Kähler–Ricci flow of cusp singularities on quasi projective varieties

Let M‾ be a compact complex manifold with smooth Kähler metric η and let D be a smooth divisor on M‾. Let M=M‾∖D and let ωˆ be a Carlson–Griffiths type metric on M. We study complete solutions to Kähler Ricci flow (1.1) on M which are comparable to ωˆ starting from a smooth initial metric ω₀=η+i∂∂¯ϕ₀ where ϕ₀∈C^∞(M). When ω₀≥cωˆ on M for some c>0 and ϕ₀ has zero Lelong number, we construct a smooth solution ω(t) to (1.1) on M×[0,T_[ω0]) where T_[ω0]:=sup⁡{T:[η]+T(c₁(K_M‾)+c₁(O_D))∈K_M} so that ω(t)≥([Formula presented]−[Formula presented])ωˆ for all t≤[Formula presented] where Kˆ is a non-negative upper bound on the bisectional curvatures of ωˆ (see Theorem 1.2). In particular, we do not assume ω₀ has bounded curvature. If ω₀ has bounded curvature and is asymptotic to ωˆ in an appropriate sense, we construct a complete bounded curvature solution on M×[0,T_[ω0]) (see Theorem 1.3). These generalize some of the results of Lott–Zhang in [13]. On the other hand if we only assume ω₀≥cη on M for some c>0 and ϕ₀ is bounded on M, we construct a smooth solution to (1.1) on M×[0,T_[ω0]) which is equivalent to ωˆ for all positive times. This includes as a special case when ω₀ is smooth on M‾ in which case the solution becomes instantaneously complete on M under (1.1) (see Theorem 1.1).