The sheaf of nonvanishing meromorphic functions in the projective algebraic case is not acyclic
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Abstract
Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>X</mml:mi> <mml:mo stretchy="false">/</mml:mo> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:math> be a projective algebraic manifold, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi mathvariant="script">M</mml:mi> <mml:mi>X</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:math> be the sheaf of nonvanishing meromorphic functions on X in the analytic topology. We prove a number of nonvanishing results for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>•</mml:mo> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:msubsup> <mml:mi mathvariant="script">M</mml:mi> <mml:mi>X</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> <mml:mo stretchy="false">)</mml:mo> </mml:math> . In particular, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi mathvariant="script">M</mml:mi> <mml:mi>X</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:math> is acyclic iff <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="normal">dim</mml:mi> <mml:mspace width="0.2em"/> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:math> .
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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