Shrinking Kähler-Ricci solitons on toric fano fibrations
We prove that a smooth toric Fano fibration admits a complete gradient shrinking Kähler-Ricci soliton.
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arXiv subjects
Tristan C. Collins
Publications and source records attributed to Tristan C. Collins.
The deformed Vortex equations and equivariant stability conditions
We study the higher rank deformed Hermitian-Yang-Mills (dHYM) equations for $SU(2)$-equivariant holomorphic vector bundles over $X\times \mathbb{P}^1$ for $X$ a compact Riemann surface. For a class of vector bundles of vortex type, we establish the equivalence between existence of solutions to higher rank dHYM equations and an appropriate notion of algebro-geometric stability, called $Z$-stability. We show that $Z$-stability is implied by, and conjecturally equivalent to, Bridgeland stability in $D^{b}{\rm Coh}^{SU(2)}(X\times \mathbb{P}^1)$.
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Non-Kähler Special Lagrangian submanifolds and SYZ mirror symmetry
We determine purely algebraic equations to identify \textit{SLags} generated by invariant distributions in a class of non-Kähler Calabi-Yau manifolds. We determine SLag distributions, determine which leaves integrate to compact submanifolds and study the deformation theory, which we find to be unobstructed. We apply our results to the Iwasawa manifold, the completely solvable 6-dimensional Nakamura manifold and the complex parallelizable Nakamura manifold. Through these examples we find families of topologically distinct \textit{SLags}, including the existence of SLag torus fibrations. Following the proposal of Lau-Tseng-Yau, we compute the non-Kähler SYZ mirrors of Nakamura manifolds, together with their refined symplectic Bott-Chern cohomologies. As a consequence, we find the existence of semi-flat non-Kähler mirror pairs which are not diffeomorphic.
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Homogeneous optimal transport maps between oblique cones
We construct homogeneous optimal transport maps for the quadratic cost between convex cones with homogeneous, possibly degenerate, densities when the cones satisfy an obliqueness condition. The existence of such maps plays a central role in the boundary regularity theory for optimal transport maps between convex domains. Our results are also relevant for the existence of complete Calabi-Yau metrics on certain quasi-projective varieties.
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Bridgeland/Weak Stability Conditions under Spherical Twist Associated to A Torsion Sheaf
In this paper, we study the action of an autoequivalence, the spherical twist associated to a torsion sheaf, on the standard Bridgeland stability conditions and a generalized weak stability condition on the derived category of a K3 surface. As a special case, we construct a Bridgeland stability condition associated to a non-nef divisor, which conjecturally lies in the geometric component but outside the geometric chamber. We also discuss the destabilizing objects and stability of certain line bundles at the weak stability condition associated to a nef divisor.
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Log Calabi--Yau manifolds: holomorphic tensors, stability and universal cover
We study various geometric properties of log Calabi-Yau manifolds, i.e. log smooth pairs $(X,D)$ such that $K_X+D=0$. More specifically, we focus on the two cases where $X$ is a Fano manifold and $D$ is either smooth or has two proportional components. Despite the existence of a complete Ricci flat Kähler metric on $X\setminus D$ in both cases, we will show that the geometric properties of the pair $(X,D)$ are vastly different, e.g. validity of Bochner principle, local triviality of the quasi-Albanese map, polystability of $T_X(-\log D)$ and compactifiability of the universal cover of $X\setminus D$. When $D$ has two components we show that the universal cover of $X\setminus D$ is a Calabi-Yau manifold of infinite topological type, and we describe the geometry at infinity from a Riemannian point of view.
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An introduction to conifold transitions
These lecture notes introduce conifold transitions between complex threefolds with trivial canonical bundle from the differential geometric point of view, and with a particular view towards aspects of mathematical physics and string theory. The lecture notes are aimed at beginning graduate students and non-experts, emphasizing explicit calculations and examples. After a brief introduction in Section 1, we recall some basic facts about Calabi-Yau manifolds in Section 2. Section 3 studies the conifold as a Calabi-Yau manifold with singularities, and introduces the local model for a conifold transition. Section 4 discusses global conifold transitions, and recalls the famous result of Friedman concerning the existence of smoothings for nodal Calabi-Yau threefolds. We give a differential geometric proof of the necessity part of Friedman's theorem. Section 5 discusses Reid's fantasy, and the web of Calabi-Yau threefolds. Section 6 discusses metric aspects of the local conifold transition, constructing explicit asymptotically conical Calabi-Yau metrics on the small resolution and the smoothing. Section 7 discusses the metric aspects of global conifold transitions, with a particular emphasis on the heterotic string.
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On a general class of free boundary Monge-Ampère equations
We solve a general class of free boundary Monge-Ampère equations given by \[ \det D^2u = λ\dfrac{f(-u)}{g(u^\star)h(\nabla u)}χ_{\{u<0\}} \; \text{ in } \mathbb{R}^n, \quad \nabla u (\mathbb{R}^n) = P \] where $P$ is a bounded convex set containing the origin, and $h>0$ on $P$. We consider applications to optimal transport with degenerate densities, Monge-Ampère eigenvalue problems, and geometric problems including a hemispherical Minkowski problem and free boundary Kähler-Ricci solitons on toric Fano manifolds.
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Boundary regularity of optimal transport maps on convex domains
We study the regularity of optimal transport maps between convex domains with quadratic cost. For nondegenerate $C^α$-densities, we prove $C^{1, 1-\varepsilon}$-regularity of the potentials up to the boundary. If in addition the boundary is $C^{1, α}$, we improve this to $C^{2, α}$-regularity. We also investigate pointwise $C^{1, 1}$-regularity at boundary points. We obtain a complete characterization of pointwise $C^{1, 1}$-regularity for planar polytopes in terms of the geometry of tangent cones. Furthermore, we study the regularity of optimal transport maps with degenerate densities on cones, which arise from recent developments in Kähler geometry. The main new technical tool we introduce is a monotonicity formula for optimal transport maps on convex domains which characterizes the homogeneity of blow-ups.
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Stability for Line Bundles and Deformed Hermitian-Yang-Mills Equation on Some Elliptic Surfaces
We study the twisted ampleness criterion due to Collins, Jacob and Yau on surfaces, which is equivalent to the existence of solutions to the deformed Hermitian-Yang-Mills (dHYM) equation. When $X$ is a Weierstrass elliptic K3 surface, and $ω$ an ample class such that $ω$ lies in the span of a section class and the fiber class, we show that for a class of line bundles $L$ with fiber degree 1 and $ωc_1(L)>0$, the twisted ampleness of $L$ respect to $ω$, always implies the $σ_{ω, 0}$-stability (Bridgeland stability) of $L$. This answers a question by Collins and Yau for a class of examples.
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The SYZ mirror symmetry conjecture for del Pezzo surfaces and rational elliptic surfaces
We prove a version of the Strominger-Yau-Zaslow mirror symmetry conjecture for non-compact Calabi-Yau surfaces arising from, on the one hand, pairs $(\check{Y},\check{D})$ of a del Pezzo surface $\check{Y}$ and $\check{D}$ a smooth anti-canonical divisor and, on the other hand, pairs $(Y,D)$ of a rational elliptic surface $Y$, and $D$ a singular fiber of Kodaira type $I_k$. Three main results are established concerning the latter pairs $(Y,D)$. First, adapting work of Hein \cite{Hein}, we prove the existence of a complete Calabi-Yau metric on $Y\setminus D$ asymptotic to a (generically non-standard) semi-flat metric in every Kähler class. Secondly, we prove a uniqueness theorem to the effect that, modulo automorphisms, every Kähler class on $Y\setminus D$ admits a unique asymptotically semi-flat Calabi-Yau metric. This result yields a finite dimensional Kähler moduli space of Calabi-Yau metrics on $Y\setminus D$. Further, this result answers, in this setting, questions of Tian-Yau and Yau. Thirdly, building on the authors' previous work, we prove that $Y\setminus D$ equipped with an asymptotically semi-flat Calabi-Yau metric $ω_{CY}$ admits a special Lagrangian fibration whenever the de Rham cohomology class of $ω_{CY}$ is not topologically obstructed. Combining these results we define a mirror map from the moduli space of del Pezzo pairs $(\check{Y}, \check{D})$ to the complexified Kähler moduli of $(Y,D)$ and prove that the special Lagrangian fibration on $(Y,D)$ is $T$-dual to the special Lagrangian fibration on $(\check{Y}, \check{D})$ previously constructed by the authors. We give some applications of these results, including to the study of automorphisms of del Pezzo surfaces fixing an anti-canonical divisor.
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A convergence result for Mean Curvature Flow of totally real submanifolds
We establish a convergence result for the mean curvature flow starting from a totally real submanifold which is "almost minimal" in a precise, quantitative sense. This extends, and makes effective, a result of H. Li for the Lagrangian mean curvature flow.
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A free boundary Monge-Ampère equation and applications to complete Calabi-Yau metrics
Let $P$ be a convex body containing the origin in its interior. We study a real Monge-Ampère equation with singularities along $\del P$ which is Legendre dual to a certain free boundary Monge-Ampère equation. This is motivated by the existence problem for complete Calabi-Yau metrics on log Calabi-Yau pairs $(X, D)$ with $D$ an ample, simple normal crossings divisor. We prove the existence of solutions in $C^{\infty}(P)\cap C^{1,α}(\overline{P})$, and establish the strict convexity of the free boundary. When $P$ is a polytope, we obtain an asymptotic expansion for the solution near the interior of the codimension $1$ faces of $\del P$.
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The Strominger system in the square of a Kähler class
We study the Strominger system with fixed balanced class. We show that classes which are the square of a Kähler metric admit solutions to the system for vector bundles satisfying the necessary conditions. Solutions are constructed by deforming a Calabi-Yau metric and a Hermitian-Yang-Mills metric along a path inside the given cohomology class.
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Recent progress on SYZ mirror symmetry for some non-compact Calabi-Yau surfaces
We survey the authors recent works, joint with A. Jacob, on Strominger-Yau-Zaslow mirror symmetry for rational elliptic surfaces and del Pezzo surfaces. We discuss some applications, including the Torelli theorem for $ALH^*$ gravitational instantons and explain how our results can be used to prove that all $ALH^*$ gravitational instantons can be compactified to a weak del Pezzo surface, recovering a recent result of Hein-Sun-Viaclovsky-Zhang.
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Uniqueness of some cylindrical tangent cones to special Lagrangians
We show that if an exact special Lagrangian $N\subset \mathbb{C}^n$ has a multiplicity one, cylindrical tangent cone of the form $\mathbb{R}^{k}\times \mathbf{C}$ where $\mathbf{C}$ is a special Lagrangian cone with smooth, connected link, then this tangent cone is unique provided $\mathbf{C}$ satisfies an integrability condition. This applies, for example, when $\mathbf{C}= \mathbf{C}_{HL}^{m}$ is the Harvey-Lawson $T^{m-1}$ cone for $m\ne 8,9$.
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Complete Calabi-Yau metrics in the complement of two divisors
We construct new complete Calabi-Yau metrics on the complement of an anticanonical divisors $D$ in a Fano manifold of dimension at least three, when $D$ consists of two transversely intersecting smooth divisors. The asymptotic geometry is modeled on a generalization of the Calabi ansatz, related to the non-archimedean Monge-Ampère equation.
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Stability and the deformed Hermitian-Yang-Mills equation
We survey some recent progress on the deformed Hermitian-Yang-Mills (dHYM) equation. We discuss the role of geometric invariant theory (GIT) in approaching the solvability of the dHYM equation, following work of the first author and S.-T. Yau. We compare the GIT picture with the conjectural picture for dHYM involving Bridgeland stability. In particular, following Arcara-Miles, we show that on the blow-up of $\mathbb{P}^2$ any line bundle admitting a solution of the deformed Hermitian-Yang-Mills equation is Bridgeland stable, but not conversely. Finally, we survey some recent progress on heat flows associated to the dHYM equation.
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