### Abstract

We define a variant of intersection space theory that applies to many compact complex and real analytic spaces X, including all complex projective varieties; this is a significant extension to a theory which has so far only been shown to apply to a particular subclass of spaces with smooth singular sets. We verify existence of these so-called algebraic intersection spaces and show that they are the (reduced) chain complexes of known topological intersection spaces in the case that both exist. We next analyze "local duality obstructions," which we can choose to vanish, and verify that algebraic intersection spaces satisfy duality in the absence of these obstructions. We conclude by defining an untwisted algebraic intersection space pairing, whose signature is equal to the Novikov signature of the complement in X of a tubular neighborhood of the singular set.

Original language | English (US) |
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Pages (from-to) | 1-38 |

Number of pages | 38 |

Journal | Journal of Topology and Analysis |

DOIs | |

State | Accepted/In press - Jan 1 2019 |

### Fingerprint

### Keywords

- Intersection spaces
- complex varieties
- duality
- intersection homology
- signature

### ASJC Scopus subject areas

- Analysis
- Geometry and Topology

### Cite this

*Journal of Topology and Analysis*, 1-38. https://doi.org/10.1142/S1793525319500778

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**Algebraic intersection spaces.** / Geske, Christian August.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Algebraic intersection spaces

AU - Geske, Christian August

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We define a variant of intersection space theory that applies to many compact complex and real analytic spaces X, including all complex projective varieties; this is a significant extension to a theory which has so far only been shown to apply to a particular subclass of spaces with smooth singular sets. We verify existence of these so-called algebraic intersection spaces and show that they are the (reduced) chain complexes of known topological intersection spaces in the case that both exist. We next analyze "local duality obstructions," which we can choose to vanish, and verify that algebraic intersection spaces satisfy duality in the absence of these obstructions. We conclude by defining an untwisted algebraic intersection space pairing, whose signature is equal to the Novikov signature of the complement in X of a tubular neighborhood of the singular set.

AB - We define a variant of intersection space theory that applies to many compact complex and real analytic spaces X, including all complex projective varieties; this is a significant extension to a theory which has so far only been shown to apply to a particular subclass of spaces with smooth singular sets. We verify existence of these so-called algebraic intersection spaces and show that they are the (reduced) chain complexes of known topological intersection spaces in the case that both exist. We next analyze "local duality obstructions," which we can choose to vanish, and verify that algebraic intersection spaces satisfy duality in the absence of these obstructions. We conclude by defining an untwisted algebraic intersection space pairing, whose signature is equal to the Novikov signature of the complement in X of a tubular neighborhood of the singular set.

KW - Intersection spaces

KW - complex varieties

KW - duality

KW - intersection homology

KW - signature

UR - http://www.scopus.com/inward/record.url?scp=85060021166&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85060021166&partnerID=8YFLogxK

U2 - 10.1142/S1793525319500778

DO - 10.1142/S1793525319500778

M3 - Article

AN - SCOPUS:85060021166

SP - 1

EP - 38

JO - Journal of Topology and Analysis

JF - Journal of Topology and Analysis

SN - 1793-5253

ER -