Multigraded Betti numbers of Möbius domino ideals

Abstract The 2-Fano varieties, defined by De Jong and Starr, satisfy some higher-dimensional analogous properties of Fano varieties. We consider (weak) k-Fano varieties and conjecture the polyhedrality of the cone of pseudoeffective k-cycles for those varieties, in analogy with the case k = 1. Then we calculate some Betti numbers of a large class of k-Fano varieties to prove some special case of the conjecture. In particular, the conjecture is true for all 2-Fano varieties of index at least n − 2, and we complete the classification of weak 2-Fano varieties answering Questions 39 and 41 in [2].

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Topology of Subvarieties of Complex Semi-abelian Varieties

International Mathematics Research Notices ◽

10.1093/imrn/rnaa242 ◽

2020 ◽

Author(s):

Yongqiang Liu ◽

Laurentiu Maxim ◽

Botong Wang

Keyword(s):

Euler Characteristic ◽

Morse Theory ◽

Characteristic Property ◽

Abelian Varieties ◽

Betti Numbers ◽

Local Systems ◽

Affine Manifolds ◽

Cyclic Covers ◽

Rank One ◽

Generic Vanishing

Abstract We use the non-proper Morse theory of Palais–Smale to investigate the topology of smooth closed subvarieties of complex semi-abelian varieties and that of their infinite cyclic covers. As main applications, we obtain the finite generation (except in the middle degree) of the corresponding integral Alexander modules as well as the signed Euler characteristic property and generic vanishing for rank-one local systems on such subvarieties. Furthermore, we give a more conceptual (topological) interpretation of the signed Euler characteristic property in terms of vanishing of Novikov homology. As a byproduct, we prove a generic vanishing result for the $L^2$-Betti numbers of very affine manifolds. Our methods also recast June Huh’s extension of Varchenko’s conjecture to very affine manifolds and provide a generalization of this result in the context of smooth closed sub-varieties of semi-abelian varieties.

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L2-BETTI NUMBERS OF DISCRETE MEASURED GROUPOIDS

International Journal of Algebra and Computation ◽

10.1142/s0218196705002748 ◽

2005 ◽

Vol 15 (05n06) ◽

pp. 1169-1188 ◽

Cited By ~ 9

Author(s):

ROMAN SAUER

Keyword(s):

Probability Space ◽

Betti Numbers ◽

Free Action ◽

Countable Group ◽

Equivalence Relations ◽

Dimension Theory ◽

Discrete Groups ◽

Type Ii ◽

Orbit Equivalent ◽

Definition Of

There are notions of L2-Betti numbers for discrete groups (Cheeger–Gromov, Lück), for type II1-factors (recent work of Connes-Shlyakhtenko) and for countable standard equivalence relations (Gaboriau). Whereas the first two are algebraically defined using Lück's dimension theory, Gaboriau's definition of the latter is inspired by the work of Cheeger and Gromov. In this work we give a definition of L2-Betti numbers of discrete measured groupoids that is based on Lück's dimension theory, thereby encompassing the cases of groups, equivalence relations and holonomy groupoids with an invariant measure for a complete transversal. We show that with our definition, like with Gaboriau's, the L2-Betti numbers [Formula: see text] of a countable group G coincide with the L2-Betti numbers [Formula: see text] of the orbit equivalence relation [Formula: see text] of a free action of G on a probability space. This yields a new proof of the fact the L2-Betti numbers of groups with orbit equivalent actions coincide.

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