Higher genus minimal surfaces in S3S^3 and stable bundles

We consider compact minimal surfaces $f\colon M\to S^3$ of genus 2 which are homotopic to an embedding. We assume that the associated holomorphic bundle is stable. We prove that these surfaces can be constructed from a globally defined family of meromorphic connections by the DPW method. The poles of the meromorphic connections are at the Weierstrass points of the Riemann surface of order at most 2. For the existence proof of the DPW potential we give a characterization of stable extensions $0\to S^{-1}\to V\to S\to 0$ of spin bundles $S$ by its dual $S^{-1}$ in terms of an associated element of $P H^0(M;K^2).$ We also consider the family of holomorphic structures associated to a minimal surface in $S^3.$ For surfaces of genus $g\geq2$ the holonomy of the connections is generically non-abelian and therefore the holomorphic structures are generically stable

Special Ulrich bundles on non-special surfaces with pg=q=0p_g=q=0

Let $S$ be a surface with $p_g(S)=q(S)=0$ and endowed with a very ample line bundle $\mathcal O_S(h)$ such that $h^1\big(S,\mathcal O_S(h)\big)=0$. We show that $S$ supports special (often stable) Ulrich bundles of rank $2$, extending a recent result by A. Beauville. Moreover, we show that such an $S$ supports families of dimension $p$ of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large $p$ except for very few cases. We also show that the same is true for linearly normal non-special surface in $\mathbb P^4$ of degree at least $4$, Enriques surface and anticanonical rational surface.Comment: 17 pages, to appear in the International Journal of Mathematic

DPW Potentials for Compact Symmetric CMC Surfaces in S3\mathbb{S}^3

Inspired by the work of Heller [12], we show that there exists a DPW potential for the Lawson surface $\xi_{k-1, l-1}$ from which it is possible to reconstruct the minimal immersion $f: \xi_{k-1, l-1} \to \mathbb{S}^3$ via the DPW method. Moreover, we extend the result to surfaces immersed in the 3-sphere with constant mean curvature which satisfy a certain symmetric condition.Comment: 20 pages, to appear in Journal of Geometry and Physic

Surgery of real symplectic fourfolds and Welschinger invariants

A surgery of a real symplectic manifold $X_{\mathbb R}$ along a real Lagrangian sphere $S$ is a modification of the symplectic and real structure on $X_{\mathbb R}$ in a neigborhood of $S$. Genus 0 Welschinger invariants of two real symplectic $4$-manifolds differing by such a surgery have been related in a previous work in collaboration with N. Puignau. In the present paper, we explore some particular situations where these general formulas greatly simplify. As an application, we complete the computation of genus 0 Welschinger invariants of all del~Pezzo surfaces, and of all $\mathbb R$-minimal real conic bundles. As a by-product, we establish the existence of some new relative Welschinger invariants. We also generalize our results to the enumeration of curves of higher genus, and give relations between hypothetical invariants defined in the same vein as a previous work by Shustin.Comment: 28 pages, 2 figures. V2: Major edition (hopefully simplifications) of the first version, references precised. V3: Minor edition

On the Equivalence Problem for Toric Contact Structures on S^3-bundles over S^2$

We study the contact equivalence problem for toric contact structures on $S^3$-bundles over $S^2$. That is, given two toric contact structures, one can ask the question: when are they equivalent as contact structures while inequivalent as toric contact structures? In general this appears to be a difficult problem. To find inequivalent toric contact structures that are contact equivalent, we show that the corresponding 3-tori belong to distinct conjugacy classes in the contactomorphism group. To show that two toric contact structures with the same first Chern class are contact inequivalent, we use Morse-Bott contact homology. We treat a subclass of contact structures which include the Sasaki-Einstein contact structures $Y^{p,q}$ studied by physicists. In this subcase we give a complete solution to the contact equivalence problem by showing that $Y^{p,q}$ and $Y^{p'q'}$ are inequivalent as contact structures if and only if $p\neq p'$.Comment: 61 page

Generalized Lazarsfeld-Mukai bundles and a conjecture of Donagi and Morrison

Let S be a K3 surface and assume for simplicity that it does not contain any (-2)-curve. Using coherent systems, we express every non-simple Lazarsfeld-Mukai bundle on S as an extension of two sheaves of some special type, that we refer to as generalized Lazarsfeld-Mukai bundles. This has interesting consequences concerning the Brill-Noether theory of curves C lying on S. From now on, let g denote the genus of C and A be a complete linear series of type g^r_d on C such that d<= g-1 and the corresponding Brill-Noether number is negative. First, we focus on the cases where A computes the Clifford index; if r>1 and with only some completely classified exceptions, we show that A coincides with the restriction to C of a line bundle on S. This is a refinement of Green and Lazarsfeld's result on the constancy of the Clifford index of curves moving in the same linear system. Then, we study a conjecture of Donagi and Morrison predicting that, under no hypothesis on its Clifford index, A is contained in a g^s_e which is cut out from a line bundle on S and satisfies e<= g-1. We provide counterexamples to the last inequality already for r=2. A slight modification of the conjecture, which holds for r=1,2, is proved under some hypotheses on the pair (C,A) and its deformations. We show that the result is optimal (in the sense that our hypotheses cannot be avoided) by exhibiting, in the Appendix, some counterexamples obtained jointly with Andreas Leopold Knutsen.Comment: 28 pages, final version, to appear in Adv. Math. with an Appendix joint with Andreas Leopold Knutse