Hodge: source and evidence

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1 {\small
2 \begin{thebibliography}{99}
3 \setlength{\itemsep}{0.18em}
4 \bibitem{ClayHodge}
5 Clay Mathematics Institute.
6 \newblock The Hodge Conjecture, Millennium Prize Problems.
7 \newblock \url{https://www.claymath.org/millennium/hodge-conjecture/}.
8 
9 \bibitem{VoisinHodge}
10 C. Voisin.
11 \newblock \emph{Hodge Theory and Complex Algebraic Geometry I and II}.
12 \newblock Cambridge University Press, 2002--2003.
13 
14 \bibitem{CDK}
15 E. Cattani, P. Deligne, and A. Kaplan.
16 \newblock On the locus of Hodge classes.
17 \newblock \emph{Journal of the American Mathematical Society}, 8:483--506, 1995.
18 
19 \bibitem{AtiyahHirzebruch}
20 M. Atiyah and F. Hirzebruch.
21 \newblock Analytic cycles on complex manifolds.
22 \newblock \emph{Topology}, 1:25--45, 1962.
23 
24 \bibitem{CharbonneauStern}
25 B. Charbonneau and M. Stern.
26 \newblock Asymptotic Hodge theory of vector bundles.
27 \newblock arXiv:1111.0591.
28 
29 \bibitem{BeauvilleDonagi}
30 A. Beauville and R. Donagi.
31 \newblock La vari\'et\'e des droites d'une hypersurface cubique de dimension 4.
32 \newblock \emph{C. R. Acad. Sci. Paris S\'er. I Math.}, 301:703--706, 1985.
33 
34 \bibitem{DebarreKuznetsov}
35 O. Debarre and A. Kuznetsov.
36 \newblock Gushel-Mukai varieties: linear spaces and periods.
37 \newblock \emph{Kyoto Journal of Mathematics}, 59:897--953, 2019.
38 
39 \bibitem{SeedI}
40 Komorebi P.
41 \newblock \emph{Projector-Resolved Seed Calculus I: Typed Seeds, Operator Grammars, and Exact Reconstruction}.
42 \newblock Internal research manuscript, 2026.
43 
44 \bibitem{SeedII}
45 Komorebi P.
46 \newblock \emph{Projector-Resolved Seed Calculus II: Polyphase Refinement, Weighted Transfer, and Dyadic-Odd Resolution}.
47 \newblock Internal research manuscript, 2026.
48 
49 \bibitem{SeedIII}
50 Komorebi P.
51 \newblock \emph{Projector-Resolved Seed Calculus III: Nonlinear Operator Trees, Obstruction Packets, and Holonomy}.
52 \newblock Internal research manuscript, 2026.
53 
54 \bibitem{KangFermat}
55 S.-J. Kang.
56 \newblock Refined motivic dimension of some Fermat varieties.
57 \newblock \emph{Bulletin of the Australian Mathematical Society}, 93(2):223--230, 2016.
58 \newblock doi:10.1017/S0004972715001379.
59 
60 \bibitem{ShiodaKatsura}
61 T. Shioda and T. Katsura.
62 \newblock On Fermat varieties.
63 \newblock \emph{Tohoku Mathematical Journal}, 31:97--115, 1979.
64 
65 \bibitem{VanGeemenHalfTwist}
66 B. van Geemen.
67 \newblock Half twists of Hodge structures of CM-type.
68 \newblock \emph{Journal of the Mathematical Society of Japan}, 53:813--833, 2001.
69 
70 \bibitem{VanGeemenIzadi}
71 B. van Geemen and E. Izadi.
72 \newblock Half twists and the cohomology of hypersurfaces.
73 \newblock arXiv:math/0008170, 2000.
74 
75 
76 \bibitem{DaSilvaFermat}
77 G. da Silva Jr.
78 \newblock Notes on the Hodge conjecture for Fermat varieties.
79 \newblock \emph{Experimental Results}, 2:e22, 2021.
80 \newblock doi:10.1017/exp.2021.14.
81 
82 \bibitem{OtsuboYamazaki}
83 N. Otsubo and T. Yamazaki.
84 \newblock Motivic Gauss and Jacobi sums.
85 \newblock arXiv:2402.06072, 2024, published in \emph{Annales de la Facult\'e des Sciences de Toulouse}, 35(1):95--135, 2026.
86 
87 \bibitem{WashingtonCyclotomic}
88 L. C. Washington.
89 \newblock \emph{Introduction to Cyclotomic Fields}, second edition.
90 \newblock Graduate Texts in Mathematics 83, Springer, 1997.
91 
92 \bibitem{FermatNotes}
93 Komorebi P.
94 \newblock Fermat fourfold packet notes: conductor polyphase, Klein--Delsarte closure, Kummer phases, and CM half-twists.
95 \newblock Internal research package, 2026.
96 
97 \bibitem{DeligneHodgeStatement}
98 P. Deligne.
99 \newblock The Hodge Conjecture.
100 \newblock Clay Mathematics Institute, official problem description, \S1--2.
101 \newblock \url{https://www.claymath.org/wp-content/uploads/2022/06/hodge.pdf}.
102 
103 
104 \bibitem{DeligneHodgeII}
105 P. Deligne.
106 \newblock Th\'eorie de Hodge II.
107 \newblock \emph{Publications Math\'ematiques de l'IH\'ES}, 40:5--57, 1971.
108 \newblock doi:10.1007/BF02684692.
109 \newblock Corollary 3.2.17, p.~39, Lemma 4.2.3, p.~44.
110 
111 
112 \bibitem{ArapuraUniruled}
113 D. Arapura.
114 \newblock Varieties with very little transcendental cohomology.
115 \newblock arXiv:0706.2506v1, 17 June 2007, Corollary 4.2.
116 
117 
118 \bibitem{ConteMurre}
119 A. Conte and J. P. Murre.
120 \newblock The Hodge conjecture for fourfolds admitting a covering by rational curves.
121 \newblock \emph{Mathematische Annalen}, 238:79--88, 1978.
122 % Original not freshly extracted, the checked primary reproof is ArapuraUniruled.
123 
124 
125 \bibitem{BDPP}
126 S. Boucksom, J.-P. Demailly, M. P\u{a}un, and T. Peternell.
127 \newblock The pseudo-effective cone of a compact K\"ahler manifold and varieties of negative Kodaira dimension.
128 \newblock arXiv:math/0405285v1, 14 May 2004, Theorem 2.6.
129 
130 
131 \bibitem{Aoki}
132 N. Aoki.
133 \newblock Some new algebraic cycles on Fermat varieties.
134 \newblock \emph{Journal of the Mathematical Society of Japan}, 39(3):385--396, 1987.
135 \newblock doi:10.2969/jmsj/03930385.
136 \newblock Theorems 1-4 and 2-1, p.~388.
137 
138 
139 \bibitem{ShiodaRan}
140 T. Shioda, The Hodge conjecture for Fermat varieties,
141 \emph{Mathematische Annalen}, 245:175--184, 1979,
142 Z. Ran, Cycles on Fermat hypersurfaces,
143 \emph{Compositio Mathematica}, 42:121--142, 1981.
144 \newblock Concatenation and complementary-pair removal are stated in
145 Aoki, Theorem 1-4, p.~388.
146 % These original papers were not independently proof-audited here.
147 
148 
149 \bibitem{ShiodaFermat}
150 T. Shioda.
151 \newblock The Hodge conjecture for Fermat varieties.
152 \newblock \emph{Mathematische Annalen}, 245(2):175--184, 1979.
153 \newblock doi:10.1007/BF01428804.
154 
155 
156 \bibitem{VillaflorPeriods}
157 R. Villaflor Loyola.
158 \newblock Periods of complete intersection algebraic cycles.
159 \newblock arXiv:1812.03964v5, 30 March 2021, Theorems 1.1--1.2.
160 
161 
162 \bibitem{FermatLengths2026}
163 M. Miranda, H. Movasati, L. Rufino, and R. Villaflor.
164 \newblock Lengths of Hodge characters in Fermat varieties.
165 \newblock arXiv:2609.27301v2, 26 September 2026.
166 % Current preprint context, its integer programs were not replayed here.
167 
168 
169 \bibitem{MarkmanWeil}
170 E. Markman.
171 \newblock Cycles on abelian $2n$-folds of Weil type from secant sheaves on abelian $n$-folds.
172 \newblock arXiv:2502.03415v2, 8 June 2025.
173 \newblock Theorem 1.5.1 and Corollary 1.6.1.
174 
175 
176 \bibitem{MarkmanICM2026}
177 E. Markman.
178 \newblock Secant Sheaves and Weil Classes on Abelian Varieties.
179 \newblock \emph{Proceedings of the International Congress of Mathematicians 2026},
180 vol.~3, pp.~586--602.
181 \newblock doi:10.1137/25M1803796.
182 \newblock Theorem 1.2 and Corollary 1.3.
183 
184 
185 \bibitem{BrosnanDegeneration}
186 P. Brosnan.
187 \newblock Discriminants of Hermitian forms, maximal degenerations and Kontsevich's tropical approach to the Hodge conjecture.
188 \newblock arXiv:2609.14169v1, 12 September 2026.
189 % Exact Proposition4 unavailable in this audit, retain conditional scope.
190 
191 
192 \bibitem{SecantPreprintAudit}
193 D. Bhattacharjee and U. Bhattacharya.
194 \newblock Relative Secant Cycles and Hodge Classes.
195 \newblock \emph{Preprints.org}, 202602.0462, version~5, 5 August 2026.
196 \newblock \url{https://www.preprints.org/manuscript/202602.0462/v5}.
197 \newblock \S4.3, Proposition 2 (local eigencharacter identification audited).
198 
199 
200 \bibitem{SharifiDirichlet}
201 R. Sharifi.
202 \newblock Iwasawa Theory, Chapter 4: Cyclotomic fields.
203 \newblock Author's online notes, accessed 9 October 2026.
204 \newblock Lemma 4.1.18, Corollary 4.1.22, Proposition 4.2.13, Theorem 4.2.14.
205 \newblock \url{https://www.math.ucla.edu/~sharifi/notes/iwasawa-ch04.html}.
206 
207 
208 \bibitem{DLMFDirichlet}
209 NIST Digital Library of Mathematical Functions.
210 \newblock \S25.15, Dirichlet $L$-functions, equations 25.15.5 and 25.15.9.
211 \newblock \url{https://dlmf.nist.gov/25.15}, accessed 9 October 2026.
212 
213 
214 \bibitem{Jumagulov2026}
215 R. Jumagulov.
216 \newblock The Hodge conjecture for Fermat fourfolds of odd degree at most 199.
217 \newblock arXiv:2608.18134v1, Appendix A and the standard-family discussion.
218 \newblock Used for context and target selection. No global census from this source is certified by this edition.
219 \end{thebibliography}
220 }

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