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K3 and Enriques Surfaces Read chapter Read first chapter

2013 | OriginalPaper | Chapter

Transcendental Methods in the Study of Algebraic Cycles with a Special Emphasis on Calabi–Yau Varieties

Author : James D. Lewis

Published in: Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds

Publisher: Springer New York

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Abstract

We review the transcendental aspects of algebraic cycles, and explain how this relates to Calabi–Yau varieties. More precisely, after presenting a general overview, we begin with some rudimentary aspects of Hodge theory and algebraic cycles. We then introduce Deligne cohomology, as well as the generalized higher cycles due to Bloch that are connected to higher K-theory, and associated regulators. Finally, we specialize to the Calabi–Yau situation, and explain some recent developments in the field.

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previous chapter K3 and Enriques Surfaces
next chapter Two Lectures on the Arithmetic of K3 Surfaces
Footnotes
1
Strict compatibility means that \(h({F}^{r}V _{1,\mathbf{C}}) = h(V _{1,\mathbf{C}}) \cap {F}^{r}V _{2,\mathbf{C}}\) and \(h(W_{\ell}V _{1,\mathbf{A}\otimes \mathbf{Q}}) = h(V _{1,\mathbf{A}\otimes \mathbf{Q}}) \cap W_{\ell}V _{2,\mathbf{A}\otimes \mathbf{Q}}\) for all r and ℓ. A nice explanation of Deligne’s proof of this fact can be found in [44], where a quick summary goes as follows: For any A-MHS V, V C has a C-splitting into a bigraded direct sum of complex vector spaces \({I}^{p,q} := {F}^{p} \cap W_{p+q} \cap \big [\overline{{F}^{q}} \cap W_{p+q} +\sum _{i\geq 2}\overline{{F}^{q-i+1}} \cap W_{p+q-i}\big]\), where one shows that \({F}^{r}V _{\mathbf{C}} = \oplus _{p\geq r} \oplus _{q}{I}^{p,q}\) and \(W_{\ell}V _{\mathbf{C}} = \oplus _{p+q\leq \ell}{I}^{p,q}\). Then by construction of I p, q , one has \(h({I}^{p,q}(V _{1,\mathbf{C}}) \subseteq {I}^{p,q}(V _{2,\mathbf{C}})\). Hence h preserves both the Hodge and complexified weight filtrations. Now use the fact that A ⊗ Q is a field to deduce that h preserves the weight filtration over A ⊗ Q.
 
2
The fact that a smooth connected Γ will suffice (as opposed to a [connected] chain of curves) in the definition of algebraic equivalence follows from the transitive property of algebraic equivalence (see [36] (p. 180)).
 
3
We remind the reader that for singular homology H  ∗  sing (U, Z) and ignoring twists, Poincaré duality gives the isomorphism \(H_{c}^{i}(U,\mathbf{Z}) \simeq H_{2d-i}^{sing}(U,\mathbf{Z})\), where H c i (U, Z) is cohomology with compact support; whereas \({H}^{i}(U,\mathbf{Z}) \simeq H_{2d-i}^{BM}(U,\mathbf{Z})\).
 
4
A special thanks to Rob de Jeu for supplying us this idea.
 
5
Matt Kerr informed us of an alternate and slick approach to this example via the definition given in Example 4.7(ii). Namely one need only add Tame\(\{z_{1}/z_{0},z_{2}/z_{0}\} = (-f_{0}^{-1},\ell_{0}) + (f_{1}^{-1},\ell_{1}) + (f_{2}^{-1},\ell_{2})\) to ξ to get the 2-torsion class ( − 1, ℓ 0), which is the same as ξ in CH2(P 2, 1).
 
6
Indeed first consider \((a,b) \in \Omega _{X}^{r} \oplus \Omega _{X}^{r-1}\ \mathop{\mapsto }\limits \delta \ (-da,db - a) \in \Omega _{X}^{r+1} \oplus \Omega _{X}^{r}\). Then δ(a, b) = (0, 0) ⇔ da = 0 & a = db ⇔ a = db. Therefore \(\ker \delta /\mathrm{Im}(0,d) \simeq \Omega _{X}^{r-1}/d\Omega _{X}^{r-2} = {\mathcal{H}}^{r-1}(\mathbf{A}_{\mathcal{D}}(r))\). Next, for j ≥ 1, \((a,b) \in \Omega _{X}^{r+j} \oplus \Omega _{X}^{r+j-1}\), δ(a, b) = 0 ⇔ (a, b) = δ( − b, 0).
 
7
The reader familiar with Deligne homology will see this definition as the same thing up to twist. Indeed this definition already incorporates Poincaré duality.
 
8
Y is a normal crossing divisor, which in local analytic coordinates (z 1, …, z d ) on X, Y is given by z 1⋯z ℓ  = 0, and so Ω X 1⟨Y ⟩ has local frame \(\big\{dz_{1}/z_{1},\ldots,dz_{\ell}/z_{\ell},dz_{\ell+1},\ldots,dz_{d}\big\}\).
 
9
For compactly supported ω ∈ E U, c 2d − 1, and \(f \in \mathcal{O}_{U}^{\times }(U)\),
$$\displaystyle{\int _{U}\frac{df} {f} \wedge \omega =\int _{U}d\big(\log f\wedge \omega \big) -\int _{U\setminus {f}^{-1}[-\infty,0]}\log f \wedge d\omega = 2\pi \mathrm{i}\int _{{f}^{-1}[-\infty,0]}\omega + d[\log f](\omega ),}$$
where we use the principal branch of log.
 
10
The decision to consider the factor (2πi) − 1 is somewhat “political”, as reflected in the remark on page 2 of [32]. From a cohomological point of view, one works with Z(2) coefficient periods, whereas homologically, is it with Z(1) coefficients. This is neatly illustrated via the Poincaré duality isomorphism in (16).
 
11
Alternatively, taking Re\(\big({(2\pi \mathrm{i})}^{-1}\tilde{R}\big)\) gives the formula in (15), viz., with the factor (2π) − 1, right on the nose.
 
12
M. Asakura informed me of his work in [1], which includes this theorem as a special case. Further he provides an upper bound for the rank of the dlog image for variants of the family in Example 8.25.
 
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Metadata
Title
Transcendental Methods in the Study of Algebraic Cycles with a Special Emphasis on Calabi–Yau Varieties
Author
James D. Lewis
Copyright Year
2013
Publisher
Springer New York
Book
Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds

Print ISBN: 978-1-4614-6402-0

Electronic ISBN: 978-1-4614-6403-7

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-1-4614-6403-7_2

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