Positivity of Segre-MacPherson classes
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866910899941408768 |
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| author | Aluffi, Paolo Mihalcea, Leonardo C. Schürmann, Jörg Su, Changjian |
| author_facet | Aluffi, Paolo Mihalcea, Leonardo C. Schürmann, Jörg Su, Changjian |
| contents | Let $X$ be a complex nonsingular variety with globally generated tangent bundle. We prove that the signed Segre-MacPherson (SM) class of a constructible function on $X$ with effective characteristic cycle is effective. This observation has a surprising number of applications to positivity questions in classical situations, unifying previous results in the literature and yielding several new results. We survey a selection of such results in this paper. For example, we prove general effectivity results for SM classes of subvarieties which admit proper (semi-)small resolutions and for regular or affine embeddings. Among these, we mention the effectivity of (signed) Segre-Milnor classes of complete intersections if $X$ is projective and an alternation property for SM classes of Schubert cells in flag manifolds; the latter result proves and generalizes a variant of a conjecture of Fehér and Rimányi. Among other applications we prove the positivity of Behrend's Donaldson-Thomas invariant for a closed subvariety of an abelian variety and the signed-effectivity of the intersection homology Chern class of the theta divisor of a non-hyperelliptic curve; and we extend the (known) non-negativity of the Euler characteristic of perverse sheaves on a semi-abelian variety to more general varieties dominating an abelian variety. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1902_00762 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Positivity of Segre-MacPherson classes Aluffi, Paolo Mihalcea, Leonardo C. Schürmann, Jörg Su, Changjian Algebraic Geometry Representation Theory Primary 14C17, 14M17, Secondary 32C38, 32S60 Let $X$ be a complex nonsingular variety with globally generated tangent bundle. We prove that the signed Segre-MacPherson (SM) class of a constructible function on $X$ with effective characteristic cycle is effective. This observation has a surprising number of applications to positivity questions in classical situations, unifying previous results in the literature and yielding several new results. We survey a selection of such results in this paper. For example, we prove general effectivity results for SM classes of subvarieties which admit proper (semi-)small resolutions and for regular or affine embeddings. Among these, we mention the effectivity of (signed) Segre-Milnor classes of complete intersections if $X$ is projective and an alternation property for SM classes of Schubert cells in flag manifolds; the latter result proves and generalizes a variant of a conjecture of Fehér and Rimányi. Among other applications we prove the positivity of Behrend's Donaldson-Thomas invariant for a closed subvariety of an abelian variety and the signed-effectivity of the intersection homology Chern class of the theta divisor of a non-hyperelliptic curve; and we extend the (known) non-negativity of the Euler characteristic of perverse sheaves on a semi-abelian variety to more general varieties dominating an abelian variety. |
| title | Positivity of Segre-MacPherson classes |
| topic | Algebraic Geometry Representation Theory Primary 14C17, 14M17, Secondary 32C38, 32S60 |
| url | https://arxiv.org/abs/1902.00762 |