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The Lefschetz defect of Fano varieties

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Abstract

Smooth, complex Fano varieties are a very natural class of projective varieties, and have been intensively studied both classically and in recent times, due to their significance in the framework of the Minimal Model Program. The Lefschetz defect is an invariant of smooth Fano varieties that has been recently introduced in [7]; it is related to the Picard number \(\rho _X\) of X, and to the Picard number of prime divisors in X. This paper is a survey on this new invariant and its properties: we explain the definition of the Lefschetz defect \(\delta _X\) and its origin, the known results, and several examples, especially in dimensions 3 and 4; we do not give new theoretical results. In particular, we explain how the results on the Lefschetz defect allow to recover the classification of Fano 3-folds with \(\rho _X\ge 5\), due to Mori and Mukai. We also review the known examples of Fano 4-folds with \(\rho _X\ge 6\), that are remarkably few: apart products and toric examples, there are only 6 known families, with \(\rho _X\in \{6,7,8,9\}\).

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Notes

  1. A smooth projective variety X is of Fano type if there exists an effective \(\mathbb {Q}\)-divisor \(\Delta \) such that the pair \((X,\Delta )\) is log Fano, namely \((X,\Delta )\) has klt singularities and \(-(K_X+\Delta )\) is ample, see [27].

  2. For instance, if \(L_1\) is globally generated, then the vector bundle \(L_1\oplus \mathcal {O}_T\oplus \mathcal {O}_T\) is globally generated, thus H is too, and by Bertini \(S_1\) is smooth.

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  1. Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123, Torino, Italy

    C. Casagrande

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Casagrande, C. The Lefschetz defect of Fano varieties. Rend. Circ. Mat. Palermo, II. Ser 72, 3061–3075 (2023). https://doi.org/10.1007/s12215-022-00846-4

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