We study the existence of special Lagrangian submanifolds of log Calabi–Yau manifolds equipped with the complete Ricci-flat Kähler metric constructed by Tian and Yau. We prove that if X is a Tian–Yau manifold and if the compact Calabi–Yau manifold at infinity admits a single special Lagrangian, then X admits infinitely many disjoint special Lagrangians. In complex dimension 2, we prove that if Y is a del Pezzo surface or a rational elliptic surface and D∈|-KY| is a smooth divisor with D^2=d , then X=Y\D admits a special Lagrangian torus fibration, as conjectured by Strominger–Yau–Zaslow and Auroux. In fact, we show that X admits twin special Lagrangian fibrations, confirming a prediction of Leung and Yau. In the special case that Y is a rational elliptic surface or Y=P^2, we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-Kähler rotation, X can be compactified to the complement of a Kodaira type I_d fiber appearing as a singular fiber in a rational elliptic surface π: Y →P^1.

We prove the representability theorem in derived analytic geometry. The theorem as- serts that an analytic moduli functor is a derived analytic stack if and only if it is compatible with Postnikov towers, has a global analytic cotangent complex, and its truncation is an analytic stack. Our result applies to both derived complex analytic geometry and derived nonarchimedean analytic geometry (rigid analytic geometry). The representability theorem is of both philosophical and prac- tical importance in derived geometry. The conditions of representability are natural expectations for a moduli functor. So the theorem confirms that the notion of derived analytic space is natural and sufficiently general. On the other hand, the conditions are easy to verify in practice. So the theo- rem enables us to enhance various classical moduli spaces with derived structures, thus providing plenty of down-to-earth examples of derived analytic spaces. For the purpose of proof, we study analytification, square-zero extensions, analytic modules and cotangent complexes in the context of derived analytic geometry. We will explore applications of the representability theorem in our subsequent works. In particular, we will establish the existence of derived mapping stacks via the representability theorem.

Let k be the algebraic closure of a finite field of odd characteristic p, and X a smooth projective scheme over the Witt ring W (k) which is geometrically connected in characteristic zero. We introduce the notion of Higgs-de Rham flow1 and prove that the category of periodic Higgs-de Rham flows over X/W(k) is equivalent to the category of Fontaine modules, hence further equivalent to the category of crystalline representations of the e ́tale fundamental group π_1(X_K ) of the generic fiber of X, after Fontaine-Laffaille and Faltings, where K is the fraction field of W(k). Moreover, we prove that every semistable Higgs bundle over the special fiber X_k of X of rank ≤ p initiates a semistable Higgs–de Rham flow and thus those of rank ≤ p − 1 with trivial Chern classes induce k-representations of π_1(X_K ). A fundamental construction in this paper is the inverse Cartier transform over a truncated Witt ring. In characteristic p, it was constructed by Ogus–Vologodsky in the nonabelian Hodge theory in positive characteristic; in the affine local case, our construction is related to the local Ogus–Vologodsky correspondence of Shiho.

We establish a compact analogue of the P=W conjecture. For a projective irreducible holomorphic symplectic variety with a Lagrangian fibration, we show that the perverse numbers associated with the fibration match perfectly with the Hodge numbers of the total space. This builds a new connection between the topology of Lagrangian fibrations and the Hodge theory of hyper-Kähler manifolds. We present two applications of our result: one on the cohomology of the base and fibers of a Lagrangian fibration, and the other on the refined Gopakumar–Vafa invariants of a K3 surface. Furthermore, we show that the perverse filtration associated with a Lagrangian fibration is multiplicative under cup product.

If 𝑈:[0,+∞[×𝑀 is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation ∂_𝑡𝑈+𝐻(𝑥,∂_𝑥 𝑈)=0, where 𝑀 is a not necessarily compact manifold, and 𝐻 is a Tonelli Hamiltonian, we prove the set Σ(𝑈), of points in ]0,+∞[×𝑀 where 𝑈 is not differentiable, is locally contractible. Moreover, we study the homotopy type of Σ(𝑈). We also give an application to the singularities of the distance function to a closed subset of a complete Riemannian manifold.