Algebraic Cycles Research Articles

Homological Mirror Symmetry, coisotropic branes and $$P=W$$ P = W

We discuss the $$P=W$$ conjecture and suggest and a new approach to it using the theory of coisotropic branes and algebraic cycles. Then we show a way to produce many examples of these coisotropic branes.

On complete intersections in varieties with finite-dimensional motive

Let X be a complete intersection inside a variety M with finite-dimensional motive and for which the Lefschetz-type conjecture B (M) holds. We show how conditions on the niveau filtration on the homology of X influence directly the niveau on the level of Chow groups. This leads to a generalization of Voisin's result. The latter states that if M has trivial Chow groups and if X has non-trivial variable cohomology parametrized by c-dimensional algebraic cycles, then the cycle class maps A(k)(X) -> H-2k (X) are injective for k < c. We give variants involving group actions, which lead to several new examples with finite-dimensional Chow motives.

Algebraic cycles and Todorov surfaces

Motivated by the Bloch–Beilinson conjectures, Voisin has formulated a conjecture about 0-cycles on self-products of surfaces of geometric genus one. We verify Voisin’s conjecture for the family of Todorov surfaces with K2=2 and fundamental group Z/2Z. As a by-product, we prove that certain Todorov surfaces have finite-dimensional motive.

Algebraic Cycles and Residues of Degree 8 L-functions of GSp(4) × GL(2)

Abstract We prove a cohomological formula for noncritical residues of degree 8 automorphic $L$-functions of $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ in the spirit of Beilinson conjecture. We rely on the cohomological interpretation of an automorphic period integral and on the study of Novodvorsky’s integral representation of the $L$-functions.

Une propriété de continuité associée aux classes de cohomologie de Hodge

The obstructions, for a closed smooth differential form of bidimension (p,p) on a projective manifold, to be cohomologous to an algebraic cycle with complex coefficients, are calculated in terms of the Chow transformation. They can be expressed as an orthogonality condition, on the manifold itself, with families parametrized by the Grassmannian of currents that are completely determined. Each of these currents is ddc-closed and with support in the intersection of the manifold and of the projective subspace associated with the parameter. By the theory of harmonic forms, a period is thus associated with that differential form for each parameter. We study the set of periods, obtained when the parameter varies, and we arrive at a continuity on the Grassmannian, when the cohomology class is rational. The same property can be obtained by going to the space of divisors of the Grassmannian and by using a characterization of Chow forms. We proceed here directly, by calculating the periods by means of the Atiyah–Hirzebruch theorem. This global continuity implies the orthogonality for all parameter.

Arbeitsgemeinschaft: Higher Gross Zagier Formulas

The aim of this Arbeitsgemeinschaft is to go over the proof of the higher Gross–Zagier formula established in the paper [YZ15]. The formula relates arbitrary order central derivative of the base change L -function of an unramifed automorphic representation of PGL _2 over a function field to the self-intersection number of a certain algebraic cycle on the moduli stack of Shtukas.

Algebraic Limit Cycles in Piecewise Linear Differential Systems

This paper is devoted to study the algebraic limit cycles of planar piecewise linear differential systems. In particular, we present examples exhibiting two explicit hyperbolic algebraic limit cycles, as well as some one-parameter families with a saddle-node bifurcation of algebraic limit cycles. We also show that all degrees for algebraic limit cycles are allowed.

Algebraic Limit Cycles on Quadratic Polynomial Differential Systems

AbstractAlgebraic limit cycles in quadratic polynomial differential systems started to be studied in 1958, and a few years later the following conjecture appeared: quadratic polynomial differential systems have at most one algebraic limit cycle. We prove that a quadratic polynomial differential system having an invariant algebraic curve with at most one pair of diametrically opposite singular points at infinity has at most one algebraic limit cycle. Our result provides a partial positive answer to this conjecture.

Algebraic Cycles on Quadric Sections of Cubics in ℙ4 Under the Action of Symplectomorphisms – Erratum

Algebraic Cycles on Quadric Sections of Cubics in ℙ⁴ Under the Action of Symplectomorphisms – Erratum

A note on Grothendieck’s standard conjectures of type $C^+$ and $D$

Grothendieck conjectured in the sixties that the even Kunneth projector (with respect to a Weil cohomology theory) is algebraic and that the homological equivalence relation on algebraic cycles coincides with the numerical equivalence relation. In this note we extend these celebrated conjectures from smooth projective schemes to the broad setting of smooth proper dg categories. As an application, we prove that Grothendieck's original conjectures are invariant under homological projective duality. This leads to a proof of Grothendieck's conjectures in the case of intersections of quadrics, linear sections of determinantal varieties, and intersections of bilinear divisors. Along the way, we prove also the case of quadric fibrations.

Relative dynamical degrees of correspondences over a field of arbitrary characteristic

Abstract Let 𝕂 {\mathbb{K}} be an algebraically closed field of arbitrary characteristic, X and Y irreducible possibly singular algebraic varieties over 𝕂 {\mathbb{K}} . Let f : X ⊢ X {f:X\vdash X} and g : Y ⊢ Y {g:Y\vdash Y} be dominant correspondences, and π : X ⇢ Y {\pi:X\dashrightarrow Y} a dominant rational map which semi-conjugate f and g, i.e. so that π ∘ f = g ∘ π {\pi\circ f=g\circ\pi} . We define relative dynamical degrees λ p ( f | π ) ≥ 1 {\lambda_{p}(f|\pi)\geq 1} for any p = 0 , … , dim ⁡ ( X ) - dim ⁡ ( Y ) {p=0,\dots,\dim(X)-\dim(Y)} . These degrees measure the relative growth of positive algebraic cycles, satisfy a product formula when Y is smooth and g is a multiple of a rational map, and are birational invariants. More generally, a weaker product formula is proven for more general semi-conjugacies, and for any generically finite semi-conjugacy ( φ , ψ ) {(\varphi,\psi)} from π 2 : ( X 2 , f 2 ) → ( Y 2 , g 2 ) {\pi_{2}:(X_{2},f_{2})\rightarrow(Y_{2},g_{2})} to π 1 : ( X 1 , f 1 ) → ( Y 1 , g 1 ) {\pi_{1}:(X_{1},f_{1})\rightarrow(Y_{1},g_{1})} we have λ p ( f 1 | π 1 ) ≥ λ p ( f 2 | π 2 ) {\lambda_{p}(f_{1}|\pi_{1})\geq\lambda_{p}(f_{2}|\pi_{2})} for all p. Many of our results are new even when 𝕂 = ℂ {\mathbb{K}=\mathbb{C}} . Self-correspondences are abundant, even on varieties having not many self rational maps, hence these results can be applied in many situations. In the last section of the paper, we will discuss recent new applications of this to algebraic dynamics, in particular to pullback on l-adic cohomology groups in positive characteristics.

Algebraic cycles and EPW cubes

AbstractLet X be a hyperkähler variety with an anti‐symplectic involution ι. According to Beauville's conjectural “splitting property”, the Chow groups of X should split in a finite number of pieces such that the Chow ring has a bigrading. The Bloch–Beilinson conjectures predict how ι should act on certain of these pieces of the Chow groups. We verify part of this conjecture for a 19‐dimensional family of hyperkähler sixfolds that are “double EPW cubes” (in the sense of Iliev–Kapustka–Kapustka–Ranestad). This has interesting consequences for the Chow ring of the quotient , which is an “EPW cube” (in the sense of Iliev–Kapustka–Kapustka–Ranestad).

Algebraic limit cycles for quadratic polynomial differential systems

We prove that for a quadratic polynomial differential system having three pairs of diametrally opposite equilibrium points at infinity that are positively rationally independent, has at most one algebraic limit cycle. Our result provides a partial positive answer to the following conjecture: Quadratic polynomial differential systems have at most one algebraic limit cycle.

A zoo of geometric homology theories

The theories in our zoo are all bordism groups, which generalize the case of smooth manifolds by allowing singularities. There are many concepts of manifolds with singularities one could use here. For our pupose the objects the author introduced some years ago, which are called stratifolds, work particularly well. The zoo comes from forcing certain strata indexed by the subset $A$ to be empty. Special cases are ordinary singular homology and singular bordism. Despite their simple construction computations of these groups seem to be very complicated. We give a few simple examples. Thus there are no interesting applications so far and the zoo looks a bit like a curiosity. But one never knows for what these theories might be good in the future. We mention a concrete question which might be useful in connection with the Griffith group consisting of algebraic cycles in a smooth algebraic variety over the complex numbers which vanish in singular homology. I dedicate these notes to my friend Egbert Brieskorn. Egbert is (in a very different way like our common teacher Hirzebruch) a person which had a great influence on me. When I had to make a complicated decision I often had him in front of my eyes and asked myself: What would Egbert suggest? Conversations with him were always intense and fruitful. I miss him very much.

On an inductive approach to the standard conjecture for a fibred complex variety with strong semistable degeneracies

We prove that Grothendieck's standard conjecture of Lefschetz type on the algebraicity of the operators and of Hodge theory holds for a 4-dimensional smooth projective complex variety fibred over a smooth projective curve provided that every degenerate fibre is a union of smooth irreducible components of multiplicity with normal crossings, the standard conjecture holds for a generic geometric fibre , there is at least one degenerate fibre and the rational cohomology rings and of the irreducible components of every degenerate fibre are generated by classes of algebraic cycles. We obtain similar results for -dimensional fibred varieties with algebraic invariant cycles (defined by the smooth part of the structure morphism ) or with a degenerate fibre all of whose irreducible components possess the property .

Algebraic cycles and local anomalies in F-theory

We introduce a set of identities in the cohomology ring of elliptic fibrations which are equivalent to the cancellation of gauge and mixed gauge-gravitational anomalies in F-theory compactifications to four and six dimensions. The identities consist in (co)homological relations between complex codimension-two cycles. The same set of relations, once evaluated on elliptic Calabi-Yau three-folds and four-folds, is shown to universally govern the structure of anomalies and their Green-Schwarz cancellation in six- and four-dimensional F-theory vacua, respectively. We furthermore conjecture that these relations hold not only within the cohomology ring, but even at the level of the Chow ring, i.e. as relations among codimension-two cycles modulo rational equivalence. We verify this conjecture in non-trivial examples with Abelian and non-Abelian gauge groups factors. Apart from governing the structure of local anomalies, the identities in the Chow ring relate different types of gauge backgrounds on elliptically fibred Calabi-Yau four-folds.

Gauge backgrounds and zero-mode counting in F-theory

Computing the exact spectrum of charged massless matter is a crucial step towards understanding the effective field theory describing F-theory vacua in four dimensions. In this work we further develop a coherent framework to determine the charged massless matter in F-theory compactified on elliptic fourfolds, and demonstrate its application in a concrete example. The gauge background is represented, via duality with M-theory, by algebraic cycles modulo rational equivalence. Intersection theory within the Chow ring allows us to extract coherent sheaves on the base of the elliptic fibration whose cohomology groups encode the charged zero-mode spectrum. The dimensions of these cohomology groups are computed with the help of modern techniques from algebraic geometry, which we implement in the software gap. We exemplify this approach in models with an Abelian and non-Abelian gauge group and observe jumps in the exact massless spectrum as the complex structure moduli are varied. An extended mathematical appendix gives a self-contained introduction to the algebro-geometric concepts underlying our framework.

The Cubic Polynomial Differential Systems with two Circles as Algebraic Limit Cycles

Abstract In this paper we characterize all cubic polynomial differential systems in the plane having two circles as invariant algebraic limit cycles.

Rational mixed Tate motivic graphs

In this paper, we study the combinatorics of a subcomplex of the Bloch-Kriz cycle complex [4] used to construct the category of mixed Tate motives. The algebraic cycles we consider properly contain the subalgebra of cycles that correspond to multiple logarithms (as defined in [12]). We associate an algebra of graphs to our subalgebra of algebraic cycles. We give a purely graphical criterion for admissibilty. We show that sums of bivalent graphs correspond to coboundary elements of the algebraic cycle complex. Finally, we compute the Hodge realization for an infinite family of algebraic cycles represented by sums of graphs that are not describable in the combinatorial language of [12].

Symmetric correspondences on quadrics

We prove a result comparing the rationality of some elementary algebraic cycles introduced by Alexander Vishik, defined on orthogonal grassmannians, with the rationality of some algebraic cycles defined on fiber products of the corresponding quadric.