Qualitative aspects of counting real rational curves on real K3 surfaces

Kharlamov, Viatcheslav¹ ; Răsdeaconu, Rareş²

¹ IRMA UMR 7501, Université de Strasbourg, 7 Rue René-Descartes, 67084 Strasbourg Cedex, France
² Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, United States

Geometry & topology, Tome 21 (2017) no. 1, pp. 585-601.

Résumé

We study qualitative aspects of the Welschinger-like $\mathbb{Z}$–valued count of real rational curves on primitively polarized real K3 surfaces. In particular, we prove that, with respect to the degree of the polarization, at logarithmic scale, the rate of growth of the number of such real rational curves is, up to a constant factor, the rate of growth of the number of complex rational curves. We indicate a few instances when the lower bound for the number of real rational curves provided by our count is sharp. In addition, we exhibit various congruences between real and complex counts.

Détail

DOI : 10.2140/gt.2017.21.585

Classification : 14N99, 14P99, 14J28
Keywords: $K3$ surfaces, real rational curves, Yau–Zaslow formula, Welschinger invariants

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