Qualitative aspects of counting real rational curves on real K3 surfaces

Abstract

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.Comment: corrected typos, accepted for publicatio

On the Smith–Thom deficiency of Hilbert squares

“…We conclude this section with the following results generalizing Proposition 3.2 in [6]. The proof presented in [6, Proposition 3.2] extends literally to higher dimensions and will be omitted.…”

“…We conclude this section with the following results generalizing Proposition 3.2 in [6]. The proof presented in [6, Proposition 3.2] extends literally to higher dimensions and will be omitted. Proposition Let $$X$$ be a compact complex manifold of dimension $$n$$ equipped with a real structure $$\operatorname{c}$$.…”

“…As it was observed in our previous paper [6], many deformation classes of algebraic surfaces do not contain any real representative $$X$$ whose Hilbert square $$X^{[2]}$$ is maximal. Here, we refine results obtained in [6] and generalize them to higher dimensions. The main results are as follows.…”

“…We follow here the same strategy as in the proof of [6, Theorem 1.1]. As a first step, we show the following: Proposition Let $$X$$ be a real nonsingular projective variety of dimension $$n\geqslant 2$$.…”

“…More examples of real projective manifolds with maximal Hilbert square will be presented elsewhere [7], in a different context.…”