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In this paper, we investigate the geometry of 4-dimensional complete gradientshrinking Ricci solitons with half positive isotropic curvature (half PIC) orhalf nonnegative isotropic curvature. Our first main result is a certain formof curvature estimates for such Ricci shrinkers, including a quadraticcurvature lower bound estimate for noncompact ones with half PIC. As aconsequence, we obtain a new and more direct proof of the classificationresult, first observed by Li-Ni-Wang [35], for gradient shrinkingK\"ahler-Ricci solitons of complex dimension two with nonnegative isotropiccurvature. Moreover, based on a strong maximum principle argument, we classify4-dimensional complete gradient shrinking Ricci solitons with half nonnegativeisotropic curvature (except the half PIC case). Finally, we treat the half PICcase under an additional assumption on the Ricci tensor.

Four-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature