A surgery for generalized complex structures on 4-manifolds

Topology, 2007

A study of symplectic actions of a finite group G on smooth 4-manifolds is initiated. The central new idea is the use of G-equivariant Seiberg-Witten-Taubes theory in studying the structure of the fixed-point set of these symmetries. The main result in this paper is a complete description of the fixed-point set structure (and the action around it) of a symplectic cyclic action of prime order on a minimal symplectic 4-manifold with c 2 1 = 0. Comparison of this result with the case of locally linear topological actions is made. As an application of these considerations, the triviality of many such actions on a large class of 4-manifolds is established. In particular, we show the triviality of homologically trivial symplectic symmetries of a K3 surface (in analogy with holomorphic automorphisms). Various examples and comments illustrating our considerations are also included.

Bulletin mathématiques de la Société des sciences mathématiques de Roumanie

We compute the deformations, in the sense of generalized complex structures, of the standard complex structure on a complex 2-torus. We get a smooth complete family depending on six complex parameters and, in particular, we obtain the well-known smooth complete family of complex deformations depending on four parameters.

International Mathematics Research Notices, 2006

Journal of Geometry and Physics, 2013

We show that symplectic forms taming complex structures on compact manifolds are related to special types of almost generalized Kähler structures. By considering the commutator Q of the two associated almost complex structures J ± , we prove that if either the manifold is 4dimensional or the distribution Im Q is involutive, then the manifold can be expressed locally as a disjoint union of twisted Poisson leaves.

arXiv (Cornell University), 2019

Let (X, ω) be a symplectic rational surface. We study the space of tamed almost complex structures Jω using a fine decomposition via smooth rational curves and a relative version of the infinite dimensional Alexander-Pontrjagin duality. This decomposition provides new understandings of both the variation and stability of the symplectomorphism group Symp(X, ω) when deforming ω. In particular, we compute the rank of π 1 (Symp(X, ω)) with χ(X) ≤ 7 in terms of the number Nω of (-2)-symplectic sphere classes.

International Journal of Mathematics, 2015

In [Small exotic 4-manifolds, Algebr. Geom. Topol.8 (2008) 1781–1794], the first author constructed the first known example of exotic minimal symplectic[Formula: see text] and minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to [Formula: see text]. The construction in [Small exotic 4-manifolds, Algebr. Geom. Topol.8 (2008) 1781–1794] uses Yukio Matsumoto's genus two Lefschetz fibrations on [Formula: see text] over 𝕊2 along with the fake symplectic 𝕊2 × 𝕊2 construction given in [Construction of symplectic cohomology 𝕊2 × 𝕊2, Proc. Gökova Geom. Topol. Conf.14 (2007) 36–48]. The main goal in this paper is to generalize the construction in [Small exotic 4-manifolds, Algebr. Geom. Topol.8 (2008) 1781–1794] using the higher genus versions of Matsumoto's fibration constructed by Mustafa Korkmaz and Yusuf Gurtas on [Formula: see text] for any k ≥ 2 and n = 1, and k ≥ 1 and n ≥ 2, respectively. Using our symplectic building blocks, we also construct new symple...

Communications in Contemporary Mathematics, 2009

We show that a minimal surface of general type has a canonical symplectic structure (unique up to symplectomorphism) which is invariant for smooth deformation. We show that the symplectomorphism type is also invariant for deformations which allow certain normal singularities, provided one remains in the same smoothing component. We use this technique to show that the Manetti surfaces yield examples of surfaces of general type which are not deformation equivalent but are canonically symplectomorphic.

Publications mathématiques de l'IHÉS, 2013

This is the first of a series of papers about quantization in the context of derived algebraic geometry. In this first part, we introduce the notion of n-symplectic structures, a generalization of the notion of symplectic structures on smooth varieties and schemes, meaningful in the setting of derived Artin n-stacks (see [HAG-II, To2]). We prove that classifying stacks of reductive groups, as well as the derived stack of perfect complexes, carry canonical 2-symplectic structures. Our main existence theorem states that for any derived Artin stack F equipped with an n-symplectic structure, the derived mapping stack Map(X, F) is equipped with a canonical (n − d)-symplectic structure as soon a X satisfies a Calabi-Yau condition in dimension d. These two results imply the existence of many examples of derived moduli stacks equipped with n-symplectic structures, such as the derived moduli of perfect complexes on Calabi-Yau varieties, or the derived moduli stack of perfect complexes of local systems on a compact and oriented topological manifold. We explain how the known symplectic structures on smooth moduli spaces of simple objects (e.g. simple sheaves on Calabi-Yau surfaces, or simple representations of π 1 of compact Riemann surfaces) can be recovered from our results, and that they extend canonically as 0-symplectic structures outside of the smooth locus of simple objects. We also deduce new existence statements, such as the existence of a natural (−1)-symplectic structure (whose formal counterpart has been previously constructed in [Co, Co-Gw]) on the derived mapping scheme Map(E, T * X), for E an elliptic curve and T * X the cotangent space of a smooth scheme X. Canonical (−1)-symplectic structures are also shown to exist on Lagrangian intersections, moduli of sheaves (or complexes of sheaves) on Calabi-Yau 3-folds, and moduli of representations of π 1 of compact topological 3-manifolds. More generally, moduli sheaves on higher dimensional varieties are shown to carry canonical n-symplectic structures (with n depending on the dimension). * Partially supported by NSF RTG grant DMS-0636606 and NSF grants DMS-0700446 and DMS-1001693. † Partially supported by the ANR grant ANR-09-BLAN-0151 (HODAG). 3. Let M be a compact oriented topological manifold of dimension d. Then, the (derived) moduli stack of perfect complexes of local systems on X admits a canonical (2 − d)shifted symplectic structure. Future parts of this work will be concerned with the dual notion of Poisson (and n-Poisson) structures in derived algebraic geometry, formality (and n-formality) theorems, and finally with quantization. p-Forms, closed p-forms and symplectic forms in the derived setting A symplectic form on a smooth scheme X (over some base ring k, of characteristic zero), is the datum of a closed 2-form ω ∈ H 0 (X, Ω 2,cl X/k), which is moreover required to be non-degenerate, i.e. it induces an isomorphism Θ ω : T X/k ≃ Ω 1 X/k between the tangent and cotangent bundles. In our context X will no longer be a scheme, but rather a derived Artin stack in the sense of [HAG-II, To2], the typical example being an X that is the solution to some derived moduli problem (e.g. of sheaves, or complexes of sheaves on smooth and proper schemes, see [To-Va, Corollary 3.31], or of maps between proper schemes as in [HAG-II, Corollary 2.2.6.14]). In this context, differential 1-forms are naturally sections in a quasi-coherent complex L X/k , called the cotangent complex (see [Il, To2]), and the quasi-coherent complex of p-forms is defined to be ∧ p L X/k. The p-forms on X are then naturally defined as sections of ∧ p L X/k , i.e. the set of p-forms on X is defined to be the (hyper)cohomology group H 0 (X, ∧ p L X/k). More generally, elements in H n (X, ∧ p L X/k) are called p-forms of degree n on X (see Definition 1.11 and Proposition 1.13). The first main difficulty is to define the notion of closed p-forms and of closed p-forms of degree n in a meaningful manner. The key idea of this work is to interpret p-forms, i.e. sections of ∧ p L X/k , as functions on the derived loop stack LX of [To2, To-Ve-1, Ben-Nad] by means of the HKR theorem of [To-Ve-2] (see also [Ben-Nad]), and to interpret closedness as the condition of being S 1-equivariant. One important aspect here is that S 1-equivariance must be understood in the sense of homotopy theory, and therefore closedness defined as above is not simply a property of p-form but consists of an extra structure (see Definition 1.9). This picture is accurate (see Remark 1.8 and 1.15), but technically difficult to work with 1. We have therefore chosen a different presentation, by introducing local constructions for affine derived schemes, that are then glued over X to obtain global definitions for any derived Artin stack X. To each commutative dg-algebra A over k, we define a graded complex, called the weighted negative cyclic complex of A over k, explicitly constructed using the derived de Rham complex of A. Elements of weight p and of degree n − p of this complex are by definition closed p-forms of degree n on Spec A (Definition 1.7). For a general derived Artin stack X, closed p-forms are defined by smooth descent techniques (Definition 1.11). This definition of closed p-forms has a more explicit local nature, but can be shown to coincide with the original idea of S 1-equivariant functions on the loop stack LX (using, for instance, results from [To-Ve-2, Ben-Nad]). By definition a closed p-form ω of degree n on X has an underlying p-form of degree n (as we already mentioned this underlying p-form does not determine the closed p-form

Tohoku Mathematical Journal, 2015

We study the existence of strong Kähler with torsion (SKT) metrics and of symplectic forms taming invariant complex structures J on solvmanifolds G/Γ providing some negative results for some classes of solvmanifolds. In particular, we show that if either J is invariant under the action of a nilpotent complement of the nilradical of G or J is abelian or G is almost abelian (not of type (I)), then the solvmanifold G/Γ cannot admit any symplectic form taming the complex structure J , unless G/Γ is Kähler. As a consequence, we show that the family of non-Kähler complex manifolds constructed by Oeljeklaus and Toma cannot admit any symplectic form taming the complex structure.

arXiv (Cornell University), 2008

For any compact almost complex manifold (M, J), the last two authors [8] defined two subgroups H + J (M ), H - J (M ) of the degree 2 real de Rham cohomology group H 2 (M, R). These are the sets of cohomology classes which can be represented by J-invariant, respectively, Janti-invariant real 2-forms. In this note, it is shown that in dimension 4 these subgroups induce a cohomology decomposition of H 2 (M, R). This is a specifically 4-dimensional result, as it follows from a recent work of Fino and Tomassini [6]. Some estimates for the dimensions of these groups are also established when the almost complex structure is tamed by a symplectic form and an equivalent formulation for a question of Donaldson is given.