Let X, Y be smooth projective varieties over <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textbf{C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">C</mi> </math></EquationSource> </InlineEquation>. Let K be a bounded complex of coherent sheaves on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$X\times Y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>×</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> and let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Phi _K :\textsf{D}^b_{\textsf{Coh}}(X) \rightarrow \textsf{D}^b_{\textsf{Coh}}(Y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Φ</mi> <mi>K</mi> </msub> <mo>:</mo> <msubsup> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">Coh</mi> <mi>b</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <msubsup> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">Coh</mi> <mi>b</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the resulting Fourier–Mukai functor. There is a well-known criterion due to Bondal–Orlov for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Phi _K$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Φ</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation> to be fully faithful. This criterion was recently extended to smooth Deligne–Mumford stacks with projective coarse moduli schemes by Lim–Polischuk. We extend this to all smooth, proper Deligne–Mumford stacks over arbitrary fields of characteristic 0. Along the way, we establish a number of foundational results for bounded derived categories of proper and tame morphisms of noetherian algebraic stacks (e.g., coherent duality).

A generalized Bondal–Orlov full faithfulness criterion for Deligne–Mumford stacks

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2026年5月28日

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Jack Hall,
Kyle Priver

摘要

Let X, Y be smooth projective varieties over $\textbf{C}$ $C$ . Let K be a bounded complex of coherent sheaves on $X\times Y$ $X \times Y$ and let $\Phi _K :\textsf{D}^b_{\textsf{Coh}}(X) \rightarrow \textsf{D}^b_{\textsf{Coh}}(Y)$ $Φ_{K} : D_{Coh}^{b} (X) \to D_{Coh}^{b} (Y)$ be the resulting Fourier–Mukai functor. There is a well-known criterion due to Bondal–Orlov for $\Phi _K$ $Φ_{K}$ to be fully faithful. This criterion was recently extended to smooth Deligne–Mumford stacks with projective coarse moduli schemes by Lim–Polischuk. We extend this to all smooth, proper Deligne–Mumford stacks over arbitrary fields of characteristic 0. Along the way, we establish a number of foundational results for bounded derived categories of proper and tame morphisms of noetherian algebraic stacks (e.g., coherent duality).