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Siegel operators for holomorphic differential forms
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We give a geometric interpretation of the Siegel operators for holomorphic differential forms on Siegel modular varieties. This involves extension of the differential forms over a toroidal compactification, and we show that the Siegel operator essentially describes the restriction and descent to the boundary Kuga variety via holomorphic Leray filtration. As a consequence, we obtain equivalence of various notions of "vanishing at boundary'' for holomorphic forms. We also study the case of orthogonal modular varieties.
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Siegel modular forms arising from higher Chow cycles
For abelian varieties of dimension at most three, higher Chow cycles yield meromorphic Siegel modular forms of weight Sym^4 det^-1, and the K-theory elevator matches the Siegel operator under rank-one degeneration.
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