Kalyan Banerjee, Jaya NN Iyer, James D. Lewis
We introduce a homological Lefschetz conjecture on (rational) Chow groups, which can be deduced from some well known conjectures, and illustrate it by a series of key examples. We then prove the injectivity of the push-forward morphism on Chow groups, induced by the closed embedding of the Theta divisor in it's Jacobian $J(C)$. Here $C$ is a smooth irreducible complex projective curve.
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We prove a moving lemma for higher Chow groups with modulus, in the sense of Binda-Kerz-Saito, of projective schemes when the modulus is given by a very ample divisor. This provides one of the first cases of moving lemmas for cycles with modulus, not covered by the additive higher Chow groups. We apply this to prove a contravariant functoriality of higher Chow groups with modulus. We use our moving techniques to show that the higher Chow groups of a line bundle over a scheme, with the 0-section as the modulus, vanishes.
In this note we prove that the kernel of the push-forward homomorphism on $d$-cycles modulo rational equivalence, induced by the closed embedding of an ample divisor linearly equivalent to some multiple of the theta divisor inside the Jacobian variety $J(C)$ is trivial. Here $C$ is a smooth projective curve of genus $g$.
In this text we prove that if an abelian variety $A$ admits of an embedding into the Jacobian of a smooth projective curve $C$, and if we consider $\Th_A$ to be the divisor $\Th_C\cap A$, where $\Th_C$ denotes the theta divisor of $J(C)$, then the embedding of $\Th_A$ into $A$ induces an injective push-forward homomorphism at the level of Chow groups. We show that this is the case for every principally polarized abelian varieties.
Pushforward of Siegel flag varieties in the Chow ring
本文计算了 Siegel 情形下子旗簇在 Chow 环中的类，并指出其对应于阿贝尔簇紧化模空间 tautological 环中支撑在环面边界上的类生成元，猜想该对应关系并非巧合。
In this article we generalize the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of $\Sym^m C$ into $\Sym^n C$ for $m\leq n$, where $C$ is a smooth projective curve, to symmetric powers of a smooth projective variety of higher dimension. We also prove the analog of this theorem for product of symmetric powers of smooth projective varieties. As an application we prove the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of self product of theta divisor into the self product of the…
We define tautological Chow classes on the moduli space of curves with linear series. In the case where the forgetful morphism to the moduli space of curves has relative dimension zero, we describe the images of these classes in the Chow group of Mgbar. As an application, we compute the (virtual) slopes of several different classes of divisors on Mgbar.
For any smooth complex projective variety X and smooth very ample hypersurface Y in X, we develop the technique of genus zero relative Gromov-Witten invariants of Y in X in algebro-geometric terms. We prove an equality of cycles in the Chow groups of the moduli spaces of relative stable maps that relates these relative invariants to the Gromov-Witten invariants of X and Y. Given the Gromov-Witten invariants of X, we show that these relations are sufficient to compute all relative invariants, as well as all genus zero Gromov-Witten invariants of Y whose homology and cohomology classes are induc…
Let $X$ be a smooth projective variety over a field of characteristic $0$ and $L$ an ample divisor. In this paper we study the Weak Lefschetz conjecture for Chow groups using the technique employed by Grothendieck in his study of the problem for Picard groups, and using Bloch's formula to interpret Chow groups in terms of Milnor $K$- theory we prove the infinitesimal part of the conjecture.
Motivated by the study of rationally connected fibrations (and the MRC quotient) we study different notions of birationally simple fibrations. We say a fibration of smooth projective varieties is Chow constant if pushforward induces an isomorphism on the Chow group of 0-cycles. Likewise we say a fibration is cohomologically constant if pullback induces an isomorphism on holomorphic p-forms for all p. Our main result is the construction of maximal Chow constant and cohomologically constant fibrations. The paper is largely self contained and we prove a number of basic properties of these fibrati…
Let $\Sym^m X$ denote the $m$-th symmetric power of a smooth projective curve $X$. Let $\wt{\Sym^m X}$ be the blow up of $\Sym^m X$ along some non-singular subvariety. In this note we are going to discuss when the pushforward homomorphism induced by the natural morphism from $\wt{\Sym^m X}$ to $\Sym^n X$ is injective at the level of Chow groups for $m\leq n$. Also we are going to prove that base changes of embeddings of one symmetric power into another, with respect to closed embeddings induces an injection at the level of Chow groups.
We define an `enriched' notion of Chow groups for algebraic varieties, agreeing with the conventional notion for complete varieties, but enjoying a functorial push-forward for arbitrary maps. This tool allows us to glue intersection-theoretic information across elements of a stratification of a variety; we illustrate this operation by giving a direct construction of Chern-Schwartz-MacPherson classes of singular varieties, providing a new proof of an old (and long since settled) conjecture of Deligne and Grothendieck.
Let $X$ be a smooth complex projective variety with trivial Chow groups. (By trivial, we mean that the cycle class is injective.) We show (assuming the Lefschetz standard conjecture) that if the vanishing cohomology of a general complete intersection $Y$ of ample hypersurfaces in $X$ has geometric coniveau $\geq c$, then the Chow groups of cycles of dimension $\leq c-1$ of $Y$ are trivial. The generalized Bloch conjecture for $Y$ is this statement with "geometric coniveau" replaced by "Hodge coniveau".
Chow groups of surfaces of lines in cubic fourfolds
研究四次三次折叠中与固定直线相交的直线曲面的动机分解，定义Beauville-Voisin类的类似物，并利用Shen和Vial引入的Bloch-Beilinson滤过自然分裂研究到所有直线Fano簇的前推映射。
A natural Chow lemma for smooth threefolds and fourfolds
本文针对维数至多四的光滑真整簇，构造了沿光滑中心的有限爆破序列，以射影端点为终点，并保持所有极大拟射影开集的交集，该构造在函子性主化过程下自然且同构不变。
We prove an unconditional (but slightly weakened) version of the main result of our earlier paper with the same title, which was, starting from dimension $4$, conditional to the Lefschetz standard conjecture. Let $X$ be a variety with trivial Chow groups, (i.e. the cycle class map to cohomology is injective on $CH(X)_\mathbb{Q}$). We prove that if the cohomology of a general very ample hypersurface $Y$ in $X$ is ``parameterized by cycles of dimension $c$'', then the Chow groups $CH_{i}(Y)_\mathbb{Q}$ are trivial for $i\leq c-1$.
We prove analogs of Whitehead's theorem (from algebraic topology) for both the Chow groups and for the Grothendieck group of coherent sheaves: a morphism between smooth projective varieties whose pushforward is an isomorphism on the Chow groups, or on the Grothendieck group of coherent sheaves, is an isomorphism. As a corollary, we show that there are no nontrivial naive $\mathbb{A}^1$-homotopy equivalences between smooth projective varieties.
We introduce a smooth projective variety $T_{d,n}$ which compactifies the space of configurations of $n$ distinct points on affine $d$-space modulo translation and homothety. The points in the boundary correspond to $n$-pointed stable rooted trees of $d$-dimensional projective spaces, which for $d = 1$, are $(n+1)$-pointed stable rational curves. In particular, $T_{1,n}$ is isomorphic to $\bar{M}_{0,n+1}$, the moduli space of such curves. The variety $T_{d,n}$ shares many properties with $\bar{M}_{0,n}$. For example, as we prove, the boundary is a smooth normal crossings divisor whose componen…
Let X be a smooth projective variety. Starting with a finite set of cycles on powers X^m of X, we consider the Q-vector subspaces of the Q-linear Chow groups of the X^m obtained by iterating the algebraic operations and pullback and push forward along those morphisms X^l -> X^m for which each component X^l -> X is a projection. It is shown that these Q-vector subspaces are all finite-dimensional, provided that the Q-linear Chow motive of X is a direct summand of that of an abelian variety.
Divisibility and torsion in higher Chow groups over arithmetic fields
本文研究算术域上光滑概形的高阶Chow群的可除性与无挠性，并应用于有限域、局部域和整体域。
A theorem of Batyrev's asserts that if two nonsingular varieties V,W are birational, and their canonical bundles agree after pull-back to a resolution of indeterminacies of a birational map between them, then the Betti numbers of V and W coincide. We prove that, in the same hypotheses, the total homology Chern classes of V and W are push-forwards of the same class in the Chow group of the resolution. For example, it follows that the push-forward of the total Chern class of a crepant resolution of a singular variety is independent of the resolution.
We give effective bounds on the generation of pushforwards of log-pluricanonical bundles twisted by ample line bundles. This gives a partial answer to a conjecture proposed by Popa and Schnell. We prove two types of statements: first, more in the spirit of the general conjecture, we show generic global generation with predicted bound when the dimesnion of the variety if less than 4 and more generally, with a quadratic Angehrn-Siu type bound. Secondly, assuming that the relative canonical bundle is relatively semi-ample, we make a very precise statement. In particular, when the morphism is smoo…
Homological Milnor-Witt modules and Chow-Witt groups over general bases
本文提出了一种一般基上的同调Milnor-Witt模理论，扩展了Chow-Witt群，并证明了广义Bloch公式和有限性结果。
We recall P. Balmer's definition of tensor triangular Chow group for a tensor triangulated category $\mathcal{K}$ and explore some of its properties. We give a proof that for a suitably nice scheme $X$ it recovers the usual notion of Chow group from algebraic geometry when we put $\mathcal{K} = \mathrm{D^{perf}}(X)$. Furthermore, we identify a class of functors for which tensor triangular Chow groups behave functorially and show that (for suitably nice schemes) proper push-forward and flat pull-back of algebraic cycles can be interpreted as being induced by the derived inverse and direct image…
We prove a formula for the push-forward class of Bott-Samelson resolutions in the algebraic cobordism ring of the flag bundle. We specialise our formula to connective K-theory providing a geometric interpretation to the double beta-polynomials of Fomin and Kirillov by computing the fundamental classes of Schubert varieties. As a corollary we obtain a Thom-Porteous formula generalising those of the Chow ring and of the Grothendieck ring of vector bundles.
Filtrations of the Chow group of zero-cycles on abelian varieties and behavior under isogeny
本文研究阿贝尔簇零维Chow群上的Bloch-Beilinson型滤过在同源下的行为，证明乘以n映射在商群上诱导乘以n^r，并在椭圆曲线乘积情形与已知滤过一致，同时为Raskind-Spiess和Colliot-Thélène猜想提供新证据。
The classical cycle class map for a smooth complex variety sends cycles in the Chow ring to cycles in the singular cohomology ring. We study two cycle class maps for smooth real varieties: the map from the I-cohomology ring to singular cohomology induced by the signature, and a new cycle class map defined on the Chow-Witt ring. For both maps, we establish basic compatibility results like compatibility with pullbacks, pushforwards and cup products. As a first application of these general results, we show that both cycle class maps are isomorphisms for cellular varieties.
A birational version of K-stability for big classes
本文提出大线丛上光滑射影簇的统一K稳定性理论，扩展了已有理论，并证明统一K稳定性在拉回和某些推前下保持，从而在b除子层面定义明确。
Let $k$ be an uncountable algebraically closed field of characteristic $0$, and let $X$ be a smooth projective connected variety of dimension $2p$, appropriately embedded into $\mathbb P^m$ over $k$. Let $Y$ be a hyperplane section of $X$, and let $A^p(Y)$ and $A^{p+1}(X)$ be the groups of algebraically trivial algebraic cycles of codimension $p$ and $p+1$ modulo rational equivalence on $Y$ and $X$ respectively. Assume that, whenever $Y$ is smooth, the group $A^p(Y)$ is regularly parametrized by an abelian variety $A$ and coincides with the subgroup of degree $0$ classes in the Chow group $CH^…
The eleventh cohomology group of $\bar{\mathcal{M}}_{g,n}$
本文证明 $\bar{\mathcal{M}}_{g,n}$ 的第十一有理上同调群几乎总为零，并利用 $H^{11}(\bar{\mathcal{M}}_{1,11})$ 构造出具有非零奇上同调和非重言闭链类的新模空间例子。
In this text we prove that if an abelian variety $A$ admits an embedding into the Jacobian of a smooth projective curve $C$, and if we consider $Θ_A$ to be the divisor $Θ_C\cap A$, where $Θ_C$ denotes the theta divisor of $J(C)$, then the embedding of $Θ_A$ into $A$ induces an injective push-forward homomorphism (under certain conditions) at the level of Chow groups. We show that this is the case for every Prym varietiy arising from an unramified double cover of smooth projective curves. As a consequence we prove that there does not exist a universal codimension two cycle on the product of a v…
Virtual localization revisited
本文在拟光滑导出概形或Deligne-Mumford栈上推广了Edidin-Graham的局部化逆映射描述，并证明了无全局分解假设的Graber-Pandharipande虚拟局部化公式。
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