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Zbl 1018.14001
Colombo, Elisabetta; Pirola, Gian Pietro; Tortora, Alfonso
Hodge-Gaussian maps. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 30, No.1, 125-146 (2001). ISSN 0391-173X

This paper is related to the topic of the variations of Hodge structures and the higher differentials of the period map. The authors define a family of maps, that they propose to call Hodge-Gaussian maps, for any line bundle $L$ on any compact Kähler manifold $X$. Namely, let $I_k(L)$ be the kernel of the multiplication map $m_k : \text{Sym}^k H^0(L) \rightarrow H^0(L^k)$. For all $h\leq k$, the authors define a map $$\rho : I_k(L)\rightarrow \text{Hom}\left(H^{p,q}(L^{-h}), H^{p+1,q-1}(L^{k-h})\right).$$ When $L$ is the canonical bundle, the map $\rho$ computes a second fundamental form associated to the deformations of $X$. If $X$ is a curve, then $\rho$ is a lifting of the Wahl map $I_2(L)\rightarrow H^0(L^2\otimes K_C^2)$. The authors also show how to generalize the construction of $\rho$ to the cases of harmonic bundles and of couples of vector bundles.
[Vladimir L.Popov (Moskow)]

MSC 2000:: *14D07 Variation of Hodge structures
32G20 Period matrices
14C30 Transcendental methods
14H15 Families, analytic moduli (curves)

Keywords: variations of Hodge structures; higher differentials of the period map; Hodge-Gaussian maps; vector bundle; Wahl map

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