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We show that the underlying complex manifold of a complete non-compacttwo-\linebreak dimensional shrinking gradient K\"ahler-Ricci soliton$(M,\,g,\,X)$ with soliton metric $g$ with bounded scalar curvature$\operatorname{R}_{g}$ whose soliton vector field $X$ has an integral curvealong which $\operatorname{R}_{g}\not\to0$ is biholomorphic to either$\mathbb{C}\times\mathbb{P}^{1}$ or to the blowup of this manifold at onepoint. Assuming the existence of such a soliton on this latter manifold, weshow that it is toric and unique. We also identify the corresponding solitonvector field. Given these possibilities, we then prove a strong form of theFeldman-Ilmanen-Knopf conjecture for finite time Type I singularities of theK\"ahler-Ricci flow on compact K\"ahler surfaces, leading to a classificationof the bubbles of such singularities in this dimension.

On finite time Type I singularities of the K\"ahler-Ricci flow on compact K\"ahler surfaces