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Fano manifolds of Calabi-Yau Hodge type. (English) Zbl 1314.14085
Given a smooth projective variety $$X$$, the authors define it to be Calabi-Yau Hodge types if it satisfies the following three conditions.
1.
The middle dimensional Hodge structure is numerically similar to that of a Calabi-Yau threefold, that is $$h^{n+2,n-1}(X)=1$$, and $$h^{n+p+1,n-p}(X) = 0$$ for $$p \geq 2$$.
2.
For any generator $$\omega \in H^{n+2,n-1}(X)$$, the contraction map $$H^1(X, TX) \rightarrow H^{n-1}(X, \Omega^{n+1}_X)$$ is an isomorphism.
3.
The Hodge numbers $$h^{k, 0}(X) = 0$$ for $$1 \leq k \leq 2n$$.
The author study some basic properties of varieties of this type, give examples among complete intersections and hypersurfaces in homogeneous spaces, and study the derived categories of some of the examples.
Reviewer: Zhiyu Tian (Bonn)

##### MSC:
 14J45 Fano varieties 14J40 $$n$$-folds ($$n>4$$) 14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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##### References:
 [1] Beauville, A., Determinantal hypersurfaces, Mich. Math. J., 48, 39-64, (2000) · Zbl 1076.14534 [2] Bialynicki-Birula, A., Some theorems on actions of algebraic groups, Ann. Math., 98, 480-497, (1973) · Zbl 0275.14007 [3] Borcea, C., Smooth global complete intersections in certain compact homogeneous complex manifolds, J. Reine Angew. Math., 344, 65-70, (1983) · Zbl 0511.14027 [4] Bertin, J.; Peters, C., Variations of Hodge structures, Calabi-Yau manifolds and mirror symmetry, (Bertin, J.; Demailly, J.-P.; Illusie, J.-L.; Peters, C., Introduction to Hodge Theory, SMF/AMS Texts and Monographs, vol. 8, (2002)), 151-228 [5] Candelas, P.; Derrick, E.; Parkes, L., Generalized Calabi-Yau manifolds and the mirror of a rigid manifold, Nucl. Phys. B, 407, 115-154, (1993) · Zbl 0899.32011 [6] Cynk, S., Cohomologies of a double covering of a non-singular algebraic 3-fold, Math. Z., 240, 731-743, (2002) · Zbl 1004.14007 [7] Donagi, R.; Markman, E., Cubics, integrable systems, and Calabi-Yau threefolds, (Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry, Isr. Math. Conf. Proc., vol. 9, (1996)), 199-221 · Zbl 0878.14031 [8] Favero, D.; Iliev, A.; Katzarkov, L., On the Griffiths groups of Fano manifolds of Calabi-Yau Hodge type, Pure Appl. Math. Q., 9, 4, (2013), (Special Issue: In memory of Andrey Todorov) [9] Freed, D., Special Kähler manifolds, Commun. Math. Phys., 203, 31-52, (1999) · Zbl 0940.53040 [10] Gelfand, I. M.; Kapranov, M. M.; Zelevinsky, A. V., Discriminants, resultants, and multidimensional determinants, (1994), Birkhäuser · Zbl 0827.14036 [11] Griffiths, P. A., On the periods of certain rational integrals I, II, Ann. Math., 90, 460-495, (1969), 496-541 · Zbl 0215.08103 [12] Igusa, J., A classification of spinors up to dimension twelve, Am. J. Math., 92, 997-1028, (1970) · Zbl 0217.36203 [13] Iliev, A.; Manivel, L., The Chow ring of the Cayley plane, Compos. Math., 141, 146-160, (2005) · Zbl 1071.14056 [14] Iliev, A.; Manivel, L., On cubic hypersurfaces of dimension seven and eight, Proc. Lond. Math. Soc. (3), 108, 2, 517-540, (2014) · Zbl 1304.14053 [15] Kostant, B., Lie algebra cohomology and generalized Schubert cells, Ann. Math., 77, 72-144, (1963) · Zbl 0134.03503 [16] Kuznetsov, A., Derived categories of cubic and $$V_{14}$$ threefolds, Proc. Steklov Inst. Math., 246, 171-194, (2004) · Zbl 1107.14028 [17] Kuznetsov, A., Homological projective duality, Publ. Math. IHÉS, 105, 157-220, (2007) · Zbl 1131.14017 [18] Nagel, J., The Abel-Jacobi map for complete intersections, Indag. Math. (N.S.), 8, 1, 95-113, (1997) · Zbl 0888.14016 [19] Popov, V. L., Classification of the spinors of dimension fourteen, Usp. Mat. Nauk, 32, 199-200, (1977) · Zbl 0327.15028 [20] Schimmrigk, R., Mirror symmetry and string vacua from a special class of Fano varieties, Int. J. Mod. Phys. A, 11, 3049-3096, (1996) · Zbl 1044.32504 [21] Snow, D., Cohomology of twisted holomorphic forms on Grassmann manifolds and quadric hypersurfaces, Math. Ann., 276, 159-176, (1986) · Zbl 0596.32016 [22] Steenbrink, J., Intersection form for quasi-homogeneous singularities, Compos. Math., 34, 211-223, (1977) · Zbl 0347.14001 [23] Voisin, C., Symétrie miroir, Panoramas et Synthèses, vol. 2, (1996), SMF · Zbl 0849.14001
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