Degenerate Complex Monge–Ampère Equations by Vincent Guedj, Ahmed Zeriahi

By Vincent Guedj, Ahmed Zeriahi

Complicated Monge–Ampère equations were probably the most strong instruments in Kähler geometry in view that Aubin and Yau’s classical works, culminating in Yau’s technique to the Calabi conjecture. A remarkable software is the development of Kähler-Einstein metrics on a few compact Kähler manifolds. in recent times degenerate advanced Monge–Ampère equations were intensively studied, requiring extra complicated tools.

The major target of this e-book is to offer a self-contained presentation of the hot advancements of pluripotential concept on compact Kähler manifolds and its program to Kähler–Einstein metrics on mildly singular kinds. After reviewing easy homes of plurisubharmonic services, Bedford–Taylor’s neighborhood conception of advanced Monge–Ampère measures is constructed. so as to remedy degenerate advanced Monge–Ampère equations on compact Kähler manifolds, effective homes of quasi-plurisubharmonic services are explored, periods of finite energies outlined and diverse greatest ideas verified. After proving Yau’s celebrated theorem in addition to its contemporary generalizations, the implications are then used to unravel the (singular) Calabi conjecture and to build (singular) Kähler–Einstein metrics on a few kinds with gentle singularities.

The booklet is on the market to complicated scholars and researchers of advanced research and differential geometry.

Keywords: Pluripotential conception, plurisubharmonic features, advanced Monge–Ampère operators, generalized capacities, vulnerable suggestions, a priori estimates, canonical Kähler metrics, singular forms

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Degenerate Complex Monge–Ampère Equations

Advanced Monge–Ampère equations were probably the most robust instruments in Kähler geometry due to the fact Aubin and Yau’s classical works, culminating in Yau’s technique to the Calabi conjecture. A awesome software is the development of Kähler-Einstein metrics on a few compact Kähler manifolds. in recent times degenerate advanced Monge–Ampère equations were intensively studied, requiring extra complex instruments.

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U; z/ 2 PSH. / 7 ! z/ 2 R [ f 1g is upper semi-continuous. In particular, if U PSH. / is a compact family of plurisubharmonic functions, its upper envelope U WD supfuj u 2 U g is upper semi-continuous, hence plurisubharmonic in . uj / be a sequence in PSH. / converging Proof. u; z0 / 2 PSH. / to u and let r > 0 and ı > 0 be small enough. uj / is locally uniformly bounded from above. a;2r/ for jz aj < r. a;2r/ since U is compact hence bounded. We can thus assume without loss of generality that uj Ä 0.

W/ w2C defines a subharmonic function with logarithmic growth in C. 2) Show that if ' is a subharmonic function with logarithmic growth in C, then there exists c 2 R such that ' D ' C c, where D '=2 . 3) Approximating by Dirac masses, show that every subharmonic function with logarithmic growth in C can be approximated in L1 by functions of the type j 1 log jPj j, where Pj is a polynomial of degree j . 10. aj / 2 CN be a bounded P sequence which is dense in the unit disc and let "j > 0 be positive reals such that j "j < C1.

20. The Riesz measure of u 2 SH. / is u D 1 u: 2 Using the complex coordinate z D x C iy, the real differential operator acting on smooth functions f W ! C by df D @f @f dx C dy @x @y splits as d D @ C @, where the complex differential operators @ and @ are defined by @f D @f dz @z and @f D @f d z: @z We extend these differential operators to distributions: if f is a distribution, then df is defined as above, but it has to be understood in the sense of currents of degree 1 on : it is a differential form of degree 1 with distribution coefficients.

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