Complex structures on product manifolds

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:math> {M}_{i} , for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:math> i=1 , 2, be a Kähler manifold, and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G be a compact Lie group acting on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:math> {M}_{i} by Kähler isometries. Suppose that the action admits a momentum map <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:math> {\mu }_{i} , and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>N</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>≔</m:mo> <m:msubsup> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {N}_{i}:= {\mu }_{i}^{-1}\left(0) be a regular-level set. When the action of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>N</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:math> {N}_{i} is proper and free, the Meyer-Marsden-Weinstein quotient <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>P</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>≔</m:mo> <m:msub> <m:mrow> <m:mi>N</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>∕</m:mo> <m:mi>G</m:mi> </m:math> {P}_{i}:= {N}_{i}/G is a Kähler manifold and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>π</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>:</m:mo> <m:msub> <m:mrow> <m:mi>N</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>→</m:mo> <m:msub> <m:mrow> <m:mi>P</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:math> {\pi }_{i}:{N}_{i}\to {P}_{i} is a principal fiber bundle with base <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>P</m:mi> </m:mrow>