Author : Paolo Aluffi
Release : 2022
Genre : MATHEMATICS
Kind : eBook
Book Rating : 855/5 ( reviews)

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Book Synopsis Facets of Algebraic Geometry by : Paolo Aluffi

Download or read book Facets of Algebraic Geometry written by Paolo Aluffi. This book was released on 2022. Available in PDF, EPUB and Kindle. Book excerpt: "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 Feh'er and Rim'anyi. 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) nonnegativity of the Euler characteristic of perverse sheaves on a semi-abelian variety to more general varieties dominating an abelian variety"--