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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Notes
- 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.
References
M. Asakura, On dlog Image of K 2 of Elliptic Surface Minus Singular Fibers, preprint (2006) [arXiv:math/0511190v4]
H. Bass, J. Tate, in The Milnor Ring of a Global Field, in Algebraic K-Theory II. Lecture Notes in Mathematics, vol. 342 (Springer, New York, 1972), pp. 349–446
A. Beilinson, Notes on absolute Hodge cohomology, in Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory. Contemporary Mathematics, vol. 55, Part 1 (AMS, Providence 1986), pp. 35–68
A. Beilinson, Higher regulators of modular curves, in Applications of K-Theory to Algebraic Geometry and Number Theory, Boulder, CO, 1983. Contemporary Mathematics, vol. 55 (AMS, Providence, 1986), pp. 1–34
S. Bloch, Lectures on Algebraic Cycles. Duke University Mathematics Series, vol. IV (Duke University, Durham, 1980)
S. Bloch, Algebraic cycles and higher K-theory. Adv. Math. 61, 267–304 (1986)
S. Bloch, V. Srinivas, Remarks on correspondences and algebraic cycles. Am. J. Math. 105, 1235–1253 (1983)
J. Carlson, Extension of mixed Hodge structures, in Journées de Géométrie Algébrique d’Angers 1979 (Sijthoff and Nordhoff, The Netherlands, 1980), pp. 107–127
X. Chen, J.D. Lewis, Noether-Lefschetz for K 1 of a certain class of surfaces. Bol. Soc. Mat. Mexicana (3) 10(1), 29–41 (2004)
X. Chen, J.D. Lewis, The Hodge\(-\mathcal{D}-\)conjecture for K3 and Abelian surfaces. J. Algebr. Geom. 14(2), 213–240 (2005)
X. Chen, J.D. Lewis, Density of rational curves on K3 surfaces. Math. Ann. [arXiv:1004.5167] (2011)
X. Chen, C. Doran, M. Kerr, J.D. Lewis, Higher normal functions, derivatives of normal functions, and elliptic fibrations (2011) (submitted) [arXiv:1108.2223]
C.H. Clemens, Homological equivalence, modulo algebraic equivalence, is not finitely generated. Publ. I.H.E.S. 58, 19–38 (1983)
A. Collino, Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic jacobians. J. Algebr. Geom. 6, 393–415 (1997)
R. de Jeu, J.D. Lewis, Beilinson’s Hodge conjecture for smooth varieties, J. of K-Theor. [arXiv:1104.4364] (2011)
P. Deligne, Théorie de Hodge, II, III. Inst. Hautes Études Sci. Publ. Math. 40, 5–57 (1971); 44, 5–77 (1974)
P. Elbaz-Vincent, S. Müller-Stach, Milnor K-theory of rings, higher Chow groups and applications. Invent. Math. 148, 177–206 (2002)
H. Esnault, K.H. Paranjape, Remarks on absolute de Rham and absolute Hodge cycles. C. R. Acad. Sci. Paris t 319, Serie I, 67–72 (1994)
H. Esnault, E. Viehweg, Deligne-Beilinson cohomology, in Beilinson’s Conjectures on Special Values of L-Functions, ed. by Rapoport, Schappacher, Schneider. Perspectives in Mathematics, vol. 4 (Academic, New York, 1988), pp. 43–91
R. Friedman, R. Laza, Semi-algebraic horizontal subvarieties of Calabi-Yau type, preprint 2011 [arXive:1109.5632v1]
B.B. Gordon, J.D. Lewis, Indecomposable higher Chow cycles, in The Arithmetic and Geometry of Algebraic Cycles, Banff, AB, 1998. Nato Science Series C: Mathematical and Physical Sciences, vol. 548 (Kluwer, Dordrecht, 2000), pp. 193–224
M. Green, Griffiths’ infinitesimal invariant and the Abel-Jacobi map. J. Differ. Geom. 29, 545–555 (1989)
M. Green, P. Griffiths, The regulator map for a general curve, in Symposium in Honor of C.H. Clemens, Salt Lake City, UT, 2000. Contemporary Mathematics, vol. 312 (American Mathematical Society, Providence, 2002), pp. 117–127
M. Green, S. Müller-Stach, Algebraic cycles on a general complete intersection of high multi-degree of a smooth projective variety. Comp. Math. 100(3), 305–309 (1996)
P. Griffiths, J. Harris, Principles of Algebraic Geometry (Wiley, New York, 1978)
P.A. Griffiths, On the periods of certain rational integrals: I and II. Ann. Math. 90, 460–541 (1969)
U. Jannsen, Deligne cohomology, Hodge\(-\mathcal{D}-\)conjecture, and motives, in Beilinson’s Conjectures on Special Values of L-Functions, ed. by Rapoport, Schappacher, Schneider. Perspectives in Mathematics, vol. 4 (Academic, New York, 1988), pp. 305–372
U. Jannsen, in Mixed Motives and Algebraic K-Theory. Lecture Notes in Mathematics, vol. 1000 (Springer, Berlin, 1990)
B. Kahn, Groupe de Brauer et (2, 1)-cycles indecomposables. Preprint (2011)
K. Kato, Milnor K-theory and the Chow group of zero cycles. Contemp. Math. Part I 55, 241–253 (1986)
M. Kerr, J.D. Lewis, The Abel-Jacobi map for higher Chow groups, II. Invent. Math. 170(2), 355–420 (2007)
M. Kerr, J.D. Lewis, S. Müller-Stach, The Abel-Jacobi map for higher Chow groups. Compos. Math. 142(2), 374–396 (2006)
M. Kerz, The Gersten conjecture for Milnor K-theory. Invent. Math. 175(1), 1–33 (2009)
J. King, Log complexes of currents and functorial properties of the Abel-Jacobi map. Duke Math. J. 50(1), 1–53 (1983)
M. Levine, Localization on singular varieties. Invent. Math. 31, 423–464 (1988)
J.D. Lewis, in A Survey of the Hodge Conjecture, 2nd edn. Appendix B by B. Brent Gordon. CRM Monograph Series, vol. 10 (American Mathematical Society, Providence, 1999), pp. xvi+368
J.D. Lewis, Lectures on algebraic cycles. Bol. Soc. Mat. Mexicana (3) 7(2), 137–192 (2001)
J.D. Lewis, Real regulators on Milnor complexes. K-Theory 25(3), 277–298 (2002)
J.D. Lewis, Regulators of Chow cycles on Calabi-Yau varieties, in Calabi-Yau Varieties and Mirror Symmetry, Toronto, ON, 2001. Fields Institute Communications, vol. 38 (American Mathematical Society, Providence, 2003), pp. 87–117
S. Müller-Stach, Constructing indecomposable motivic cohomology classes on algebraic surfaces. J. Algebr. Geom. 6, 513–543 (1997)
S. Müller-Stach, Algebraic cycle complexes, in Proceedings of the NATO Advanced Study Institute on the Arithmetic and Geometry of Algebraic Cycles, vol. 548, ed. by J.D. Lewis, N. Yui, B. Gordon, S. Müller-Stach, S. Saito (Kluwer, Dordrecht, 2000), pp. 285–305
D. Ramakrishnan, Regulators, algebraic cycles, and values of L-functions, in Contemporary Mathematics, vol. 83 (American Mathematical Society, Providence, 1989), pp. 183–310
M. Raynaud, Courbes sur une variété abélienne et points de torsion. Invent. Math. 71, 207–233 (1983)
J.H.M. Steenbrink, in A Summary of Mixed Hodge Theory, Motives, Seattle, WA, 1991. Proceedings of Symposia in Pure Mathematics, vol. 55, Part 1 (American Mathematical Society, Providence, 1994), pp. 31–41
C. Voisin, The Griffiths group of a general Calabi-Yau threefold is not finitely generated. Duke Math. J. 102(1), 151–186 (2000)
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Partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
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Lewis, J.D. (2013). Transcendental Methods in the Study of Algebraic Cycles with a Special Emphasis on Calabi–Yau Varieties. In: Laza, R., Schütt, M., Yui, N. (eds) Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds. Fields Institute Communications, vol 67. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6403-7_2
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