Is variance homogeneity check necessary before t-test? According to statistics textbooks, t-tests require the dependent variable to be normally distributed and the variance to be homogenous across conditions.
What's the authorized standard in statistics? Is it OK or even necessary to check homogeneity before running a t-test?
 A: No, it is not necessary. Given that there is a test that accounts for heterogeneous variances (Welch's t-test), you can simply conduct it. For one, the tests for homogeneity of variance (HOV) are problematic in a number of ways. Some lack power, they - like other statistical tests - are too powerful with large sample sizes, effect sizes are missing for these tests, some are faulty under non-normality, ...
The typical approach for most applied researchers is to conduct Levene's test, then decide whether to conduct Student's t-test or Welch's t-test based on the result of Levene's test. However, Zimmerman (2004) showed through simulation that conditioning the test on the result of Levene's test distorts the p-value of the test i.e. your p-value from Student's or Welch's is not reliable when you choose which one to do based on Levene's test. Furthermore, given that Welch's test is almost as powerful as Student's test under HOV, and it is much more powerful when HOV is absent, it is advisable to "just do Welch's test".
Zimmerman, D. W. (2004). A note on preliminary tests of equality of variances. British Journal of Mathematical and Statistical Psychology, 57(1), 173–181. https://doi.org/10.1348/000711004849222
Here is another paper that gives the same basic advice:
Delacre, M., Lakens, D., & Leys, C. (2017). Why Psychologists Should by Default Use Welch’s t-test Instead of Student’s t-test. International Review of Social Psychology, 30(1), 92–101. https://doi.org/10.5334/irsp.82
A: 
According to statistics textbooks, t-tests require the dependent
  variable to be normally distributed and the variance to be homogenous
  across conditions

This is misleading. Generally, introductory statistics textbooks teach 2 (maybe 3 if you count paired stuff) two sample t-tests. Both tests assume that each of the two random samples are iid normal random samples, and are independent between each other. However they are different in that


*

*one assumes further that the two groups have equal variance, 

*one makes no additional assumptions, but the sampling distribution of your test statistic is only approximately t-distributed.


The assumption that both groups have the same variance is unverifiable. This is because this is an assumption about unobservable variance parameters. However, 1) there do exist tests that can test equality of variances between the two groups, and 2) you can sometimes reassure yourself looking at, perhaps, histograms of the two sets of data, checking to make sure they have the same variance, roughly. 
Regarding the first technique: like any hypothesis tests, there are the associated type 1 and type 2 error events. If you decide to formally test equality of variances before you test the means, since you are running two tests, you need to realize that there is some type 1 and type 2 error for your overall strategy. 
A: Not only is it not necessary, see user162986's answer, it can also imperil the interpretability of your test.
