/1672doi ach0018173 7aPBKD2bicssc a32-xx2msc1 aGuedj, Vincent,eauthor.10aDegenerate Complex Monge–Ampère Equationsh[electronic resource] /cVincent Guedj, Ahmed Zeriahi3 aZuerich, Switzerland :bEuropean Mathematical Society Publishing House,c2017 a1 online resource (496 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda0 aEMS Tracts in Mathematics (ETM)v261 aRestricted to subscribers:uhttps://www.ems-ph.org/ebooks.php aWinner of the 2016 EMS Monograph Award!
Complex Monge–Ampère equations have been one of the most powerful tools in Kähler geometry since Aubin and Yau’s classical works, culminating in Yau’s solution to the Calabi conjecture. A notable application is the construction of Kähler-Einstein metrics on some compact Kähler manifolds. In recent years degenerate complex Monge–Ampère equations have been intensively studied, requiring more advanced tools.
The main goal of this book is to give a self-contained presentation of the recent developments of pluripotential theory on compact Kähler manifolds and its application to Kähler–Einstein metrics on mildly singular varieties. After reviewing basic properties of plurisubharmonic functions, Bedford–Taylor’s local theory of complex Monge–Ampère measures is developed. In order to solve degenerate complex Monge–Ampère equations on compact Kähler manifolds, fine properties of quasi-plurisubharmonic functions are explored, classes of finite energies defined and various maximum principles established. After proving Yau’s celebrated theorem as well as its recent generalizations, the results are then used to solve the (singular) Calabi conjecture and to construct (singular) Kähler–Einstein metrics on some varieties with mild singularities.
The book is accessible to advanced students and researchers of complex analysis and differential geometry.07aComplex analysis2bicssc07aSeveral complex variables and analytic spaces2msc1 aZeriahi, Ahmed,eauthor.40uhttps://doi.org/10.4171/167423cover imageuhttps://www.ems-ph.org/img/books/guedj_mini.jpg