The Spread Philosophy in the Study of Algebraic Cycles

2017 ◽  
Author(s):  
Mark L. Green
Keyword(s):  
Hodge Structure ◽  
Hodge Theory ◽  
Algebraic Cycles ◽  
Algebraic Cycle ◽  
Integral Lattice ◽  
Group Of Automorphisms ◽  
Intersection Pairing ◽  
Variation Of Hodge Structure

This chapter discusses the spread philosophy in the study of algebraic cycles, in order to make use of a geometry by considering a variation of Hodge structure where D is the Hodge domain (or the appropriate Mumford–Tate domain) and Γ‎ is the group of automorphisms of the integral lattice preserving the intersection pairing. If we have an algebraic cycle Z on X, taking spreads yields a cycle Ƶ on X. Applying Hodge theory to Ƶ on X gives invariants of the cycle. Another related situation is algebraic K-theory. For example, to study Kₚsuperscript Milnor(k), the geometry of S can be used to construct invariants.

Hodge Theory (MN-49)

2017 ◽  
Author(s):  
Eduardo Cattani ◽  
Fouad El Zein ◽  
Phillip A. Griffiths ◽  
Lê Dung Tráng
Keyword(s):  
Summer School ◽  
Hodge Structure ◽  
Hodge Theory ◽  
Algebraic Cycles ◽  
Mixed Hodge Structure ◽  
Hodge Structures ◽  
De Rham ◽  
New Treatment ◽  
Variations Of Hodge Structure ◽  
Mixed Hodge Structures

This book provides a comprehensive and up-to-date introduction to Hodge theory—one of the central and most vibrant areas of contemporary mathematics—from leading specialists on the subject. The topics range from the basic topology of algebraic varieties to the study of variations of mixed Hodge structure and the Hodge theory of maps. Of particular interest is the study of algebraic cycles, including the Hodge and Bloch–Beilinson Conjectures. Based on lectures delivered at the 2010 Summer School on Hodge Theory at the ICTP in Trieste, Italy, the book is intended for a broad group of students and researchers. The exposition is as accessible as possible and does not require a deep background. At the same time, the book presents some topics at the forefront of current research. The book is divided between introductory and advanced lectures. The introductory lectures address Kähler manifolds, variations of Hodge structure, mixed Hodge structures, the Hodge theory of maps, period domains and period mappings, algebraic cycles (up to and including the Bloch–Beilinson conjecture) and Chow groups, sheaf cohomology, and a new treatment of Grothendieck's algebraic de Rham theorem. The advanced lectures address a Hodge-theoretic perspective on Shimura varieties, the spread philosophy in the study of algebraic cycles, absolute Hodge classes (including a new, self-contained proof of Deligne's theorem on absolute Hodge cycles), and variation of mixed Hodge structures.

Algebraic Cycles and Hodge Theory

1994 ◽  
Author(s):  
Mark L. Green ◽  
Jacob P. Murre ◽  
Claire Voisin
Keyword(s):  
Hodge Theory ◽  
Algebraic Cycles

scholarly journals Kodaira–Saito vanishing via Higgs bundles in positive characteristic

2019 ◽  
Vol 2019 (755) ◽  
pp. 293-312
Author(s):  
Donu Arapura
Keyword(s):  
Positive Characteristic ◽  
Hodge Structure ◽  
Higgs Bundles ◽  
Vanishing Theorem ◽  
Variation Of Hodge Structure ◽  
Mixed Hodge Modules ◽  
Special Case

AbstractThe goal of this paper is to give a new proof of a special case of the Kodaira–Saito vanishing theorem for a variation of Hodge structure on the complement of a divisor with normal crossings. The proof does not use the theory of mixed Hodge modules, but instead reduces it to a more general vanishing theorem for semistable nilpotent Higgs bundles, which is then proved by using some facts about Higgs bundles in positive characteristic.

scholarly journals Lowest weights in cohomology of variations of Hodge structure

2012 ◽  
Vol 206 ◽  
pp. 1-24
Author(s):  
Chris Peters ◽  
Morihiko Saito
Keyword(s):  
Open Subset ◽  
Hodge Structure ◽  
Analytic Space ◽  
Mixed Hodge Structure ◽  
Zariski Open Subset ◽  
Weight Part ◽  
Variation Of Hodge Structure ◽  
Local Monodromy ◽  
Variations Of Hodge Structure ◽  
Complex Analytic Space

AbstractLetXbe an irreducible complex analytic space withj:U ↪ Xan immersion of a smooth Zariski-open subset, and let 𝕍 be a variation of Hodge structure of weightnoverU. Assume thatXis compact Kähler. Then, provided that the local monodromy operators at infinity are quasi-unipotent,IHk(X, 𝕍) is known to carry a pure Hodge structure of weightk+n, whileHk(U, 𝕍) carries a mixed Hodge structure of weight at leastk+n. In this note it is shown that the image of the natural mapIHk(X, 𝕍) →Hk(U, 𝕍) is the lowest-weight part of this mixed Hodge structure. In the algebraic case this easily follows from the formalism of mixed sheaves, but the analytic case is rather complicated, in particular when the complementX — Uis not a hypersurface.

scholarly journals Functoriality of motivic lifts of the canonical construction

2019 ◽  
Vol 163 (1-2) ◽  
pp. 27-56 ◽  
Author(s):  
Alex Torzewski
Keyword(s):  
Full Subcategory ◽  
Hodge Structure ◽  
Compact Subgroup ◽  
Base Change ◽  
Shimura Varieties ◽  
Open Compact Subgroup ◽  
Chow Motives ◽  
Variation Of Hodge Structure

Abstract Let $$(G,{\mathfrak {X}})$$ ( G , X ) be a Shimura datum and K a neat open compact subgroup of $$G(\mathbb {A}_f)$$ G ( A f ) . Under mild hypothesis on $$(G,{\mathfrak {X}})$$ ( G , X ) , the canonical construction associates a variation of Hodge structure on $$\text {Sh}_K(G,{\mathfrak {X}})(\mathbb {C})$$ Sh K ( G , X ) ( C ) to a representation of G. It is conjectured that this should be of motivic origin. Specifically, there should be a lift of the canonical construction which takes values in relative Chow motives over $$\text {Sh}_K(G,{\mathfrak {X}})$$ Sh K ( G , X ) and is functorial in $$(G,{\mathfrak {X}})$$ ( G , X ) . Using the formalism of mixed Shimura varieties, we show that such a motivic lift exists on the full subcategory of representations of Hodge type $$\{(-1,0),(0,-1)\}$$ { ( - 1 , 0 ) , ( 0 , - 1 ) } . If $$(G,{\mathfrak {X}})$$ ( G , X ) is equipped with a choice of PEL-datum, Ancona has defined a motivic lift for all representations of G. We show that this is independent of the choice of PEL-datum and give criteria for it to be compatible with base change. Additionally, we provide a classification of Shimura data of PEL-type and demonstrate that the canonical construction is applicable in this context.

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