Volume 2015, Issue 702

A quasi-smooth derived enhancement of a Deligne–Mumford stack 𝒳 naturally endows 𝒳 with a functorial perfect obstruction theory in the sense of Behrend–Fantechi. We apply this result to moduli of maps and perfect complexes on a smooth complex projective variety. For moduli of maps , for X = S an algebraic K 3-surface, g ∈ ℕ, and β ≠ 0 in H 2 ( S ,ℤ) a curve class, we construct a derived stack ℝ𝐌¯g,nred(S;β)${\mathbb {R}\overline{\mathbf {M}}^{\text{red}}_{g,n}(S;\beta )}$ whose truncation is the usual stack 𝐌¯g,n(S;β)${\overline{\mathbf {M}}_{g,n}(S;\beta )}$ of pointed stable maps from curves of genus g to S hitting the class β, and such that the inclusion 𝐌¯g(S;β)↪ℝ𝐌¯gred(S;β)${\overline{\mathbf {M}}_{g}(S;\beta )\hookrightarrow \mathbb {R}\overline{\mathbf {M}}^{\text{red}}_{g}(S;\beta )}$ induces on 𝐌¯g(S;β)${\overline{\mathbf {M}}_{g}(S;\beta )}$ a perfect obstruction theory whose tangent and obstruction spaces coincide with the corresponding reduced spaces of Okounkov–Maulik–Pandharipande–Thomas. The approach we present here uses derived algebraic geometry and yields not only a full rigorous proof of the existence of a reduced obstruction theory – not relying on any result on semiregularity maps – but also a new global geometric interpretation. We give two further applications to moduli of complexes . For a K 3-surface S we show that the stack of simple perfect complexes on S is smooth. This result was proved with different methods by Inaba for the corresponding coarse moduli space. Finally, we construct a map from the derived stack of stable embeddings of curves (into a smooth complex projective variety X ) to the derived stack of simple perfect complexes on X with vanishing negative Ext's, and show how this map induces a morphism of the corresponding obstruction theories when X is a Calabi–Yau 3-fold. An important ingredient of our construction is a perfect determinant map from the derived stack of perfect complexes to the derived stack of line bundles whose tangent morphism is given by Illusie's trace map for perfect complexes.

An ideal I in a polynomial ring S has linear powers if all the powers I k of I have a linear free resolution. We show that the ideal of maximal minors of a sufficiently general matrix with linear entries has linear powers. The required genericity is expressed in terms of the heights of the ideals of lower order minors. In particular we prove that every rational normal scroll has linear powers.

This paper is motivated by the question of understanding the asymptotic behavior of the Betti numbers of the resolution of the ideal of a projective variety as the positivity of the embedding line bundle grows. We present a conjecture asserting that these invariants approach a Gaussian distribution, and we verify this in the case of curves. Then we work out the asymptotics of “random” Betti tables with a fixed number of rows, sampled according to a uniform choice of Boij–Söderberg coefficients. This analysis suggests that the normal distribution of Betti numbers is in any event the typical behavior from a probabilistic viewpoint.

Using previous results concerning the rank two and rank three cases, all connected simply connected Cartan schemes for which the real roots form a finite irreducible root system of arbitrary rank are determined. As a consequence one obtains the list of all crystallographic arrangements, a large subclass of the class of simplicial hyperplane arrangements. Supposing that the rank is at least three, the classification yields Cartan schemes of type A and B , an infinite family of series involving the types C and D , and 74 sporadic examples.

Let ( X , D ) be a projective log canonical pair. We show that for any natural number p , the sheaf (ΩXp(log⌊D⌋))**${(\Omega _X^p(\log \lfloor {D}\rfloor ))^{**}}$ of reflexive logarithmic p -forms does not contain a Weil divisorial subsheaf whose Kodaira–Iitaka dimension exceeds p . This generalizes a classical theorem of Bogomolov and Sommese. In fact, we prove a more general version of this result which also deals with the orbifoldes géométriques introduced by Campana. The main ingredients to the proof are the Extension Theorem of Greb–Kebekus–Kovács–Peternell, a new version of the Negativity Lemma, the minimal model program, and a residue map for symmetric differentials on dlt pairs. We also give an example showing that the statement cannot be generalized to spaces with Du Bois singularities. As an application, we give a Kodaira–Akizuki–Nakano-type vanishing result for log canonical pairs which holds for reflexive as well as for Kähler differentials.

We prove that the Frobenius structure constructed from the Gromov–Witten theory for an orbifold projective line with at most three orbifold points is uniquely determined by the Witten–Dijkgraaf–Verlinde–Verlinde equations with certain natural initial conditions.

Let S be an essentially smooth scheme over a field of characteristic exponent c . We prove that there is a canonical equivalence of motivic spectra over S MGL/( a 1 , a 2 ,...)[1/ c ] ≃ H ℤ[1/ c ] where H ℤ is the motivic cohomology spectrum, MGL is the algebraic cobordism spectrum, and the elements a n are generators of the Lazard ring. We discuss several applications including the computation of the slices of ℤ[1/ c ]-local Landweber exact motivic spectra and the convergence of the associated slice spectral sequences.

Suppose that X = G / K is the quotient of a locally compact group by a closed subgroup. If X is locally contractible and connected, we prove that X is a manifold. If the G -action is faithful, then G is a Lie group.