Some rational subvarieties of moduli spaces of stable vector bundles

While the case of curves is nowadays well understood, the situation in higher dimension remains considerably less developed. In particular, there are no general results ensuring the non-emptyness of these moduli spaces. For this reason, explicit constructions of families of µstable vector bundles dominating particular subvarieties of these moduli spaces seem to be of significant interest. Let X be a smooth complex irreducible projective variety of dimension n ≥ 2 and let L be a non-trivial globally generated line bundle on X. In this paper, our aim is to produce families of vector bundles on X with rank r ≥ 2 and determinant L, which are generated by r + 1 global sections and are µ H -stable with respect to an ample line bundle H on X. Moreover, these families give rise to subvarieties in the corresponding moduli spaces which are birational to a Grassmannian variety.

Our construction starts as follows. Let W ⊂ H 0 (L) be a (r + 1)-dimensional subspace such that the evaluation map of global sections W ⊗ O X → L is a surjective map of vector bundles on X. Denote by M W,L its kernel; it is then a vector bundle on X of rank r and determinant L -1 . Its dual is a vector bundle E W too, with rank r, determinant L, and Chern classes c = (c 1 (L), . . . , c 1 (L) n ) (see Lemma 2.6), which fit into the following exact sequence:

If M W,L is µ H semistable for an ample line bundle H on X, then so is E W and it is generated by r + 1 global sections. Vector bundles of the form M W,F (denoted as M F in the complete case W = H 0 (F )), arising as kernels of evaluation map of globally generated vector bundles F , on a smooth variety, are known in literature as kernel bundles, dual span bundles and sygyzy bundles. Their stability has been extensively studied. For a smooth curve of genus g ≥ 2, the theory is well developed at least for the complete case (see, for example, the results in [But94], [Mis08], [EL92], [CH25], [BBPN08]); there are also some results in the case of singular curves (see for example [BF20]). In higher dimension, only partial results are available, mainly in the complete case and for line bundles (see [Fle84] and [EL13] and [Cam12]). Our strategy for proving the stability of M W,L consists in reducing the problem to the stability of kernel bundles on smooth curves. More precisely, let H be an ample line bundle on X and assume that there exists a smooth curve C ⊂ X of genus g ≥ 2, given as a complete intersection of divisors of |H|, such that the restriction map of global section H 0 (X, L) → H 0 (C, L |C ) is surjective. We can prove that the restriction of M W,L to C is a kernel bundle on C and its stability implies µ H -stability of M W,L . Stability on the curve C is ensured by requiring suitable numerical assumptions on the degree of L |C . Specifically, our result holds whenever either our conditions or those established in [Mis08] are satisfied. We will say that the data (X, L, H, r) is admissible if the above mentioned assumptions are satisfied (c.f. Definition 2.1). We denote by M s H (r, L, c) the moduli space parametrizing µ Hstable vector bundles with rank r, determinant L, and Chern classes c depending on L (c.f. Definition 2.12). Our main result is the following (see Theorem 2.14):

Theorem. Let (X, L, H, r) an admissible collection, then the moduli space M s H (r, L, c) is nonempty and it contains a subvariety birational to the Grassmannian variety Gr(r + 1, H 0 (L)). This provides, in arbitrary dimension, a systematic method to construct globally generated µ H -stable vector bundles with prescribed determinant and Chern classes.

In the second part of the paper, we specialise to algebraic surfaces, and we investigate the scope of our construction through a series of examples. We exhibit admissible collections with surfaces for each Kodaira dimensions κ(S) ∈ {-∞, 0, 1, 2}. Of particular interest is the case of K3 surfaces. Indeed, when S is a K3 surface and H is an ample primitive line bundle on S, the subvariety arising from our construction turns out to be a Lagrangian subvariety of the moduli space, provided the latter is a smooth irreducible symplectic variety (see Theorem 3.11 and Remark 3.12).

1.1. Moduli spaces of stable sheaves. Let X be a smooth irreducible projective complex variety of dimension n ≥ 2 and H an ample line bundle on X. We will need to deal with moduli spaces parametrising (H-stable) vector bundles on X. In this section, we recall some well-known results on this topic. Our main reference is [HL10]. To begin with, we recall that -unlike in the case

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