×

Hodge-Gaussian maps. (English) Zbl 1018.14001

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.

MSC:

14D07 Variation of Hodge structures (algebro-geometric aspects)
32G20 Period matrices, variation of Hodge structure; degenerations
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14H15 Families, moduli of curves (analytic)
PDFBibTeX XMLCite
Full Text: arXiv Numdam EuDML

References:

[1] J. Carlson - M. Green - P. Griffiths - J. Harris , Infinitesimal variations of Hodge structure (I) , Compositio Math. 50 ( 1983 ), 109 - 205 . Numdam | MR 720288 | Zbl 0531.14006 · Zbl 0531.14006
[2] P. Deligne - P. Griffiths - J. Morgan - D. Sullivan , Real homotopy theory on Kähler manifolds , Invent. Math. 29 ( 1975 ), 245 - 274 . MR 382702 | Zbl 0312.55011 · Zbl 0312.55011 · doi:10.1007/BF01389853
[3] M. Green , Infinitesimal Methods in Hodge Theory ,, In: ” Algebraic cycles and Hodge theory ”, A. Albano et al. (eds.), Lecture Notes in Math ., vol. 1594 ( 1994 ), pp. 1 - 92 . MR 1335239 | Zbl 0846.14001 · Zbl 0846.14001
[4] P. Griffiths , Infinitesimal variations of Hodge structure (III) , Compositio Math. 50 ( 1983 ), 267 - 324 . Numdam | MR 720290 | Zbl 0576.14009 · Zbl 0576.14009
[5] P. Griffiths - J. Harris , ” Principles of algebraic geometry ”, New York , John Wiley , 1978 . MR 507725 | Zbl 0408.14001 · Zbl 0408.14001
[6] P. Griffiths - J. Harris , Algebraic geometry and local differential geometry , Ann. Sci. Ècole Norm. Sup . ( 4 ) 12 ( 1979 ), 355 - 432 . Numdam | MR 559347 | Zbl 0426.14019 · Zbl 0426.14019
[7] Y. Karpishpan , On higher-order differentials of the period map , Duke Math. J. 72 ( 1993 ), 541 - 571 . Article | MR 1253615 | Zbl 0837.14027 · Zbl 0837.14027 · doi:10.1215/S0012-7094-93-07220-1
[8] J. Landsberg , On second fundamental form of projective varieties , Invent. Math. 117 ( 1994 ), 303 - 315 . MR 1273267 | Zbl 0840.14025 · Zbl 0840.14025 · doi:10.1007/BF01232243
[9] R. Paoletti , Generalized Wahl maps and adjoint line bundles on a general curve , Pacific J. Math. 168 ( 1995 ), 313 - 334 . Article | MR 1339955 | Zbl 0838.14026 · Zbl 0838.14026
[10] G.P. Pirola , The infinitesimal variation of the spin abelian differentials and periodic minimal surfaces , Comm. Anal. Geom. 6 ( 1998 ), 393 - 426 . MR 1638858 | Zbl 0914.58007 · Zbl 0914.58007
[11] E. Sernesi , ” Topics on Families of Projective Schemes ”, Queen’s Papers no. 73 , Kingston Ontario , 1986 . MR 869062
[12] C. Simpson , Higgs bundles and local systems , Inst. Hautes Études Sci. Publ. Math. 75 ( 1992 ), 5 - 95 . Numdam | MR 1179076 | Zbl 0814.32003 · Zbl 0814.32003 · doi:10.1007/BF02699491
[13] J. Wahl , Gaussian maps on algebraic curves , J. Differential Geom. 32 ( 1990 ), 77 - 98 . MR 1064866 | Zbl 0724.14022 · Zbl 0724.14022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.