The well-studied moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient ${\mathcal {M}}^{\operatorname {GIT}}$, as a Baily–Borel compactification of a ball quotient ${(\mathcal {B}_4/\Gamma )^*}$, and as a compactified K-moduli space. From all three perspectives, there is a unique boundary point corresponding to non-stable surfaces. From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup ${\mathcal {M}}^{\operatorname {K}}\rightarrow {\mathcal {M}}^{\operatorname {GIT}}$, whereas from the ball quotient point of view, it is natural to consider the toroidal compactification ${\overline {\mathcal {B}_4/\Gamma }}\rightarrow {(\mathcal {B}_4/\Gamma )^*}$. The spaces ${\mathcal {M}}^{\operatorname {K}}$ and ${\overline {\mathcal {B}_4/\Gamma }}$ have the same cohomology, and it is therefore natural to ask whether they are isomorphic. Here, we show that this is in fact not the case. Indeed, we show the more refined statement that ${\mathcal {M}}^{\operatorname {K}}$ and ${\overline {\mathcal {B}_4/\Gamma }}$ are equivalent in the Grothendieck ring, but not K-equivalent. Along the way, we establish a number of results and techniques for dealing with singularities and canonical classes of Kirwan blowups and toroidal compactifications of ball quotients.

Designing an algorithm with a singly exponential complexity for computing semialgebraic triangulations of a given semialgebraic set has been a holy grail in algorithmic semialgebraic geometry. More precisely, given a description of a semialgebraic set $S \subset \mathbb {R}^k$ by a first-order quantifier-free formula in the language of the reals, the goal is to output a simplicial complex $\Delta $ , whose geometric realization, $|\Delta |$ , is semialgebraically homeomorphic to S. In this paper, we consider a weaker version of this question. We prove that for any $\ell \geq 0$ , there exists an algorithm which takes as input a description of a semialgebraic subset $S \subset \mathbb {R}^k$ given by a quantifier-free first-order formula $\phi $ in the language of the reals and produces as output a simplicial complex $\Delta $ , whose geometric realization, $|\Delta |$ is $\ell $ -equivalent to S. The complexity of our algorithm is bounded by $(sd)^{k^{O(\ell )}}$ , where s is the number of polynomials appearing in the formula $\phi $ , and d a bound on their degrees. For fixed $\ell $ , this bound is singly exponential in k. In particular, since $\ell $ -equivalence implies that the homotopy groups up to dimension $\ell $ of $|\Delta |$ are isomorphic to those of S, we obtain a reduction (having singly exponential complexity) of the problem of computing the first $\ell $ homotopy groups of S to the combinatorial problem of computing the first $\ell $ homotopy groups of a finite simplicial complex of size bounded by $(sd)^{k^{O(\ell )}}$ .

We construct a scheme $B(r; {\mathbb {A}}^n)$ such that a map $X \to B(r; {\mathbb {A}}^n)$ corresponds to a degree-n étale algebra on X equipped with r generating global sections. We then show that when $n=2$ , i.e., in the quadratic étale case, the singular cohomology of $B(r; {\mathbb {A}}^n)({\mathbb {R}})$ can be used to reconstruct a famous example of S. Chase and to extend its application to showing that there is a smooth affine $r-1$ -dimensional ${\mathbb {R}}$ -variety on which there are étale algebras ${\mathcal {A}}_n$ of arbitrary degrees n that cannot be generated by fewer than r elements. This shows that in the étale algebra case, a bound established by U. First and Z. Reichstein in [6] is sharp.

We show that for a smooth, projective variety X defined over a number field K, cyclic homology with coefficients in the ring , provides the right theory to obtain, using λ-operations, Serre's archimedean local factors of the complex L-function of X as regularized determinants.

For any subfield $K\subseteq \mathbb{C}$, not contained in an imaginary quadratic extension of $\mathbb{Q}$, we construct conjugate varieties whose algebras of $K$-rational ($p,p$)-classes are not isomorphic. This compares to the Hodge conjecture which predicts isomorphisms when $K$ is contained in an imaginary quadratic extension of $\mathbb{Q}$; additionally, it shows that the complex Hodge structure on the complex cohomology algebra is not invariant under the Aut($\mathbb{C}$)-action on varieties. In our proofs, we find simply connected conjugate varieties whose multilinear intersection forms on $H^{2}(-,\mathbb{R})$ are not (weakly) isomorphic. Using these, we detect nonhomeomorphic conjugate varieties for any fundamental group and in any birational equivalence class of dimension $\geq $10.

We give bounds for the Betti numbers of projective algebraic varieties in terms of their classes (degrees of dual varieties of successive hyperplane sections). We also give bounds for classes in terms of ramification volumes (mixed ramification degrees), sectional genus and, eventually, in terms of dimension, codimension and degree. For varieties whose degree is large with respect to codimension, we give sharp bounds for the above invariants and classify the varieties on the boundary, thus obtaining a generalization of Castelnuovo’s theory for curves to varieties of higher dimension.

In [M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants, Preprint (2011), arXiv:1006.2706v2[math.AG]], the authors, in particular, associate to each finite quiver Q with a set of vertices I the so-called cohomological Hall algebra ℋ, which is ℤI≥0-graded. Its graded component ℋγ is defined as cohomology of the Artin moduli stack of representations with dimension vector γ. The product comes from natural correspondences which parameterize extensions of representations. In the case of a symmetric quiver, one can refine the grading to ℤI≥0×ℤ, and modify the product by a sign to get a super-commutative algebra (ℋ,⋆) (with parity induced by the ℤ-grading). It is conjectured in [M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants, Preprint (2011), arXiv:1006.2706v2[math.AG]] that in this case the algebra (ℋ⊗ℚ,⋆) is free super-commutative generated by a ℤI≥0×ℤ-graded vector space of the form V =Vprim ⊗ℚ[x] , where x is a variable of bidegree (0,2)∈ℤI≥0×ℤ, and all the spaces ⨁ k∈ℤVprimγ,k, γ∈ℤI≥0. are finite-dimensional. In this paper we prove this conjecture (Theorem 1.1). We also prove some explicit bounds on pairs (γ,k) for which Vprimγ,k≠0 (Theorem 1.2). Passing to generating functions, we obtain the positivity result for quantum Donaldson–Thomas invariants, which was used by Mozgovoy to prove Kac’s conjecture for quivers with sufficiently many loops [S. Mozgovoy, Motivic Donaldson–Thomas invariants and Kac conjecture, Preprint (2011), arXiv:1103.2100v2[math.AG]]. Finally, we mention a connection with the paper of Reineke [M. Reineke, Degenerate cohomological Hall algebra and quantized Donaldson–Thomas invariants for m-loop quivers, Preprint (2011), arXiv:1102.3978v1[math.RT]].

In a recent paper, Bondarko [Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), Preprint (2007), 0704.4003] defined the notion of weight structure, and proved that the category DMgm(k) of geometrical motives over a perfect field k, as defined and studied by Voevodsky, Suslin and Friedlander [Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, vol. 143 (Princeton University Press, Princeton, NJ, 2000)], is canonically equipped with such a structure. Building on this result, and under a condition on the weights avoided by the boundary motive [J. Wildeshaus, The boundary motive: definition and basic properties, Compositio Math. 142 (2006), 631–656], we describe a method to construct intrinsically in DMgm(k) a motivic version of interior cohomology of smooth, but possibly non-projective schemes. In a sequel to this work [J. Wildeshaus, On the interior motive of certain Shimura varieties: the case of Hilbert–Blumenthal varieties, Preprint (2009), 0906.4239], this method will be applied to Shimura varieties.

Let Mc be the moduli space of semistable torsion-free sheaves of rank 2 with Chern classes c1 = 0 and c2 = c over a K3 surface with generic polarization. When $c=2n\ge 4$ is even, Mc is a singular projective variety which admits a symplectic form, called the Mukai form, on the smooth part. A natural question raised by O'Grady asks if there exists a desingularization on which the Mukai form extends everywhere nondegenerately. In this paper we show that such a desingularization does not exist for many even integers c by computing the stringy Euler numbers.