Some rational subvarieties of moduli spaces of stable vector bundles

The notion of µ-stability for vector bundles on curves was introduced by Mumford, and subsequently extended to higher-dimensional varieties by the foundational works of Takemoto, Gieseker and Maruyama. In particular, Maruyama proved the existence of coarse moduli spaces parametrising isomorphism classes of µ H -stable vector bundles with respect to an ample polarisation H, on a smooth projective variety (see [Mar77]).

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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