scholarly journals Variation of Hodge structure and enumerating tilings of surfaces by triangles and squares

2021 ◽  
Vol 8 ◽  
pp. 831-857
Author(s):  
Vincent Koziarz ◽  
Duc-Manh Nguyen
Keyword(s):  
Hodge Structure ◽  
Variation Of Hodge Structure

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.

Chapter I. VARIATION OF HODGE STRUCTURE

1984 ◽  
pp. 1-28 ◽  
Author(s):  
Phillip Griffiths ◽  
Loring Tu
Keyword(s):  
Hodge Structure ◽  
Variation Of Hodge Structure

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.

scholarly journals EXTENDING MIRROR CONJECTURE TO CALABI–YAU WITH BUNDLES

1999 ◽  
Vol 01 (01) ◽  
pp. 65-70 ◽  
Author(s):  
CUMRUN VAFA
Keyword(s):  
Riemann Surfaces ◽  
Hodge Structure ◽  
Stable Bundle ◽  
Type Ii ◽  
Holomorphic Maps ◽  
Variation Of Hodge Structure ◽  
Mirror Side ◽  
Calabi Yau Manifolds

We define the notion of mirror of a Calabi–Yau manifold with a stable bundle in the context of type II strings in terms of supersymmetric cycles on the mirror. This allows us to relate the variation of Hodge structure for cohomologies arising from the bundle to the counting of holomorphic maps of Riemann surfaces with boundary on the mirror side. Moreover it opens up the possibility of studying bundles on Calabi–Yau manifolds in terms of supersymmetric cycles on the mirror.

The Mumford-Tate Group of a Variation of Hodge Structure

2012 ◽  
Author(s):  
Mark Green ◽  
Phillip Griffiths ◽  
Matt Kerr
Keyword(s):  
Complex Manifold ◽  
Moduli Space ◽  
Hodge Structure ◽  
Integral Manifold ◽  
Equivalence Classes ◽  
Hodge Structures ◽  
Tate Group ◽  
Variation Of Hodge Structure ◽  
Definition Of ◽  
Variations Of Hodge Structure

This chapter deals with the Mumford-Tate group of a variation of Hodge structure (VHS). It begins by presenting a definition of VHS, which consists of a connected complex manifold and a locally liftable, holomorphic mapping that is an integral manifold of the canonical differential ideal. The moduli space of Γ‎-equivalence classes of polarized Hodge structures is also considered, along with a generic point for the VHS and the monodromy group of the VHS. Associated to a VHS is its Mumford-Tate group. The chapter proceeds by discussing the structure theorem for VHS, where S is a quasi-projective algebraic variety, referred to as global variations of Hodge structure. It concludes by describing an application of Mumford-Tate groups, along with the Noether-Lefschetz locus.

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