Max Karoubi, Maria Luiza Lapa de Souza
We give a necessary and sufficient condition for lagrangians in a symplectic vector bundle to be deformed stably into transversal lagrangians. In the case of three lagrangians, we show that the associated Grothendieck group can be identified with a Hermitian K-theory group.
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Transversality for perturbed special Lagrangian submanifolds
本文在配备广义Calabi-Yau结构的6维流形上，证明了扰动特殊拉格朗日子流形模空间的横截性定理，并表明该模空间一般由孤立点组成，为Donaldson-Segal计数方案提供了关键理论支撑。
We exhibit infinitely many, explicit special Lagrangian isolated singularities that admit no asymptotically conical special Lagrangian smoothings. The existence/ nonexistence of such smoothings is an important component of the current efforts to understand which singular special Lagrangians arise as limits of smooth special Lagrangians. We also use soft methods from symplectic geometry (the relative version of the h--principle) and tools from algebraic topology to prove (both positive and negative) results about Lagrangian desingularizations of Lagrangian submanifolds with isolated singulariti…
A Universal Obstruction to the Samuelson Condition for Tangent Lagrangian 2-Webs
研究平面曲线切线生成的辛平面中拉格朗日2-网的萨缪尔森面积条件，证明在非零曲率点附近局部切向2-网不是萨缪尔森网，障碍源于雅可比行列式沿对角线有简单零点导致对数奇异性。
Let $(X,α)$ be a Kähler manifold of dimension n, and let $[ω] \in H^{1,1}(X,\mathbb{R})$. We study the problem of specifying the Lagrangian phase of $ω$ with respect to $α$, which is described by the nonlinear elliptic equation \[ \sum_{i=1}^{n} \arctan(λ_i)= h(x) \] where $λ_i$ are the eigenvalues of $ω$ with respect to $α$. When $h(x)$ is a topological constant, this equation corresponds to the deformed Hermitian-Yang-Mills (dHYM) equation, and is related by Mirror Symmetry to the existence of special Lagrangian submanifolds of the mirror. We introduce a notion of subsolution for this equati…
This short paper shows a topological obstruction of the existence of certain Lagrangian submanifolds in symplectic $4m$-manifolds.
This is the fourth in a series of five papers math.DG/0211294, math.DG/0211295, math.DG/0302355, math.DG/0303272 studying compact special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally modelled on special Lagrangian cones C_1,...,C_n in C^m with isolated singularities at 0. Readers are advised to begin with the final paper math.DG/0303272 which surveys the series, gives examples, and applies the results to prove some conjectures. The first paper math.DG/0211294 studied the regularity of X near its singular points, and the second m…
We give topological obstructions to the existence of a closed exact Lagrangian submanifold in the cotangent bundle of a closed manifold M which is the total space of a fibration over the circle. For instance we show that the fundamental group of such a Lagrangian submanifold cannot be the free product of two non-trivial groups and that in any finite presentation of this group the difference between the number of generators and the number of relations is less than two.
Gauge theory and symplectic structures
该研究针对四维以上流形辛结构存在性障碍问题，在殆厄米特流形上引入椭圆方程组，证明其指标性质、解的横截性与唯一性，相关结果为高维规范理论提供了证据。
Schwarzian Residue of the Samuelson Obstruction in Tangent Lagrangian 2-Webs
该数学研究推导切触拉格朗日2-网中萨缪尔森障碍的施瓦茨残差表达式，揭示其与等仿曲率的关系，解释萨缪尔森条件局部失效的原因并推导其重参数化变换律。
Classical Sturm non-oscillation and comparison theorems as well as the Sturm theorem on zeros for solutions of second order differential equations have a natural symplectic version, since they describe the rotation of a line in the phase plane of the equation. In the higher dimensional symplectic version of these theorems, lines are replaced by Lagrangian subspaces and intersections with a given line are replaced by non-transversality instants with a distinguished Lagrangian subspace. Thus the symplectic Sturm theorems describe some properties of the Maslov index. Starting from the celebrated …
Metaplectic-c prequantizations and the Einstein--Brillouin--Keller quantizability condition
该研究证明每个辛流形存在辛形式，其复预量子化给出拉格朗日子流形可量化条件，当第一陈类消失时恢复经典爱因斯坦-布里渊-凯勒条件，并发展了相关障碍理论。
We address the integrability conditions of the inverse problem of the calculus of variations for time-dependent SODE using the Spencer version of the Cartan-Kähler theorem. We consider a linear partial differential operator $P$ given by the two Helmholtz conditions expressed in terms of semi-basic 1-forms and study its formal integrability. We prove that $P$ is involutive and there is only one obstruction for the formal integrability of this operator. The obstruction is expressed in terms of the curvature tensor $R$ of the induced nonlinear connection. We recover some of the classes of Lagrang…
Floer invented his theory in the mid eighties in order to prove the Arnol'd conjectures on the number of fixed point of Hamiltonian diffeomorphisms and Lagrangian intersections. Over the last thirty years, many versions of Floer homology have been constructed. In symplectic and contact dynamics and geometry they have become a principal tool, with applications that go far beyond the Arnol'd conjectures: The proof of the Conley conjecture and of many instances of the Weinstein conjecture, rigidity results on Lagrangian submanifolds and on the group of symplectomorphisms, lower bounds for the top…
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