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Variations of mixed Hodge structure attached to the deformation theory of a complex variation of Hodge structures. (English) Zbl 1246.14018

Let \(X\) be a compact Kähler manifold, \(x \in X\) a base point and \(\rho : \pi_1(X,x) \to GL_N(\mathbb{C})\) the monodromy representation of a \(\mathbb{C}\)-variations of Hodge structures. Let \(T\) be the formal universal deformation space of \(\rho\), with structure sheaf \(\hat{\mathcal{O}}_T\) and maximal ideal \(m\). Denote by \(\rho_{T,n} : \pi_1(X,x) \to GL_N(\hat{\mathcal{O}}_T/m^n)\) the restriction of the universal deformation of \(\rho\). The main result of the article is the construction of a \(\mathbb{C}\)-variations of Hodge structures on \(X\), the monodromy representation of which is exactly \(\rho_{T,n}\) and the weight filtration of which is given by the powers of \(m\). The main motivation for this result is that it is used in the proof of linear Shafarevich conjecture [P. Eyssidieux, L. Katzarkov, T. Pantev and M. Ramachandran, “Linear Shafarevich conjecture”, arXiv:0904.0693]. Also, the result is a consequence of the construction of a mixed Hodge structure on the formal local ring of \(\rho\) in the variety of representations \(R(\pi(X,x), GL_N)\).

MSC:

14D07 Variation of Hodge structures (algebro-geometric aspects)
14D15 Formal methods and deformations in algebraic geometry
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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References:

[1] Deligne, P.: Théorie de Hodge, II. Publ. Math. IHES 40, 5-58 (1971) · Zbl 0219.14007 · doi:10.1007/BF02684692
[2] Deligne, P.: Théorie de Hodge, III. Publ. Math. IHES 44, 6-77 (1975) · Zbl 0237.14003 · doi:10.1007/BF02685881
[3] Deligne, P., Griffiths, P., Morgan, J., Sullivan, D.: The real homotopy theory of Kähler manifolds. Invent. Math. 29, 245-274 (1975) · Zbl 0312.55011 · doi:10.1007/BF01389853
[4] Eyssidieux, P., Katzarkov, L., Pantev, T., Ramachandran, M.: Linear Shafarevich con- jecture. arxiv/math:0904.0693 (2009) · Zbl 1273.32015
[5] Fujiki, A.: Hyperkähler structure on the moduli space of flat bundles. In: Prospects in Complex Geometry (Katata, 1989), Lecture Notes in Math. 1468, Springer, 1-83 (1991) · Zbl 0749.32011
[6] Ginzburg, V., Kaledin, D.: Poisson deformation of symplectic quotient singularities. Adv. Math. 186, 1-57 (2004) · Zbl 1062.53074 · doi:10.1016/j.aim.2003.07.006
[7] Goldman, W., Millson, J.: The deformation theory of representations of funda- mental groups of compact Kähler manifolds. Publ. Math. IHES 67, 43-96 (1988) · Zbl 0678.53059 · doi:10.1007/BF02699127
[8] Goldman, W., Millson, J.: The homotopy invariance of the Kuranishi space. Illinois J. Math. 34, 337-367 (1990) · Zbl 0707.32004
[9] Hain, R.: The de Rham homotopy theory of complex algebraic varieties, I. K-theory 1, 271-324 (1987) · Zbl 0637.55006 · doi:10.1007/BF00533825
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