Smooth Compactification Research Articles

Line bundles on the moduli space of Lie algebroid connections over a curve

We explore algebro-geometric properties of the moduli space of holomorphic Lie algebroid (L) connections on a compact Riemann surface X of genus g≥3. A smooth compactification of the moduli space of L-connections, such that underlying vector bundle is stable, is constructed; the complement of the moduli space inside the compactification is a divisor. A criterion for the numerical effectiveness of the boundary divisor is given. We compute the Picard group of the moduli space, and analyze Lie algebroid Atiyah bundles associated with an ample line bundle. This enables us to conclude that regular functions on the space of certain Lie algebroid connections are constants. Moreover, under some condition, it is shown that the moduli space of L-connections does not admit non-constant algebraic functions. Rationally connectedness of the moduli spaces is explored.

A decomposition theorem for 0-cycles and applications to class field theory

We describe the abelian {e}tale fundamental group with modulus in terms of 0-cycles on a class of smooth quasi-projective varieties over finite fields which may not admit smooth compactifications. Using a limit process, this yields a cycle-theoretic analogue of the $K$-theoretic reciprocity theorem of Kato-Saito for such varieties. The proof is based on a decomposition theorem for the cohomological Chow group of 0-cycles on the double of a quasi-projective $R_1$-scheme over a field along a closed subscheme in terms of the Chow groups with and without modulus of the scheme. This also extends the decomposition theorem of Binda-Krishna to smooth schemes over imperfect fields.

The W(E6)$W(E_6)$‐invariant birational geometry of the moduli space of marked cubic surfaces

AbstractThe moduli space of marked cubic surfaces is one of the most classical moduli spaces in algebraic geometry, dating back to the nineteenth‐century work of Cayley and Salmon. Modern interest in was restored in the 1980s by Naruki's explicit construction of a ‐equivariant smooth projective compactification of , and in the 2000s by Hacking, Keel, and Tevelev's construction of the Kollár–Shepherd‐Barron–Alexeev (KSBA) stable pair compactification of as a natural sequence of blowups of . We describe generators for the cones of ‐invariant effective divisors and curves of both and . For Naruki's compactification , we further obtain a complete stable base locus decomposition of the ‐invariant effective cone, and as a consequence find several new ‐equivariant birational models of . Furthermore, we fully describe the log minimal model program for the KSBA compactification , with respect to the divisor , where is the boundary and is the sum of the divisors parameterizing marked cubic surfaces with Eckardt points.

Cohomology and geometry of Deligne–Lusztig varieties for GLn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{GL}_n$$\\end{document}

We give a description of the cohomology groups of the structure sheaf on smooth compactifications X¯(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{X}(w)$$\\end{document} of Deligne–Lusztig varieties X(w) for GLn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{GL}_n$$\\end{document}, for all elements w in the Weyl group. As a consequence, we obtain the modpm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{mod}\\ p^m$$\\end{document} and integral p-adic étale cohomology of X¯(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{X}(w)$$\\end{document}. Moreover, using our result for X¯(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{X}(w)$$\\end{document} and a spectral sequence associated to a stratification of X¯(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{X}(w)$$\\end{document}, we deduce the modpm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{mod}\\ p^m$$\\end{document} and integral p-adic étale cohomology with compact support of X(w). In our proof of the main theorem, in addition to considering the Demazure–Hansen smooth compactifications of X(w), we show that a similar class of constructions provide smooth compactifications of X(w) in the case of GLn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{GL}_n$$\\end{document}. Furthermore, we show in the appendix that the Zariski closure of X(w), for any connected reductive group G and any w, has pseudo-rational singularities.

A remark on the moduli space of Lie algebroid λ-connections

Let X be a compact Riemann surface of genus g ≥ 3 . Let L = ( L , [ . , . ] , ♯ ) be a holomorphic Lie algebroid over X of rank one and degree ( L ) < 2 − 2 g . We consider the moduli space of holomorphic L λ -connections over X, where λ ∈ C . We compute the Picard group of the moduli space of L λ -connections by constructing an explicit smooth compactification of the moduli space of those L λ -connections whose underlying vector bundle is stable such that the complement is a smooth divisor. We also show that the automorphism group of the moduli space of L λ -connections fits into a short exact sequence that involves the automorphism group of the moduli space of stable vector bundle over X. For λ = 1, we get Lie algebroid de Rham moduli space of L -connections and we determine its Chow group. Communicated by Manuel Reyes

Categorical action filtrations via localization and the growth as a symplectic invariant

Abstract We develop a purely categorical theory of action filtrations and their associated growth invariants. When specialized to categories of geometric interest, such as the wrapped Fukaya category of a Weinstein manifold, and the bounded derived category of coherent sheaves on a smooth algebraic variety, our categorical action filtrations essentially recover previously studied filtrations of geometric origin. Our approach is built around the notion of a smooth categorical compactification. We prove that a smooth categorical compactification induces well-defined growth invariants, which are moreover preserved under zig-zags of such compactifications. The technical heart of the paper is a method for computing these growth invariants in terms of the growth of certain colimits of (bi)modules. In practice, such colimits arise in both geometric settings of interest. The main applications are: (1) A “quantitative” refinement of homological mirror symmetry, which relates the growth of the Reeb-length filtration on the symplectic geometry side with the growth of filtrations on the algebraic geometry side defined by the order of pole at infinity (often these can be expressed in terms of the dimension of the support of sheaves). (2) A proof that the Reeb-length growth of symplectic cohomology and wrapped Floer cohomology on a Weinstein manifold are at most exponential. (3) Lower bounds for the entropy and polynomial entropy of certain natural endofunctors acting on Fukaya categories.

A smooth compactification of the space of genus two curves in projective space: via logarithmic geometry and Gorenstein curves

A smooth compactification of the space of genus two curves in projective space: via logarithmic geometry and Gorenstein curves

Smooth Compactifications of the Abel-Jacobi Section

Abstract For $\theta $ a small generic universal stability condition of degree $0$ and A a vector of integers adding up to $-k(2g-2+n)$ , the spaces $\overline {\mathcal {M}}_{g,A}^\theta $ constructed in [AP21, HMP+22] are observed to lie inside the space $\textbf {Div}$ of [MW20], and their pullback under $\textbf {Rub} \to \textbf {Div}$ of loc. cit. to be smooth. This provides smooth and modular modifications $\widetilde {\mathcal {M}}_{g,A}^\theta $ of $\overline {\mathcal {M}}_{g,n}$ on which the logarithmic double ramification cycle can be calculated by several methods.

Equations of linear subvarieties of strata of differentials

For a linear subvariety $M$ of a stratum of meromorphic differentials, we investigate its closure in the multi-scale compactification constructed by Bainbridge-Chen-Gendron-Grushevsky-M\"oller. We prove various restrictions on the type of defining linear equations in period coordinates for $M$ near its boundary, and prove that the closure is locally a toric variety. As applications, we give a fundamentally new proof of a generalization of the cylinder deformation theorem of Wright to the case of meromorphic strata, and construct a smooth compactification of the Hurwitz space of covers of the Riemann sphere.

Disk counting and wall-crossing phenomenon via family Floer theory

We use the wall-crossing formula in the non-archimedean SYZ mirror construction (arXiv: 2003.06106) to compute the Landau-Ginzburg superpotential and the one-pointed open Gromov-Witten invariants for a Chekanov-type Lagrangian torus in any smooth toric Fano compactification of C^n. It agrees with the works of Auroux, Chekanov-Schlenk, and Pascaleff-Tonkonog.

Steinberg slices and group-valued moment maps

We define a class of transversal slices in spaces which are quasi-Poisson for the action of a complex semisimple group G. This is a multiplicative analogue of Whittaker reduction. One example is the multiplicative universal centralizer Z of G, which is equipped with the usual symplectic structure in this way. We construct a smooth relative compactification Z‾ by taking the closure of each centralizer fiber in the wonderful compactification of G. By realizing this relative compactification as a transversal in a larger quasi-Poisson variety, we show that it is smooth and log-symplectic.

Classification of Kerr–de Sitter-like spacetimes with conformally flat I in all dimensions

Using asymptotic characterization results of spacetimes at conformal infinity, we prove that Kerr-Schild-de Sitter spacetimes are in one-to-one correspondence with spacetimes in the Kerr-de Sitter-like class with conformally flat $\mathscr{I}$. Kerr-Schild-de Sitter are spacetimes of Kerr-Schild form with de Sitter background that solve the $(\Lambda>0)$-vacuum Einstein equations and admit a smooth conformal compactification sharing $\mathscr{I}$ with the background metric. Kerr-de Sitter-like metrics with conformally flat $\mathscr{I}$ are a generalization of the Kerr-de Sitter metrics, defined originally in four spacetime dimensions and extended here to all dimensions in terms of their initial data at null infinity. We explicitly construct all metrics in this class as limits or analytic extensions of Kerr-de Sitter. The structure of limits is inferred from corresponding limits of the asymptotic data, which appear to be hard to guess from the spacetime metrics.

The Chern classes and Euler characteristic of the moduli spaces of Abelian differentials

AbstractFor the moduli spaces of Abelian differentials, the Euler characteristic is one of the most intrinsic topological invariants. We give a formula for the Euler characteristic that relies on intersection theory on the smooth compactification by multi-scale differentials. It is a consequence of a formula for the full Chern polynomial of the cotangent bundle of the compactification.The main new technical tools are an Euler sequence for the cotangent bundle of the moduli space of multi-scale differentials and computational tools in the Chow ring, such as a description of normal bundles to boundary divisors.

On singular log Calabi-Yau compactifications of Landau-Ginzburg models

We consider the procedure that constructs log Calabi-Yau compactifications of weak Landau-Ginzburg models of Fano varieties. We apply it to del Pezzo surfaces and coverings of projective spaces of index . For coverings of degree greater than the log Calabi-Yau compactification is singular; moreover, no smooth projective log Calabi-Yau compactification exists. We also prove, in the cases under consideration, the conjecture that the number of components of the fibre over infinity is equal to the dimension of an anticanonical system of the Fano variety. Bibliography: 46 titles.

Subvarieties of quotients of bounded symmetric domains

We present a new criterion for the complex hyperbolicity of a non-compact quotient X of a bounded symmetric domain. For each \(p \ge 1\), this criterion gives a precise condition under which the subvarieties \(V \subset X\) with \(\dim V \ge p\) are of general type, and X is p-measure hyperbolic. Then, we give several applications related to ball quotients, or to the Siegel moduli space of principally polarized abelian varieties. In the case of smooth compactifications of ball quotients, we obtain a conditional upper bound on the dimension on the exceptional locus, assuming the Green–Griffiths–Lang conjecture holds true. As another example of application, we give effective lower bounds for levels l so that the moduli spaces of genus g curves with l-level structures are of general type.

The Case Against Smooth Null Infinity I: Heuristics and Counter-Examples

This paper initiates a series of works dedicated to the rigorous study of the precise structure of gravitational radiation near infinity. We begin with a brief review of an argument due to Christodoulou (in: The Ninth Marcel Grossmann Meeting, World Scientific Publishing Company, Singapore, 2002) stating that Penrose’s proposal of smooth conformal compactification of spacetime (or smooth null infinity) fails to accurately capture the structure of gravitational radiation emitted by N infalling masses coming from past timelike infinity i^-. Modelling gravitational radiation by scalar radiation, we then take a first step towards a dynamical understanding of the non-smoothness of null infinity by constructing solutions to the spherically symmetric Einstein–Scalar field equations that arise from polynomially decaying boundary data, rphi sim t^{-1} as trightarrow -infty , on a timelike hypersurface (to be thought of as the surface of a star) and the no incoming radiation condition, rpartial _vphi =0, on past null infinity. We show that if the initial Hawking mass at i^- is nonzero, then, in accordance with the non-smoothness of {mathcal {I}}^+, the asymptotic expansion of partial _v(rphi ) near {mathcal {I}}^+ reads partial _v(rphi )=Cr^{-3}log r+{mathcal {O}}(r^{-3}) for some non-vanishing constant C. In fact, the same logarithmic terms appear already in the linear theory, i.e. when considering the spherically symmetric linear wave equation on a fixed Schwarzschild background. As a corollary, we can apply our results to the scattering problem on Schwarzschild: Putting compactly supported scattering data for the linear (or coupled) wave equation on {mathcal {I}}^- and on {mathcal {H}}^-, we find that the asymptotic expansion of partial _v(rphi ) near {mathcal {I}}^+ generically contains logarithmic terms at second order, i.e. at order r^{-4}log r.

Categorical smooth compactifications and neighborhoods of infinity

In this note we give a short overview of some of our results on derived categories of coherent sheaves, in particular on smooth categorical compactifications and on the formal punctured neighborhoods of infinity.

Categorical smooth compactifications and neighborhoods of infinity

Categorical smooth compactifications and neighborhoods of infinity

Orientations for DT invariants on quasi-projective Calabi–Yau 4-folds

For a Calabi–Yau 4-fold (X,ω), where X is quasi-projective and ω is a nowhere vanishing section of its canonical bundle KX, the (derived) moduli stack of compactly supported perfect complexes MX is −2-shifted symplectic and thus has an orientation bundle Oω→MX in the sense of Borisov–Joyce [11] necessary for defining Donaldson–Thomas type invariants of X. We extend first the orientability result of Cao–Gross–Joyce [23] to projective spin 4-folds. Then for any smooth projective compactification Y, such that D=Y﹨X is strictly normal crossing, we define orientation bundles on the stack MY×MDMY and express these as pullbacks of Z2-bundles in gauge theory, constructed using positive Dirac operators on the double of X. As a result, we relate the orientation bundle Oω→MX to a gauge-theoretic orientation on the classifying space of compactly supported K-theory. Using orientability of the latter, we obtain orientability of MX. We also prove orientability of moduli spaces of stable pairs and Hilbert schemes of proper subschemes. Finally, we consider the compatibility of orientations under direct sums.

Orientation data for moduli spaces of coherent sheaves over Calabi–Yau 3-folds

Let X be a compact Calabi–Yau 3-fold, and write M,M‾ for the moduli stacks of objects in coh(X),Dbcoh(X). There are natural line bundles KM→M, KM‾→M‾, analogues of canonical bundles. Orientation data on M,M‾ is an isomorphism class of square root line bundles KM1/2,KM‾1/2, satisfying a compatibility condition on the stack of short exact sequences. It was introduced by Kontsevich and Soibelman [35, §5] in their theory of motivic Donaldson–Thomas invariants, and is also important in categorifying Donaldson–Thomas theory using perverse sheaves.We show that natural orientation data can be constructed for all compact Calabi–Yau 3-folds X, and also for compactly-supported coherent sheaves and perfect complexes on noncompact Calabi–Yau 3-folds X that admit a spin smooth projective compactification X↪Y. This proves a long-standing conjecture in Donaldson–Thomas theory.These are special cases of a more general result. Let X be a spin smooth projective 3-fold. Using the spin structure we construct line bundles KM→M, KM‾→M‾. We define spin structures on M,M‾ to be isomorphism classes of square roots KM1/2,KM‾1/2. We prove that natural spin structures exist on M,M‾. They are equivalent to orientation data when X is a Calabi–Yau 3-fold with the trivial spin structure.We prove this using our previous paper [33], which constructs ‘spin structures’ (square roots of a certain complex line bundle KPE•→BP) on differential-geometric moduli stacks BP of connections on a principal U(m)-bundle P→X over a compact spin 6-manifold X.