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European Mathematical Society Publishing House
2024-03-29 15:06:21
6
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=18&iss=2&update_since=2024-03-29
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
18
2016
2
The KSBA compactification for the moduli space of degree two $K$3 pairs
Radu
Laza
Stony Brook University, STONY BROOK, UNITED STATES
$K$3 surfaces, moduli space of K3 surfaces, KSBA
Inspired by the ideas of the minimal model program, Shepherd-Barron, Kollár, and Alexeev have constructed a geometric compactification for the moduli space of surfaces of log general type. In this paper, we discuss one of the simplest examples that fits into this framework: the case of pairs $(X,H)$ consisting of a degree two $K3$ surface $X$ and an ample divisor $H$. Specifically, we construct and describe explicitly a geometric compactification $\overline{\mathcal P}_2$ for the moduli of degree two $K$3 pairs. This compactification has a natural forgetful map to the Baily–Borel compactification of the moduli space $\mathcal F_2$ of degree two $K$3 surfaces. Using this map and the modular meaning of $\overline{\mathcal P}_2$, we obtain a better understanding of the geometry of the standard compactifications of $\mathcal F_2$.
Algebraic geometry
225
279
10.4171/JEMS/589
http://www.ems-ph.org/doi/10.4171/JEMS/589
Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers
Robert
Lipshitz
Columbia University, NEW YORK, UNITED STATES
David
Treumann
Boston College, CHESTNUT HILL, UNITED STATES
Hochschild homology, localization, Smith theory, Heegaard Floer homology
Let $A$ be a dg algebra over $\mathbb F_2$ and let $M$ be a dg $A$-bimodule. We show that under certain technical hypotheses on $A$, a noncommutative analog of the Hodge-to-de Rham spectral sequence starts at the Hochschild homology of the derived tensor product $M \otimes_A^L M$ and converges to the Hochschild homology of $M$. We apply this result to bordered Heegaard Floer theory, giving spectral sequences associated to Heegaard Floer homology groups of certain branched and unbranched double covers.
Manifolds and cell complexes
Associative rings and algebras
Differential geometry
281
325
10.4171/JEMS/590
http://www.ems-ph.org/doi/10.4171/JEMS/590
Sets of $\beta$-expansions and the Hausdorff measure of slices through fractals
Tom
Kempton
University of Utrecht, UTRECHT, NETHERLANDS
Bernoulli convolution, $\beta$ expansion, slicing fractals, conditional measures
We study natural measures on sets of $\beta$-expansions and on slices through self similar sets. In the setting of $\beta$-expansions, these allow us to better understand the measure of maximal entropy for the random $\beta$-transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing, leading to conditions under which almost every slice through a self similar set has positive Hausdorff measure, generalising long known results about almost everywhere values of the Hausdorff dimension.
Measure and integration
Number theory
Dynamical systems and ergodic theory
Fourier analysis
327
351
10.4171/JEMS/591
http://www.ems-ph.org/doi/10.4171/JEMS/591
Complexity of intersections of real quadrics and topology of symmetric determinantal varieties
Antonio
Lerario
SISSA, TRIESTE, ITALY
Real algebraic geometry, real quadrics, homological compexity, determinantal varieties
Let $W$ be a linear system of quadrics on the real projective space $\mathbb R P^n$ and $X$ be the base locus of that system (i.e. the common zero set of the quadrics in $W$). We prove a formula relating the topology of $X$ to the one of the discriminant locus $\Sigma_W$ (i.e. the set of singular quadrics in $W$). The set $\Sigma_W$ equals the intersection of $W$ with the discriminant hypersurface for quadrics; its singularities are unavoidable (they might persist after a small perturbation of $W$) and we set $\{\Sigma_W^{(r)}\}_{r\geq 1}$ for its singular point stratification, i.e. $\Sigma_W^{(1)}=\Sigma_W$ and $\Sigma_W^{(r)}=\textrm{Sing}\big( \Sigma_W^{(r-1)}\big)$. With this notation, for a generic $W$ the mentioned formula writes: $$b(X) \leq b(\mathbb R P^n)+ \sum_{r \geq 1}b(\mathbb{P}\Sigma_W^{(r)}).$$ In the general case a similar formula holds, but we have to replace each $b(\mathbb{P}\Sigma_W^{(r)})$ with $\frac{1}{2}b(\Sigma_\epsilon^{(r)})$, where $\Sigma_\epsilon$ equals the intersection of the discriminant hypersurface with the unit sphere on the translation of $W$ in the direction of a small negative definite form. Each $\Sigma_\epsilon^{(r)}$ is a determinantal variety on the sphere $S^{k-1}$ defined by equations of degree at most $n+1$ (here $k$ denotes the dimension of $W$); we refine Milnor's bound, proving that for such affine varieties $b(\Sigma_\epsilon^{(r)})\leq O(n)^{k-1}$. Since the sum in the above formulas contains at most $O(k)^{1/2}$ terms, as a corollary we prove that if $X$ is any intersection of $k$ quadrics in $\mathbb R P^n$ then the following sharp estimate holds: $$ b(X) \leq O(n)^{k-1}.$$ This bound refines Barvinok's style estimates (recall that the best previously known bound, due to Basu, has the shape $O(n)^{2k+2}$).
Algebraic geometry
Algebraic topology
353
379
10.4171/JEMS/592
http://www.ems-ph.org/doi/10.4171/JEMS/592
Boundary estimates for certain degenerate and singular parabolic equations
Benny
Avelin
Uppsala Universitet, UPPSALA, SWEDEN
Ugo
Gianazza
Università di Pavia, PAVIA, ITALY
Sandro
Salsa
Politecnico di Milano, MILANO, ITALY
Degenerate and singular parabolic equations, Harnack estimates, boundary Harnack inequality, Carleson estimate
We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic $p$-Laplacian equation. Assuming that such solutions continuously vanish on some distinguished part of the lateral part $S_T$ of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of degenerate/singular parabolic equations of porous medium type.
Partial differential equations
381
424
10.4171/JEMS/593
http://www.ems-ph.org/doi/10.4171/JEMS/593
Equidistribution estimates for Fekete points on complex manifolds
Nir
Lev
Bar-Ilan University, RAMAT GAN, ISRAEL
Joaquim
Ortega-Cerdà
Universitat de Barcelona, BARCELONA, SPAIN
Beurling–Landau density, Fekete points, holomorphic line bundles
We study the equidistribution of Fekete points in a compact complex manifold. These are extremal point configurations defined through sections of powers of a positive line bundle. Their equidistribution is a known result. The novelty of our approach is that we relate them to the problem of sampling and interpolation on line bundles, which allows us to estimate the equidistribution of the Fekete points quantitatively. In particular we estimate the Kantorovich–Wasserstein distance of the Fekete points to the limiting measure. The sampling and interpolation arrays on line bundles are a subject of independent interest, and we provide necessary density conditions through the classical approach of Landau, that in this context measures the local dimension of the space of sections of the line bundle. We obtain a complete geometric characterization of sampling and interpolation arrays in the case of compact manifolds of dimension one, and we prove that there are no arrays of both sampling and interpolation in the more general setting of semipositive line bundles.
Several complex variables and analytic spaces
425
464
10.4171/JEMS/594
http://www.ems-ph.org/doi/10.4171/JEMS/594