On the complexity of deciding connectedness and computing Betti numbers of a complex algebraic variety

Suppose $X$ is a smooth complex algebraic variety. A necessary condition for a complex topological vector bundle on $X$ (viewed as a complex manifold) to be algebraic is that all Chern classes must be algebraic cohomology classes, that is, lie in the image of the cycle class map. We analyze the question of whether algebraicity of Chern classes is sufficient to guarantee algebraizability of complex topological vector bundles. For affine varieties of dimension ${\leqslant}3$, it is known that algebraicity of Chern classes of a vector bundle guarantees algebraizability of the vector bundle. In contrast, we show in dimension ${\geqslant}4$ that algebraicity of Chern classes is insufficient to guarantee algebraizability of vector bundles. To do this, we construct a new obstruction to algebraizability using Steenrod operations on Chow groups. By means of an explicit example, we observe that our obstruction is nontrivial in general.

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Locally free (ℙn)-modules

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100070924 ◽

1992 ◽

Vol 112 (2) ◽

pp. 233-245 ◽

Cited By ~ 3

Author(s):

S. C. Coutinho ◽

M. P. Holland

Keyword(s):

Projective Space ◽

Algebraic Variety ◽

Differential Operators ◽

Module Theory ◽

Rings Of Differential Operators ◽

Complex Algebraic Variety ◽

Free Modules

The purpose of this paper is to study the structure of locally free modules over the ring of differential operators on projective space. Let be a non-singular, complex, algebraic variety. Denote by the sheaf of rings of differential operators over and by its ring of global sections. A -module M is called locally free if the associated sheaf ⊗ M is locally free as a sheaf of -modules. Locally free modules arise naturally in -module theory as inverse images of determined modules; see [1] for definitions and examples.

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On the algebraicity of the zero locus of an admissible normal function

Compositio Mathematica ◽

10.1112/s0010437x1300729x ◽

2013 ◽

Vol 149 (11) ◽

pp. 1913-1962 ◽

Cited By ~ 4

Author(s):

Patrick Brosnan ◽

Gregory Pearlstein

Keyword(s):

Galois Group ◽

Algebraic Variety ◽

Hodge Structure ◽

Normal Function ◽

Mixed Hodge Structure ◽

Nilpotent Orbits ◽

Smooth Complex ◽

Complex Algebraic Variety ◽

Zero Locus

AbstractWe show that the zero locus of an admissible normal function on a smooth complex algebraic variety is algebraic. In Part II of the paper, which is an appendix, we compute the Tannakian Galois group of the category of one-variable admissible real nilpotent orbits with split limit. We then use the answer to recover an unpublished theorem of Deligne, which characterizes the ${\mathrm{sl} }_{2} $-splitting of a real mixed Hodge structure.

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Geometric motivic Poincaré series of quasi-ordinary singularities

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004110000101 ◽

2010 ◽

Vol 149 (1) ◽

pp. 49-74 ◽

Cited By ~ 2

Author(s):

HELENA COBO PABLOS ◽

PEDRO D. GONZÁLEZ PÉREZ

Keyword(s):

Algebraic Variety ◽

Poincaré Series ◽

Explicit Description ◽

Hypersurface Singularity ◽

Grothendieck Ring ◽

Rational Form ◽

Poincare Series ◽

Newton Polyhedra ◽

Complex Algebraic Variety

AbstractThegeometric motivic Poincaré seriesof a germ (S, 0) of complex algebraic variety takes into account the classes in the Grothendieck ring of the jets of arcs through (S, 0). Denef and Loeser proved that this series has a rational form. We give an explicit description of this invariant when (S, 0) is an irreducible germ ofquasi-ordinary hypersurface singularityin terms of the Newton polyhedra of thelogarithmic jacobian ideals. These ideals are determined by thecharacteristic monomialsof a quasi-ordinary branch parametrizing (S, 0).

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Parity sheaves and Smith theory

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2021-0018 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Spencer Leslie ◽

Gus Lonergan

Keyword(s):

Fixed Point ◽

Prime Number ◽

Algebraic Variety ◽

Reductive Group ◽

Derived Category ◽

Geometric Construction ◽

Affine Grassmannian ◽

Complex Algebraic Variety ◽

Smith Theory

Abstract Let p be a prime number and let X be a complex algebraic variety with an action of ℤ / p ⁢ ℤ {\mathbb{Z}/p\mathbb{Z}} . We develop the theory of parity complexes in a certain 2-periodic localization of the equivariant constructible derived category D ℤ / p ⁢ ℤ b ⁢ ( X , ℤ p ) {D^{b}_{\mathbb{Z}/p\mathbb{Z}}(X,\mathbb{Z}_{p})} . Under certain assumptions, we use this to define a functor from the category of parity sheaves on X to the category of parity sheaves on the fixed-point locus X ℤ / p ⁢ ℤ {X^{\mathbb{Z}/p\mathbb{Z}}} . This may be thought of as a categorification of Smith theory. When X is the affine Grassmannian associated to some complex reductive group, our functor gives a geometric construction of the Frobenius-contraction functor recently defined by M. Gros and M. Kaneda via the geometric Satake equivalence.

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On tangent cones at infinity of algebraic varieties

Journal of Algebra and Its Applications ◽

10.1142/s0219498818501438 ◽

2018 ◽

Vol 17 (08) ◽

pp. 1850143 ◽

Cited By ~ 2

Author(s):

Công-Trình Lê ◽

Tien-Son Phạm

Keyword(s):

Algebraic Variety ◽

Tangent Cone ◽

Tangent Cones ◽

Algebraic Varieties ◽

Combinatorial Topology ◽

Princeton University ◽

Analytic Varieties ◽

Complex Algebraic Variety ◽

Local Properties

In this paper, we define the geometric and algebraic tangent cones at infinity of algebraic varieties and establish the following version at infinity of Whitney’s theorem [Local properties of analytic varieties, in Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse) (Princeton University Press, Princeton, N. J., 1965), pp. 205–244; Tangents to an analytic variety, Ann. of Math. 81 (1965) 496–549]: The geometric and algebraic tangent cones at infinity of complex algebraic varieties coincide. The proof of this fact is based on a geometric characterization of the geometric tangent cone at infinity using the global Łojasiewicz inequality with explicit exponents for complex algebraic varieties. Moreover, we show that the tangent cone at infinity of a complex algebraic variety is actually the part at infinity of this variety [G.-M. Greuel and G. Pfister, A Singular Introduction to Commutative Algebra, 2nd extended edn. (Springer, Berlin, 2008)]. We also show that the tangent cone at infinity of a complex algebraic variety can be computed using Gröbner bases.

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Extremal Betti numbers of graded ideals

Results in Mathematics ◽

10.1007/bf03322739 ◽

2003 ◽

Vol 43 (3-4) ◽

pp. 235-244 ◽

Cited By ~ 8

Author(s):

Marilena Crupi ◽

Rosanna Utano

Keyword(s):

Betti Numbers

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Betti numbers and pseudoeffective cones in 2-Fano varieties

Advances in Geometry ◽

10.1515/advgeom-2021-0004 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Giosuè Emanuele Muratore

Keyword(s):

Large Class ◽

Betti Numbers ◽

Fano Varieties ◽

Higher Dimensional ◽

Special Case ◽

Answering Questions

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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A general form of the Second Main Theorem for meromorphic mappings from a p-Parabolic manifold to a projective algebraic variety

Indian Journal of Pure and Applied Mathematics ◽

10.1007/s13226-021-00095-8 ◽

2021 ◽

Author(s):

Wei Chen ◽

Nguyen Van Thin

Keyword(s):

Algebraic Variety ◽

Second Main Theorem ◽

Meromorphic Mappings ◽

The Second Main Theorem

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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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