scholarly journals Answer to a question by Fujita on Variation of Hodge Structures

2018 ◽  
Author(s):  
Fabrizio Catanese ◽  
Michael Dettweiler
Keyword(s):  
Hodge Structures ◽  
Variation Of Hodge Structures

scholarly journals Symmetry and Variation of Hodge Structures

2004 ◽  
Vol 8 (2) ◽  
pp. 363-390 ◽  
Author(s):  
I. C. Bauer ◽  
F. Catanese
Keyword(s):  
Hodge Structures ◽  
Variation Of Hodge Structures

scholarly journals Non-Kähler Mirror Symmetry of the Iwasawa Manifold

2018 ◽  
Vol 2020 (23) ◽  
pp. 9471-9538
Author(s):  
Dan Popovici
Keyword(s):  
Mirror Symmetry ◽  
Compact Complex Manifold ◽  
Joint Work ◽  
Hodge Structures ◽  
Four Dimensions ◽  
Yukawa Couplings ◽  
Universal Family ◽  
Variation Of Hodge Structures ◽  
Frölicher Spectral Sequence ◽  
Mirror Map

Abstract We propose a new approach to the mirror symmetry conjecture in a form suitable to possibly non-Kähler compact complex manifolds whose canonical bundle is trivial. We apply our methods by proving that the Iwasawa manifold $X$, a well-known non-Kähler compact complex manifold of dimension $3$, is its own mirror dual to the extent that its Gauduchon cone, replacing the classical Kähler cone that is empty in this case, corresponds to what we call the local universal family of essential deformations of $X$. These are obtained by removing from the Kuranishi family the two “superfluous” dimensions of complex parallelisable deformations that have a similar geometry to that of the Iwasawa manifold. The remaining four dimensions are shown to have a clear geometric meaning including in terms of the degeneration at $E_2$ of the Frölicher spectral sequence. On the local moduli space of “essential” complex structures, we obtain a canonical Hodge decomposition of weight $3$ and a variation of Hodge structures, construct coordinates and Yukawa couplings while implicitly proving a local Torelli theorem. On the metric side of the mirror, we construct a variation of Hodge structures parametrised by a subset of the complexified Gauduchon cone of the Iwasawa manifold using the sGG property (which means that all the Gauduchon metrics are strongly Gauduchon) of all the small deformations of this manifold proved in earlier joint work of the author with L. Ugarte. Finally, we define a mirror map linking the two variations of Hodge structures and we highlight its properties.

scholarly journals Formulae in noncommutative Hodge theory

2019 ◽  
Vol 15 (1) ◽  
pp. 249-299 ◽  
Author(s):  
Nick Sheridan
Keyword(s):  
Mirror Symmetry ◽  
Hodge Theory ◽  
Cyclic Homology ◽  
Homological Mirror Symmetry ◽  
Hodge Structures ◽  
Explicit Formulae ◽  
Variation Of Hodge Structures

AbstractWe prove that the cyclic homology of a saturated $$A_\infty $$A∞ category admits the structure of a ‘polarized variation of Hodge structures’, building heavily on the work of many authors: the main point of the paper is to present complete proofs, and also explicit formulae for all of the relevant structures. This forms part of a project of Ganatra, Perutz and the author, to prove that homological mirror symmetry implies enumerative mirror symmetry.

Symmetric differentials and variations of Hodge structures

2018 ◽  
Vol 2018 (743) ◽  
pp. 133-161 ◽  
Author(s):  
Yohan Brunebarbe
Keyword(s):  
Cotangent Bundle ◽  
Symmetric Power ◽  
Hodge Structures ◽  
Normal Crossing ◽  
Complex Projective Variety ◽  
Symmetric Differentials ◽  
Variation Of Hodge Structures ◽  
Complex Variation ◽  
Complex Projective ◽  
Simple Normal Crossing

Abstract Let D be a simple normal crossing divisor in a smooth complex projective variety X. We show that the existence on X-D of a non-trivial polarized complex variation of Hodge structures with integral monodromy implies that the pair (X,D) has a non-zero logarithmic symmetric differential (a section of a symmetric power of the logarithmic cotangent bundle). When the corresponding period map is generically immersive, we show more precisely that the logarithmic cotangent bundle is big.

scholarly journals Vector bundles on curves coming from variation of Hodge structures

2016 ◽  
Vol 27 (07) ◽  
pp. 1640001 ◽  
Author(s):  
Fabrizio Catanese ◽  
Michael Dettweiler
Keyword(s):  
Vector Bundles ◽  
Negative Answer ◽  
Shimura Curves ◽  
Surfaces Of General Type ◽  
Simple Argument ◽  
Hodge Structures ◽  
Genus 2 ◽  
Canonical Surface ◽  
Variation Of Hodge Structures ◽  
Del Pezzo

Fujita’s second theorem for Kähler fibre spaces over a curve asserts, that the direct image [Formula: see text] of the relative dualizing sheaf splits as the direct sum [Formula: see text], where [Formula: see text] is ample and [Formula: see text] is unitary flat. We focus on our negative answer [F. Catanese and M. Dettweiler, Answer to a question by Fujita on variation of Hodge structures, to appear in Adv. Stud. Pure Math.] to a question by Fujita: is [Formula: see text] semiample? We give here an infinite series of counterexamples using hypergeometric integrals and we give a simple argument to show that the monodromy representation is infinite. Our counterexamples are surfaces of general type with positive index, explicitly given as abelian coverings with group [Formula: see text] of a Del Pezzo surface [Formula: see text] of degree 5 (branched on the union of the lines of [Formula: see text], which form a bianticanonical divisor), and endowed with a semistable fibration with only three singular fibres. The simplest such surfaces are the three ball quotients considered in [I. C. Bauer and F. Catanese, A volume maximizing canonical surface in 3-space, Comment. Math. Helv. 83(1) (2008) 387–406.], fibred over a curve of genus 2, and with fibres of genus 4. These examples are a larger class than the ones corresponding to Shimura curves in the moduli space of Abelian varieties.

Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)

2017 ◽  
Author(s):  
Claire Voisin
Keyword(s):  
Recent Work ◽  
Chow Groups ◽  
Ring Structure ◽  
Complete Intersections ◽  
Algebraic Varieties ◽  
Chow Ring ◽  
Hodge Structures ◽  
Chow Rings ◽  
Geometric Methods ◽  
The Right

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

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