What's the current state of these conjectures?

Who is working on them?

In "Standard conjectures on algebraic cycles" Grothendieck says:

"They would form the basis of the so-called "theory of motives" which is a systematic theory of "arithmetic properties" of algebraic varieties, as embodied in their groups of classes of cycles for numerical equivalence. ... Alongside the problem of resolution of singularities, the proof of the standard conjectures seems to me to be the most urgent task in algebraic geometry."

  • 7
    $\begingroup$ I suggest The standard conjectures by S. Kleiman, in the Proceedings of the AMS Summer Conference on Motives (this volume). I don't think there has been any notable progress since then. $\endgroup$
    – abx
    Jul 15 '14 at 6:25
  • 5
    $\begingroup$ Now we know: (1) that numerical motives are abelian semisimple (2) much on dependencies between conjectures. So, there is not much progress here. In my opinion, the triangulated approach to motives (of Voevodsky and others) is much more important for motives, since it allows to work with them without proving standard conjectures. $\endgroup$ Jul 15 '14 at 7:33
  • 2
    $\begingroup$ Milne in "Polarizations and Grothendieck’s Standard Conjectures" says: "This conjecture (Lefschetz standard conjecture) is known for curves (trivial), abelian varieties (Lieberman 1968, Kleiman 1968), surfaces and Weil cohomologies for which dimH1(V) = 2dimPic0(V ) (Grothendieck), generalized flag manifolds (trivial), complete intersections (trivial), and products of such varieties (see Kleiman 1994). For abelian varieties, it is even known that the operator Λ is defined by a Lefschetz class, i.e., a class in the Q-algebra generated by divisor classes (Milne 1999a)." $\endgroup$
    – user55909
    Jul 16 '14 at 23:19
  • 3
    $\begingroup$ Milne in "Polarizations and Grothendieck’s Standard Conjectures" says: "In fact, no progress seems to have been made on these conjectures since they were first formulated: the lists of known cases in Kleiman 1968 and in Kleiman 1994 are identical." $\endgroup$
    – user55909
    Jul 16 '14 at 23:23
  • 1
    $\begingroup$ @MikhailBondarko — From a practical point of view, you might be right. But I think we have not really understood what motives are, as long as we have not answered the standard conjectures on algebraic cycles. $\endgroup$
    – jmc
    Apr 7 '15 at 12:18

For future references. Feel free to edit to include new cases, or any improvements.

As for 2015, the standard conjectures on algebraic cycles is unconditionally (at lest) known for $X$:

Lefschetz standard conjecture (Grothendieck conjectures $A(X)$ and $B(X)$)

  • a curve (trivial).
  • a surface with $H^1(X)=2\cdot\mathrm{Pic}^0(X)$ (Grothendieck).
  • an abelian variety (Liebermann).
  • a generalized flag manifold $G/P$ (Schubert).
  • a smooth varieties which is complete intersections in some projective space (trivial).
  • a Grassmannian (Liebermann?).
  • for which $H^*(X)$ is isomorphic to the Chow ring $A^*(X)$.
  • a smooth projective moduli space of sheaves on rational Poisson surfaces.
  • a uniruled threefold (Arapura).
  • a unirational fourfold (Arapura).
  • the moduli space of stable vector bundles over a smooth projective curve (Arapura).
  • the Hilbert scheme $S^{[n]}$ of a smooth projective surface (Arapura).
  • a smooth projective variety of $K3^{[n]}$-type (Charles and Markman).

Weak Lefschetz standard conjecture (Grothendieck conjecture $C(X)$)

  • all of the above, since $B(X) \Rightarrow C(X)$ (Grothendieck).
  • defined by equations with coefficients in a finite field (Katz and Messing).

Hodge standard conjecture (Grothendieck conjecture $Hdg(X)$)

  • in characteritic $0$ (Hodge).
  • a surface (Segre, Grothendieck).

Main references:

Alexander Grothendieck, "Standard Conjectures on Algebraic Cycles" (1969).

Steven Kleiman, "The standard conjectures" (1994).

Francois Charles, Eyal Markman, The standard conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of $K3$ surfaces (2011)

References for proofs:

Beniamino Segre, Intorno ad teorema di Hodge sulla teoria della base per le curve di una sperficie algebraica (1937)

Alexander Grothendieck, "Sur une note de Mattuck-Tate" (1958)

D. I. Lieberman, "Higher Picard Varieties" (1968)

N. Katz and W. Messing, "Some consequences of the Riemann hypothesis for varieties over finite fields" (1973)

Donu Arapura, Motivation for Hodge cycles (2005)

  • 1
    $\begingroup$ Not an expert at all, but I think there has been new cases proved recently, e.g. in arxiv.org/abs/1009.0413 (see references within as well). $\endgroup$
    – dhy
    Oct 18 '15 at 23:33
  • $\begingroup$ @dhy Thanks, that was very helpful! $\endgroup$
    – Myshkin
    Nov 3 '15 at 16:04

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